Hodge theory of classifying stacks

TL;DR

This paper computes the Hodge and de Rham cohomology of classifying stacks for reductive groups over various fields, revealing new connections with representation theory and comparing algebraic and topological cohomology.

Contribution

It provides explicit calculations of cohomology for classifying stacks in algebraic geometry, extending understanding to fields of small characteristic and linking to representation theory.

Findings

Cohomology calculations for classifying stacks over multiple fields.

New results connecting algebraic cohomology with representation theory.

Comparison between algebraic and topological classifying space cohomology.

Abstract

We compute the Hodge and de Rham cohomology of the classifying space BG (defined as etale cohomology on the algebraic stack BG) for reductive groups G over many fields, including fields of small characteristic. These calculations have a direct relation with representation theory, yielding new results there. The calculations are closely analogous to, but not always the same as, the cohomology of classifying spaces in topology.

Equations

$$H^i(BG,{\Omega }^j)\cong H^{i-j}(G,S^j({\mathfrak{g}}^{\ast })).$$

$${\dim }_{{\text{F}}_2}{H}_{\mathrm{dR}}^{32}(B\mathrm{Spin}(11)/{\text{F}}_2)>{\dim }_{{\text{F}}_2}{H}^{32}(B\mathrm{Spin}(11{)}_{\text{C}},{\text{F}}_2)$$

$$\begin{array}[]{cl}A_{l}&2,3,\ldots,l+1\\ B_{l}&2,4,6,\ldots,2l\\ C_{l}&2,4,6,\ldots,2l\\ D_{l}&2,4,6,\ldots,2l-2;l\\ G_{2}&2,6\\ F_{4}&2,6,8,12\\ E_{6}&2,5,6,8,9,12\\ E_{7}&2,6,8,10,12,14,18\\ E_{8}&2,8,12,14,18,20,24,30\end{array}$$

$$U\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$E_1^{ij}=H_{\mathrm{et}}^j(U_X^{i+1},F)\Rightarrow H_{\mathrm{et}}^{i+j}(X,F).$$

$$H^i(BG,{\Omega }^j)\to H^i(X,{\Omega }^j).$$

$$\mathrm{Spec}(R)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H^i(BH,{\Omega }^j)\cong H^i(EG/H,{\Omega }^j).$$

$$H_G^i(X,{\Omega }^j)\cong H_G^i(X,{\Lambda }^j{L}_{[X/G]}),$$

$$0\to {\Omega }_X^j\to {\Omega }_X^{j-1}\otimes {\mathfrak{g}}^{\ast }\to \cdots \to S^j({\mathfrak{g}}^{\ast })\to 0,$$

$$H^i(BG,{\Omega }^j)\cong H^{i-j}(G,S^j({\mathfrak{g}}^{\ast })).$$

$$0\to H^0(U,{\Omega }^1)\to H^0(E,{\Omega }^1{)}^G\to H^0(U,{\mathfrak{g}}^{\ast })\to 0$$

$$0\to {\Omega }^1\to S\to {\mathfrak{g}}^{\ast }\to 0$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\to {\Omega }_X^j\to {\Omega }_X^{j-1}\otimes {\mathfrak{g}}^{\ast }\to \cdots \to S^j({\mathfrak{g}}^{\ast })\to 0$$

$$0\to {\Lambda }^j(S)\to {\Lambda }^{j-1}(S)\otimes {\mathfrak{g}}^{\ast }\to \cdots \to S^j({\mathfrak{g}}^{\ast })\to 0.$$

$$H_G^{\ast }(X,{\Omega }_X^i\otimes S^{j-i}({\mathfrak{g}}^{\ast }))\to H_G^{\ast }(X,{\Lambda }^i(S)\otimes S^{j-i}({\mathfrak{g}}^{\ast }))$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H^0(X\times G^{r+1},{\Omega }_X^i\otimes O_{G^{r+1}}\otimes S^{j-i}({\mathfrak{g}}^{\ast }))\to H^0(X\times G^{r+1},{\Omega }_{X\times G^{r+1}}^i\otimes S^{j-i}({\mathfrak{g}}^{\ast }))$$

$${\Omega }^{i-l}(X){\otimes }_R{\Omega }^l(G^{r+1}){\otimes }_R{S}^{j-i}({\mathfrak{g}}^{\ast }).$$

$$H^i(EY,M)\cong H^i(R,M).$$

$$0\to M(Y)\to M(Y^2)\to \cdots .$$

$$(F\phi )(y_0,\dots ,y_{r-1})=\phi (1,y_0,\dots ,y_{r-1})$$

$${\Omega }^{i-l}(X){\otimes }_R{\Omega }^l(G^{r+1}){\otimes }_R{S}^{j-i}({\mathfrak{g}}^{\ast })$$

$$H^i(BG,{\Omega }^j\otimes M)\cong H^{i-j}(G,S^j({\mathfrak{g}}^{\ast })\otimes M).$$

$$H^i(BG,{\Omega }^j)\cong H^{i-j}(G,S^j(l_G)).$$

$$E/G\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\to {\Omega }^j(E/G)\to {\Omega }^j(E^2/G)\to \cdots .$$

$$0\to H^0(E,{\pi }^{\ast }({\Omega }_{E/G}^j))\to H^0(E^2,{\pi }^{\ast }({\Omega }_{E^2/G}^j))\to \cdots ,$$

$${\pi }^{\ast }({\Omega }_{X/G}^1)\to {\Omega }_X^1\to l_G,$$

Full text

For example, this cohomology vanishing holds for every symplectic group $Sp(2n)$ in characteristic 2 and for the exceptional group $G_{2}$ in characteristic 3; these are “bad primes” but not torsion primes.

Finally, we begin the problem of computing the Hodge cohomology and de Rham cohomology of $BG$, especially at torsion primes. At non-torsion primes, we have a satisfying result, proved using ideas from topology (Theorem 10.2):

At torsion primes $p$, it is an intriguing question how the de Rham cohomology of $BG_{\text{\bf F}_{p}}$ is related to the mod $p$ cohomology of the topological space $BG_{\text{\bf C}}$. We show that these graded rings are isomorphic for $G=SO(n)$ with $p=2$ (Theorem 12.1). On the other hand, we find that

[TABLE]

(Theorem 13.1). It seems that no existing results on integral $p$-adic Hodge theory address the relation between these two rings (because the stack $BG$ is not proper over Z), but the theory may soon reach that point. In particular, the results of Bhatt-Morrow-Scholze suggest that the de Rham cohomology $H^{i}_{\operatorname{dR}}(BG/\text{\bf F}_{p})$ may always be an upper bound for the mod $p$ cohomology of the topological space $BG_{\text{\bf C}}$ [4].

This work was supported by NSF grant DMS-1303105. Bhargav Bhatt convinced me to change some definitions in an earlier version of this paper: Hodge and de Rham cohomology of a smooth stack are now defined as etale cohomology. Thanks to Johan de Jong, Eric Primozic, and Raphaël Rouquier for their comments. Finally, I am grateful to Jungkai Chen for arranging my visit to National Taiwan University, where this work was completed.

1 Notation

The *fundamental degrees *of a reductive group $G$ over a field $k$ are the degrees of the generators of the polynomial ring $S(X^{*}(T)\otimes_{\text{\bf Z}}\text{\bf Q})^{W}$ of invariants under the Weyl group $W$, where $X^{*}(T)$ is the character group of a maximal torus $T$. For $k$ of characteristic zero, the fundamental degrees of $G$ can also be viewed as the degrees of the generators of the polynomial ring $O(\mathfrak{g})^{G}$ of invariant functions on the Lie algebra. Here are the fundamental degrees of the simple groups [12, section 3.7, Table 1]:

[TABLE]

For a commutative ring $R$ and $j\geq 0$, write $\Omega^{j}$ for the sheaf of differential forms over $R$ on any scheme over $R$. For an algebraic stack $X$ over $R$, $\Omega^{j}$ is a sheaf of abelian groups on the big etale site of $X$. (In particular, for every scheme $Y$ over $X$ of “size” less than a fixed limit ordinal $\alpha$ [25, Tag 06TN], we have an abelian group $\Omega^{j}(Y/R)$, and these groups form a sheaf in the etale topology.) We define Hodge cohomology $H^{i}(X,\Omega^{j})$ to mean the etale cohomology of this sheaf [25, Tag 06XI]. In the same way, we define de Rham cohomology of a stack, $H^{i}_{\operatorname{dR}}(X/R)$, as etale cohomology with coefficients in the de Rham complex over $R$. (If $X$ is an algebraic space, then the cohomology of a sheaf $F$ on the big etale site of $X$ coincides with the cohomology of the restriction of $F$ to the small etale site, the latter being the usual definition of etale cohomology for algebraic spaces [25, Tag 0DG6].) For example, this gives a definition of equivariant Hodge or De Rham cohomology, $H^{i}_{G}(X,\Omega^{j})$ or $H^{i}_{G,\operatorname{dR}}(X/R)$, as the Hodge or de Rham cohomology of the quotient stack $[X/G]$. Essentially the same definition was used for smooth stacks in characteristic zero by Teleman and Behrend [26, 1].

This definition of Hodge and de Rham cohomology is the “wrong” thing to consider for an algebraic stack which is not smooth over $R$. For non-smooth stacks, it would be better to define Hodge and de Rham cohomology using some version of Illusie and Bhatt’s derived de Rham cohomology, or in other words using the cotangent complex [3, section 4]. In this paper, we will only consider Hodge and de Rham cohomology for smooth stacks over a commutative ring $R$. An important example for the paper is that the classifying stack $BG$ is smooth over $R$ even for non-smooth group schemes $G$ [25, Tag 075T]:

Lemma 1.1**.**

Let $G$ be a group scheme which is flat and locally of finite presentation over a commutative ring $R$. Then the algebraic stack $BG$ is smooth over $R$. More generally, for a smooth algebraic space $X$ over $R$ on which $G$ acts, the quotient stack $[X/G]$ is smooth over $R$.

Let $X$ be an algebraic stack over $R$, and let $U$ be an algebraic space with a smooth surjective morphism to $X$. The C̆ech construction $C(U/X)$ means the simplicial algebraic space:

[TABLE]

For any sheaf $F$ of abelian groups on the big etale site of $X$, the etale cohomology of $X$ with coefficients in $F$ can be identified with the etale cohomology of the simplicial algebraic space $C(U/X)$ [25, Tag 06XJ]. In particular, there is a spectral sequence:

[TABLE]

Write $H^{i}_{\operatorname{H}}(X/R)=\oplus_{j}H^{j}(X,\Omega^{i-j})$ for the Hodge cohomology of an algebraic stack $X$ over $R$, graded by total degree.

Let $G$ be a group scheme which is flat and locally of finite presentation over a commutative ring $R$. Then the Hodge cohomology of the stack $BG$ can be viewed, essentially by definition, as the ring of characteristic classes in Hodge cohomology for principal $G$-bundles (in the fppf topology). Concretely, for any scheme $X$ over $R$, a principal $G$-bundle over $X$ determines a morphism $X\rightarrow BG$ of stacks over $R$ and hence a pullback homomorphism

[TABLE]

Note that for a scheme $X$ over $R$, $H^{i}(X,\Omega^{j})$ can be computed either in the Zariski or in the etale topology, because the sheaf $\Omega^{j}$ (on the small etale site of $X$) is quasi-coherent [25, Tag 03OY].

For any scheme $X$ over a commutative ring $R$, there is a simplicial scheme $EX$ whose space $(EX)_{n}$ of $n$-simplices is $X^{\{0,\ldots,n\}}=X^{n+1}$ [7, 6.1.3]. For a group scheme $G$ over $R$, the simplicial scheme $BG$ over $R$ is defined as the quotient of the simplicial scheme $EG$ by the free left action of $G$:

[TABLE]

If $G$ is smooth over $R$, then Hodge cohomology $H^{i}(BG,\Omega^{j})$ as defined above can be identified with the cohomology of the simplicial scheme $BG$, because this simplicial scheme is the C̆ech simplicial scheme associated to the smooth surjective morphism $\operatorname{Spec}(R)\rightarrow BG$. For $G$ not smooth, one has instead to use the C̆ech simplicial scheme associated to a smooth presentation of $BG$. See for example the calculation of the Hodge cohomology of $B\mu_{p}$ in characteristic $p$, Proposition 11.1.

It is useful that we can compute Hodge cohomology via any smooth presentation of a stack. For example, let $H$ be a closed subgroup scheme of a smooth group scheme $G$ over a commutative ring $R$, and assume that $H$ is flat and locally of finite presentation over $R$. Then $G/H$ is an algebraic space with a smooth surjective morphism $G/H\rightarrow BH$ over $R$, and so we can compute the Hodge cohomology of the stack $BH$ using the associated C̆ech simplicial algebraic space. Explicitly, that is the simplicial algebraic space $EG/H$, and so we have:

Lemma 1.2**.**

[TABLE]

Note that the cohomology theories we are considering are not $A^{1}$-homotopy invariant. Indeed, Hodge cohomology is usually not the same for a scheme $X$ as for $X\times A^{1}$, even over a field of characteristic zero. For example, $H^{0}(\operatorname{Spec}(k),O)=k$, whereas $H^{0}(A^{1}_{k},O)$ is the polynomial ring $k[x]$. In de Rham cohomology, $H^{0}_{\operatorname{dR}}(A^{1}/k)$ is just $k$ if $k$ has characteristic zero, but it is $k[x^{p}]$ if $k$ has characteristic $p>0$.

2 Equivariant Hodge cohomology and functions on

the Lie algebra

In this section, we identify the Hodge cohomology of a quotient stack with the cohomology of an explicit complex of vector bundles (Theorem 2.1). As a special case, we relate the Hodge cohomology of a classifying stack $BG$ to the cohomology of $G$ as an algebraic group (Corollary 2.2). In this section, we assume $G$ is smooth. Undoubtedly, various generalizations of the statements here are possible. In particular, we will give an analogous description of the Hodge cohomology of $BG$ for a non-smooth group $G$ in Theorem 3.1.

The main novelty is that these results hold in any characteristic. In particular, Theorem 2.1 was proved in characteristic zero by Simpson and Teleman [23, Example 6.8(c)].

Theorem 2.1**.**

Let $G$ be a smooth affine group scheme over a commutative ring $R$. Let $G$ act on a smooth affine scheme $X$ over $R$. Then there is a canonical isomorphism

[TABLE]

where $\Lambda^{j}L_{[X/G]}$ is the complex of $G$-equivariant vector bundles on $X$, in degrees 0 to $j$:

[TABLE]

associated to the map $\mathfrak{g}\rightarrow TX$.

This isomorphism expresses the cohomology over $[X/G]$ of the “big sheaf” $\Omega^{j}$, which is not a quasi-coherent sheaf on $[X/G]$, in terms of the cohomology of a complex of quasi-coherent sheaves on $[X/G]$. (Here differentials are over $R$ unless otherwise stated. The sheaf $\Omega^{j}$ on the big etale site of $[X/G]$ is not quasi-coherent for $j>0$ because, for a morphism $f\colon Y\rightarrow Z$ of schemes over $[X/G]$, the pullback map $f^{*}\Omega^{j}_{Z/R}\rightarrow\Omega^{j}_{Y/R}$ need not be an isomorphism.) Theorem 2.1 is useful already for $X=\operatorname{Spec}(R)$, where it gives the following result, proved in characteristic zero by Bott [5].

Corollary 2.2**.**

Let $G$ be a smooth affine group scheme over a commutative ring $R$. Then there is a canonical isomorphism

[TABLE]

The group on the left is an etale cohomology group of the algebraic stack $BG$ over $R$, as discussed in section 1. On the right is the cohomology of $G$ as an algebraic group, defined by $H^{i}(G,M)=\operatorname{Ext}^{i}_{G}(R,M)$ for a $G$-module $M$ [15, section 4.2].

Proof.

(Corollary 2.2) This follows from Theorem 2.1 applied to the stack $BG=[\operatorname{Spec}(R)/G]$. The deduction uses two facts. First, a quasi-coherent sheaf on $BG$ is equivalent to a $G$-module [25, Tag 06WS]. Second, for a $G$-module $M$, the cohomology of the corresponding quasi-coherent sheaf on the big etale site of $BG$ coincides with its cohomology as a $G$-module, $H^{*}(G,M)$, since both are computed by the same C̆ech complex (section 1 for the sheaf, [15, Proposition 4.16] for the module). ∎

Proof.

(Theorem 2.1) The adjoint representation of $G$ on $\mathfrak{g}$ determines a $G$-equivariant vector bundle $\mathfrak{g}$ on $X$. The action of $G$ on $X$ gives a morphism $\Omega^{1}_{X}\rightarrow\mathfrak{g}^{*}$ of $G$-equivariant quasi-coherent sheaves (in fact, vector bundles) on $X$. Consider these equivariant sheaves as quasi-coherent sheaves on $[X/G]$, according to [25, Tag 06WS].

We will define a map from the complex $\Omega^{1}_{X}\rightarrow\mathfrak{g}^{*}$ of quasi-coherent sheaves on $[X/G]$ (in degrees 0 and 1) to the sheaf $\Omega^{1}$, in the derived category $D([X/G]_{\operatorname{et}},O_{[X/G]})$ of $O_{[X/G]}$-modules on the big etale site $[X/G]_{\operatorname{et}}$. To do this, define another sheaf $S$ on the big etale site of $[X/G]$ by: for a scheme $U$ over $[X/G]$, let $E=U\times_{[X/G]}X$ (so that $E\rightarrow U$ is a principal $G$-bundle), and define $S(U)=H^{0}(E,\Omega^{1})^{G}$. There is a short exact sequence

[TABLE]

for each affine scheme $U$ over $[X/G]$. (Note that the principal $G$-bundle $E$ over $U$ together with the adjoint action of $G$ on $\mathfrak{g}^{*}$ determines a vector bundle which we call $\mathfrak{g}^{*}$ on $U$.) So we have an exact sequence

[TABLE]

of sheaves on the big etale site of $[X/G]$ (where $\mathfrak{g}^{*}$ is a vector bundle on $[X/G]$).

Thus the sheaf $\Omega^{1}$ on $[X/G]$ is isomorphic in the derived category to the complex $S\rightarrow\mathfrak{g}^{*}$ (in cohomological degrees 0 and 1) on $[X/G]$. Therefore, to produce the map in $D([X/G]_{\operatorname{et}},O_{[X/G]})$ promised above, it suffices to define a map of complexes of sheaves on $[X/G]_{\operatorname{et}}$:

[TABLE]

(As above, $\mathfrak{g}^{*}$ denotes the vector bundle on $[X/G]$ associated to the representation of $G$ on $\mathfrak{g}^{*}$, and $\Omega^{1}_{X}$ denotes the vector bundle on $[X/G]$ corresponding to the $G$-equivariant vector bundle of the same name on $X$.) It is now easy to produce the map of complexes: for any scheme $U$ over $[X/G]$, with associated principal $G$-bundle $E\rightarrow U$ and $G$-equivariant morphism $h\colon E\rightarrow X$, the map from $\Omega^{1}_{X}(U)=H^{0}(E,h^{*}\Omega^{1}_{X})^{G}$ to $S(U)=H^{0}(E,\Omega^{1}_{E})^{G}$ is the pullback, and the map from $\mathfrak{g}^{*}$ to itself is the identity.

For any $j\geq 0$, taking the $j$th derived exterior power over $O_{[X/G]}$ of this map of complexes gives a map from the Koszul complex

[TABLE]

(in degrees 0 to $j$) of vector bundles on $[X/G]$ to the big sheaf $\Omega^{j}$, in $D([X/G]_{\operatorname{et}},O_{[X/G]})$. (The description of the derived exterior power of a 2-term complex of flat modules as a Koszul complex follows from Illusie [13, Proposition II.4.3.1.6], by the same argument used for derived divided powers in [14, Lemme VIII.2.1.2.1].) We want to show that this map of complexes induces an isomorphism on cohomology over $[X/G]$.

By the exact sequence above for the big sheaf $\Omega^{1}$ on $[X/G]$, we can identify the big sheaf $\Omega^{j}$ in the derived category with a similar-looking Koszul complex:

[TABLE]

We want to show that the obvious map from the Koszul complex of vector bundles (in the previous paragraph) to this complex of big sheaves induces an isomorphism on cohomology over $[X/G]$. It suffices to show that for each $0\leq i\leq j$, the map

[TABLE]

is an isomorphism.

By section 1, we can compute both of these cohomology groups on the C̆ech simplicial space associated to the smooth surjective morphism $X\rightarrow[X/G]$. This simplicial space can be written as $(X\times EG)/G$, where all products are over $R$:

[TABLE]

Since $X$ is affine, all the spaces in this simplicial space are affine schemes. Therefore, for any $0\leq i\leq j$, $H^{*}_{G}(X,\Omega^{i}_{X}\otimes S^{j-i}(\mathfrak{g}^{*}))$ is the cohomology of the complex of $H^{0}$ of the sheaves $\Omega^{i}_{X}\otimes O_{EG}\otimes S^{j-i}(\mathfrak{g}^{*})$ over the spaces making up $(X\times EG)/G$. Likewise, $H^{*}_{G}(X,\Lambda^{i}(S)\otimes S^{j-i}(\mathfrak{g}^{*}))$ is the cohomology of the complex of $H^{0}$ of the sheaves $\Lambda^{i}(S)\otimes S^{j-i}(\mathfrak{g}^{*})$ over the spaces making up $(X\times EG)/G$.

Both of these complexes are spaces of $G$-invariants of analogous complexes of $H^{0}$ of sheaves over the spaces making up $X\times EG$. Moreover, all of these $G$-modules are induced from representations of the trivial group, because $X\times G^{r+1}\rightarrow(X\times G^{r+1})/G$ is a $G$-torsor with a section for each $r\geq 0$. Indeed, a choice of section of this $G$-torsor trivializes the torsor, and so the group of sections of a $G$-equivariant sheaf of $X\times G^{r+1}$ is the subspace of invariants tensored with $O(G)$, as a $G$-module. (Note that trivializations of these $G$-torsors cannot be made compatible with the face maps of the simplicial space, in general.) And every tensor product $O(G)\otimes_{R}M$ for a $G$-module $M$ is injective as a $G$-module [15, Proposition 3.10]. It follows that $H^{i}(G,O(G)\otimes_{R}M)=0$ for $i>0$ [15, Lemma I.4.7].

Therefore, to show that the map of complexes of $G$-invariants in the previous paragraph is a quasi-isomorphism (as we want), it suffices to show that the map of complexes of $H^{0}$ over $X\times EG$ is a quasi-isomorphism. And for that, we can forget about the $G$-action. That is, we want to show that the map of complexes with $r$th term (for $r\geq 0$)

[TABLE]

is a quasi-isomorphism.

We can write $\Omega^{i}_{X\times G^{r+1}}$ as the direct sum $\oplus_{l=0}^{i}\Omega^{i-l}_{X}\otimes\Omega^{l}_{G^{r+1}}$. Moreover, this splitting is compatible with pullback along the face maps of the simplicial scheme $X\times EG$. So the map of complexes above is the inclusion of a summand (corresponding to $l=0$). It remains to show that for every $0<l\leq i$, the $l$th summand is a complex with cohomology zero. Its $r$th term is

[TABLE]

To analyze its cohomology, we use the well-known “contractibility” of $EG$, in the following form:

Lemma 2.3**.**

Let $Y$ be an affine scheme over a ring $R$ with $Y(R)$ not empty. For any sheaf $M$ of abelian groups on the big etale site of $R$, the cohomology of the simplicial scheme $EY$ over $R$ coincides with the cohomology of $\operatorname{Spec}(R)$:

[TABLE]

Proof.

Since $Y$ is affine, $H^{*}(EY,M)$ is the cohomology of an explicit complex

[TABLE]

Choosing a point $1\in Y(R)$ gives an explicit chain homotopy from the identity map to the complex $M(R)$ in degree 0:

[TABLE]

for $\varphi\in M(Y^{r+1})$ with $r\geq 0$. ∎

Returning to the proof of Theorem 2.1: we want to show that for $l>0$, the complex with $r$th term

[TABLE]

has zero cohomology. By Lemma 1.2 (applied to the sheaf $\Omega^{l}$ on the big etale site of $R$ and the simplicial scheme $EG$), the complex above has cohomology equal to $\Omega^{i-l}(X)\otimes\Omega^{l}(\operatorname{Spec}R)\otimes S^{j-i}(\mathfrak{g}^{*})$ in degree 0 and zero in other degrees. Since $l>0$, the cohomology in degree 0 also vanishes. The proof is complete. ∎

The argument works verbatim to prove a twisted version of Corollary 2.2, where the sheaf $\Omega^{j}$ on $BG$ is tensored with the vector bundle associated to any $G$-module. The generalization will not be needed in this paper, but we state it for possible later use.

Theorem 2.4**.**

Let $G$ be a smooth affine group scheme over a commutative ring $R$. Let $M$ be a $G$-module that is flat over $R$. Then there is a canonical isomorphism

[TABLE]

3 Flat group schemes

We now describe the Hodge cohomology of the classifying stack of a group scheme $G$ which need not be smooth, generalizing Corollary 2.2. The analog of the co-Lie algebra $\mathfrak{g}^{*}$ in this generality is the co-Lie complex $l_{G}$ in the derived category of $G$-modules, defined by Illusie [14, section VII.3.1.2]. Namely, $l_{G}$ is the pullback of the cotangent complex of $G\rightarrow\operatorname{Spec}(R)$ to $\operatorname{Spec}(R)$, via the section $1\in G(R)$. (The cotangent complex $L_{X/Y}$ of a morphism $X\rightarrow Y$ of schemes is an object of the quasi-coherent derived category of $X$; if $X$ is smooth over $Y$, then $L_{X/Y}$ is the sheaf $\Omega^{1}_{X/Y}$.) The cohomology of $l_{G}$ in degree 0 is the $R$-module $\omega^{1}_{G}$, the restriction of $\Omega^{1}_{G}$ to the identity $1\in G(R)$; thus $\omega^{1}_{G}$ is the co-Lie algebra $\mathfrak{g}^{*}$ if $G$ is smooth over $R$. The complex $l_{G}$ has zero cohomology except in cohomological degrees $-1$ and 0. If $G$ is smooth, then $l_{G}$ has cohomology concentrated in degree 0.

Theorem 3.1**.**

Let $G$ be a flat affine group scheme of finite presentation over a commutative ring $R$. Then there is a canonical isomorphism

[TABLE]

Proof.

As discussed in section 1, we can compute $H^{*}(BG,\Omega^{j})$ as the etale cohomology with coefficients in $\Omega^{j}$ of the C̆ech simplicial space associated to any smooth algebraic space $U$ over $R$ with a smooth surjective morphism from $U$ to the stack $BG$. The assumption on $G$ implies that $BG$ is a quasi-compact algebraic stack over $R$, and so there is an affine scheme $U$ with a smooth surjective morphism $U\rightarrow BG$ [25, Tags 06FI and 04YA]. By Lemma 1.1, $BG$ is smooth over $R$, and so $U$ is smooth over $R$. Let $E=U\times_{BG}\operatorname{Spec}(R)$; then $E$ is a smooth $R$-space with a free $G$-action such that $U=E/G$. Also, $E$ is affine because $U$ and $G$ are affine.

By section 1, $H^{*}(BG,\Omega^{j})$ is the etale cohomology with coefficients in $\Omega^{j}$ of the simplicial algebraic space $EE/G$:

[TABLE]

By the properties of $E$ and $Y$ above, $E^{n+1}/G$ is an affine scheme for all $n\geq 0$. Since $H^{*}(BG,\Omega^{j})$ is the cohomology with coefficients in $\Omega^{j}$ of the simplicial scheme $EE/G$, this is the cohomology of the cochain complex

[TABLE]

As in the proof of Theorem 2.1, this complex is the $G$-invariants of the complex

[TABLE]

where we write $\pi$ for the morphism $E^{n+1}\rightarrow E^{n+1}/G$ for any $n\geq 0$.

For any smooth $R$-scheme $X$ with a free action of $G$, there is a canonical exact triangle in the quasi-coherent derived category of $G$-equivariant sheaves on $X$:

[TABLE]

where we write $l_{G}$ for the pullback of the co-Lie complex $l_{G}$ from the stack $BG$ over $R$ to $X$. To deduce this from Illusie’s results on the cotangent complex $L_{X/Y}$, let $Y=X/G$ and $S=\operatorname{Spec}(R)$, and use the transitivity exact triangle for $X\rightarrow Y\rightarrow S$ in the derived category of $X$ [13, II.2.1.5.2]:

[TABLE]

Since $X$ is smooth over $S$, so is $Y$ (even though $G$ need not be); so $L_{Y/S}\cong\Omega^{1}_{Y/S}$ and $L_{X/S}\cong\Omega^{1}_{X/S}$. Also, since $X\rightarrow Y$ is a $G$-torsor in the fppf topology, $L_{X/Y}$ is the pullback of an object $l_{X/Y}$ on $Y$ [14, VII.2.4.2.8]. Furthermore, $l_{X/Y}$ in the fppf topology is the pullback of $l_{G}$ via the morphism from $Y$ to the stack $BG$ corresponding to the $G$-torsor $X\rightarrow Y$ [14, VII.3.1.2.6].

Applying this to $E^{n+1}/G$ for any $n\geq 0$, we get an exact triangle

[TABLE]

in $D_{G}(EE)$, or equivalently

[TABLE]

It follows that for any $j\geq 0$, $\pi^{*}(\Omega^{j}_{EE/G})$ has a filtration in the derived category with quotients $\pi^{*}(\Omega^{j-m}_{EE/G})\otimes\Lambda^{m}(l_{G}[-1])$ for $m=0,\ldots,j$.

If $E(R)$ is nonempty, then $H^{i}(EE,\Omega^{j})\cong H^{i}(\operatorname{Spec}(R),\Omega^{j})$, by Lemma 2.3. That group is zero unless $i=j=0$, in which case it is $R$. By faithfully flat descent, the same conclusion holds under our weaker assumption that $E\rightarrow\operatorname{Spec}(R)$ is smooth and surjective. Therefore, in the filtration above, all objects but one have zero cohomology in all degrees over $EE$. We deduce that the homomorphism

[TABLE]

is an isomorphism of $G$-modules for all $i$. By Illusie’s “décalage” isomorphism [13, Proposition I.4.3.2.1(i)], we can write $S^{j}(l_{G})[-j]$ instead of $\Lambda^{j}(l_{G}[-1])$.

The cochain complex $O(EE)$ has cohomology $R$ in degree 0 and 0 otherwise, by Lemma 2.3 again. So the complex of global sections of the trivial vector bundle $S^{j}(l_{G})$ over $EE$ is isomorphic, in the derived category of $G$-modules, to the complex of $G$-modules $S^{j}(l_{G})$. We conclude that the complex of sections of $\pi^{*}(\Omega^{j}_{EE/G})$ over $EE$ is isomorphic to $S^{j}(l_{G})[-j]$ in the derived category of $G$-modules.

Finally, we observe that each $G$-module in this complex,

[TABLE]

for $n\geq 0$, is acyclic (meaning that $H^{>0}(G,M)=0$). More generally, for any affine $R$-scheme $Y$ with a free $G$-action such that $Y/G$ is affine, and any quasi-coherent sheaf $F$ on $Y/G$, $M:=H^{0}(Y,\pi^{*}F)$ is acyclic. Indeed, this holds if $Y\rightarrow Y/G$ is a trivial $G$-bundle, since then $M=O(G)\otimes F$ and so $M$ is acyclic [15, Lemma 4.7]. We can prove acyclicity in general by pulling the $G$-bundle over $Y/G$ back to a $G$-bundle over $Y$, which is trivial; then $H^{>0}(G,M)\otimes_{O(Y/G)}O(Y)$ is 0 by [15, Proposition 4.13], and so $H^{>0}(G,M)=0$ by faithfully flat descent.

We conclude that the complex computing $H^{*}(BG,\Omega^{j})$ is the same one that computes $H^{*}(G,S^{j}(l_{G})[-j])$. ∎

4 Good filtrations

In this section, we explain how known results in representation theory imply calculations of the Hodge cohomology of classifying spaces in many cases, via Theorem 3.1. This is not logically necessary for the rest of the paper: Theorem 10.1 is a stronger calculation of Hodge cohomology, based on ideas from homotopy theory.

Let $G$ be a split reductive group over a field $k$. (A textbook reference on split reductive groups is [18, Chapter 21].) A *Schur module *for $G$ is a module of the form $H^{0}(\lambda)$ for a dominant weight $\lambda$. By definition, $H^{0}(\lambda)$ means $H^{0}(G/B,L(\lambda))$, where $B$ is a Borel subgroup and $L(\lambda)$ is the line bundle associated to $\lambda$. For $k$ of characteristic zero, the Schur modules are exactly the irreducible representations of $G$. Kempf showed that the dimension of the Schur modules is independent of the characteristic of $k$ [15, Chapter II.4]. They need not be irreducible in characteristic $p$, however.

A $G$-module $M$ has a *good filtration *if there is a sequence of submodules $0\subset M_{0}\subset M_{1}\subset\cdots$ such that $M=\cup M_{j}$ and each quotient $M_{i}/M_{i-1}$ is a Schur module. One good feature of Schur modules is that their cohomology groups are known, by Cline-Parshall-Scott-van der Kallen [15, Proposition 4.13]. Namely,

[TABLE]

As a result, $H^{i}(G,M)=0$ for all $i>0$ when $M$ has a good filtration.

The following result was proved by Andersen-Jantzen and Donkin [9, Proposition and proof of Theorem 2.2], [15, II.4.22]. The statement on the ring of invariants incorporates earlier work by Kac and Weisfeiler. Say that a prime number $p$ is *good *for a reductive group $G$ if $p\neq 2$ if $G$ has a simple factor not of type $A_{n}$, $p\neq 2,3$ if $G$ has a simple factor of exceptional type, and $p\neq 2,3,5$ if $G$ has an $E_{8}$ factor.

Theorem 4.1**.**

Let $G$ be a split reductive group over a field $k$. Assume either that $G$ is a simply connected semisimple group and $\operatorname{char}(k)$ is good for $G$, or that $G=GL(n)$. Then the polynomial ring $O(\mathfrak{g})=S(\mathfrak{g}^{*})$ has a good filtration as a $G$-module, and the ring of invariants $O(\mathfrak{g})^{G}$ is a polynomial ring over $k$, with generators in the fundamental degrees of $G$.

It follows that, under these assumptions, $H^{>0}(G,S^{j}(\mathfrak{g}^{*}))$ is zero for all $j\geq 0$. Equivalently, $H^{i}(BG,\Omega^{j})=0$ for $i\neq j$, by Theorem 3.1. We prove this under the weaker assumption that $p$ is not a torsion prime in Theorem 10.1.

5 Künneth formula

The Künneth formula holds for Hodge cohomology, in the following form. The hypotheses apply to the main case studied in this paper: classifying stacks $BG$ with $G$ an affine group scheme of finite type over a field.

Proposition 5.1**.**

Let $X$ and $Y$ be quasi-compact algebraic stacks with affine diagonal over a field $k$. Then

[TABLE]

Proof.

Since $X$ and $Y$ are quasi-compact, there are affine schemes $A$ and $B$ with smooth surjective morphisms $A\rightarrow X$ and $B\rightarrow Y$ [25, Tag 04YA]. Since $X$ and $Y$ have affine diagonal, the fiber products $A^{n+1}_{X}$ and $B^{n+1}_{Y}$ are affine over the products $A^{n+1}$ and $B^{n+1}$ over $k$, and so they are affine schemes, for all $n\geq 0$.

The morphism $A\times B\rightarrow X\times Y$ is smooth and surjective. Therefore, the Hodge cohomology of $X\times Y$ is the cohomology of the C̆ech simplicial space $C(A\times B/X\times Y)$ over $k$, with coefficients in $\Omega^{*}$ (with zero differential). This space is the product $C(A/X)\times C(B/Y)$ over $k$. By the previous paragraph, these are in fact simplicial affine schemes over $k$.

The quasi-coherent sheaf $\Omega^{1}$ on the product of two affine schemes over $k$ is the direct sum of the pullbacks of $\Omega^{1}$ from the two factors. (No smoothness is needed for this calculation.) Therefore, the quasi-coherent sheaf $\Omega^{*}$ on the product affine scheme $A^{n+1}_{X}\times B^{n+1}_{Y}$ over $k$ is the tensor product of the pullbacks on $\Omega^{*}$ on those two schemes. So $H^{0}(A^{n+1}_{X}\times B^{n+1}_{Y},\Omega^{*})$ is the tensor product of $H^{0}(A^{n+1}_{X},\Omega^{*})$ and $H^{0}(B^{n+1}_{Y},\Omega^{*})$ over $k$.

The spectral sequence of the simplicial scheme $C(A/X)\times C(B/Y)$ with coefficients in $\Omega^{*}$ reduces to one row, since all the schemes here are affine. Explicitly, by the previous paragraph, the cohomology of the product simplicial scheme is the cohomology of the tensor product over $k$ of the two cosimplicial vector spaces $H^{0}(A^{n+1}_{X},\Omega^{*})$ and $H^{0}(B^{n+1}_{Y},\Omega^{*})$. By the Eilenberg-Zilber theorem, it follows that the cohomology of the product simplicial scheme is the tensor product over $k$ of the cohomology of the two factors. [17, Theorem 29.3]. Equivalently,

[TABLE]

∎

6 Parabolic subgroups

Theorem 6.1**.**

Let $P$ be a parabolic subgroup of a reductive group $G$ over a field $k$, and let $L$ be the Levi quotient of $P$ (the quotient of $P$ by its unipotent radical). Then the restriction

[TABLE]

is an isomorphism for all $i$ and $j$. Equivalently,

[TABLE]

is an isomorphism for all $a$ and $j$.

Theorem 6.1 can be viewed as a type of homotopy invariance for Hodge cohomology of classifying spaces. This is not automatic, since Hodge cohomology is not $A^{1}$-homotopy invariant for smooth varieties. Homotopy invariance of Hodge cohomology also fails in general for classifying spaces. For example, let $G_{a}$ be the additive group over a field $k$. Then the Hodge cohomology group $H^{1}(BG_{a},O)$ is not zero for any $k$, and it is a $k$-vector space of infinite dimension for $k$ of positive characteristic; this follows from Theorem 6.3, due to Cline, Parshall, Scott, and van der Kallen, together with Theorem 3.1.

Proof.

(Theorem 6.1) Let $U$ be the unipotent radical of $P$, so that $L=P/U$. It suffices to show that

[TABLE]

is an isomorphism after extending the field $k$. So we can assume that $G$ has a Borel subgroup $B$ and that $B$ is contained in $P$. Let $R$ be the set of roots for $G$. We follow the convention that the weights of $B$ acting on the Lie algebra of its unipotent radical are the *negative *roots $R^{-}$. There is a subset $I$ of the set $S$ of simple roots so that $P$ is the associated subgroup $P_{I}$, in the notation of [15, II.1.8]. More explicitly, let $R_{I}=R\cap\text{\bf Z}I$; then $P=P_{I}$ is the semidirect product $U_{I}\rtimes L_{I}$, where $L_{I}$ is the reductive group $G(R_{I})$ and $U:=U_{I}$ is the unipotent group $U((-R^{+})\setminus R_{I})$.

As a result, the weights of $P$ on $\mathfrak{p}$ are all the roots $\sum_{\alpha\in S}n_{\alpha}\alpha$ such that $n_{\alpha}\leq 0$ for $\alpha$ not in $I$. The coefficients $n_{\alpha}$ for $\alpha$ not in $I$ are all zero exactly for the weights of $P$ on $\mathfrak{p}/\mathfrak{u}$. As a result, for any $j\geq 0$, the weights of $P$ on $S^{j}(\mathfrak{p}^{*})$ are all in the root lattice, with nonnegative coefficients for the simple roots not in $I$, and with those coefficients all zero only for the weights of $P$ on the subspace $S^{j}((\mathfrak{p}/\mathfrak{u})^{*})\subset S^{j}(\mathfrak{p}^{*})$.

We now use the following information about the cohomology of $P$-modules [15, Proposition II.4.10]. For any element $\lambda$ of the root lattice $\text{\bf Z}S$, $\lambda=\sum_{\alpha\in S}n_{\alpha}\alpha$, the height $\operatorname{ht}(\lambda)$ means the integer $\sum_{\alpha\in S}n_{\alpha}$.

Proposition 6.2**.**

Let $P$ be a parabolic subgroup of a reductive group $G$ over a field, and let $M$ be a $P$-module. If $H^{j}(P,M)\neq 0$ for some $j\geq 0$, then there is a weight $\lambda$ of $M$ with $-\lambda\in\text{\bf N}R^{+}$ and $\operatorname{ht}(\lambda)\geq j$.

Given the information above about the weights of $P$ on $S^{j}(\mathfrak{p}^{*})$, it follows that the homomorphism

[TABLE]

is an isomorphism for all $a$ and $j$. Here $\mathfrak{p}/\mathfrak{u}\cong\mathfrak{l}$ is a representation of the quotient group $L=P/U$. It remains to show that the pullback

[TABLE]

is an isomorphism. This would not be true for an arbitrary representation of $L$; we will have to use what we know about the weights of $L$ on $S^{j}((\mathfrak{p}/\mathfrak{u})^{*})$.

We also use the following description of the cohomology of an additive group $V=(G_{a})^{n}$ over a perfect field $k$ [15, Proposition I.4.27]. (To prove Theorem 6.1, we can enlarge the field $k$, and so we can assume that $k$ is perfect.) The following description is canonical, with respect to the action of $GL(V)$ on $H^{*}(V,k)$. Write $W^{(j)}$ for the $j$th Frobenius twist of a vector space $W$, as a representation of $GL(W)$.

Theorem 6.3**.**

(1) If $k$ has characteristic zero, then $H^{*}(V,k)\cong\Lambda(V^{*})$, with $V^{*}$ in degree 1.

(2) If $k$ has characteristic 2, then

[TABLE]

with all the spaces $(V^{*})^{(j)}$ in degree 1.

(3) If $k$ has characteristic $p>2$, then

[TABLE]

with all the spaces $(V^{*})^{(j)}$ in the first factor in degree 1, and all the spaces $(V^{*})^{(j)}$ in the second factor in degree 2.

We also use the Hochschild-Serre spectral sequence for the cohomology of algebraic groups [15, I.6.5, Proposition I.6.6]:

Theorem 6.4**.**

Let $G$ be an affine group scheme of finite type over a field $k$, and let $N$ be a normal $k$-subgroup scheme of $G$. For every $G$-module (or complex of $G$-modules) $V$, there is a spectral sequence

[TABLE]

Theorems 6.3 and 6.4 give information about the weights of $L$ on $H^{*}(U,k)$, that is, about the action of a maximal torus $T\subset L$ on $H^{*}(U,k)$. The method is to write $U$ (canonically) as an extension of additive groups $V=(G_{a})^{n}$ and use the Hochschild-Serre spectral sequence. We deduce that as a representation of $L$, all weights of $H^{>0}(U,k)$ are in the root lattice of $G$, with nonnegative coefficients for the simple roots not in $I$, and with at least one of those coefficients positive. (This is the same sign as we have for the action of $L$ on $\mathfrak{u}^{*}$.)

Now apply the Hochschild-Serre spectral sequence to the normal subgroup $U$ in $P$:

[TABLE]

By the analysis of $S^{l}(\mathfrak{p}^{*})$ above, all the weights of $L$ on the subspace $S^{l}((\mathfrak{p}/\mathfrak{u})^{*})$ are in the root lattice of $G$, and the coefficients of all simple roots not in $I$ are equal to zero. Combining this with the previous paragraph, we find: for $l\geq 0$ and $j>0$, all weights of $L$ on $H^{j}(U,k)\otimes S^{l}((\mathfrak{p}/\mathfrak{u})^{*})$ have all coefficients of the simple roots not in $I$ nonnegative, with at least one positive. By Proposition 6.2, it follows that

[TABLE]

for all $i$ and $l$ and all $j>0$. So the spectral sequence above reduces to an isomorphism

[TABLE]

as we wanted. Theorem 6.1 is proved. ∎

7 Pushforward on Hodge cohomology

Gros constructed a cycle map $CH^{i}(X)\rightarrow H^{i}(X,\Omega^{i})$ for smooth schemes over a perfect field [11]. He also showed that the cycle map is compatible with proper pushforward, in the following sense [11, sections II.2 and II.4]

Proposition 7.1**.**

Let $f\colon X\rightarrow Y$ be a proper morphism of smooth schemes over a field $k$, and assume that $\dim(X)-\dim(Y)=N$ everywhere. Then there is a pushforward homomorphism

[TABLE]

This is compatible with the cycle map, via a commutative diagram: [11, section II.4]:

[TABLE]

8 Hodge cohomology of flag manifolds

Proposition 8.1**.**

Let $P$ be a parabolic subgroup of a split reductive group $G$ over a field $k$. Then the cycle map

[TABLE]

is an isomorphism of $k$-algebras. In particular, $H^{i}(G/P,\Omega^{j})=0$ for $i\neq j$.

This is well known for $k$ of characteristic zero, but the general result is also not difficult. Andersen gave the additive calculation of $H^{i}(G/P,\Omega^{j})$ in any characteristic [15, Proposition II.6.18]. Note that Chevalley and Demazure gave combinatorial descriptions of the Chow ring of $G/P$, which in particular show that this ring is independent of $k$, and isomorphic to the ordinary cohomology ring $H^{*}(G_{\text{\bf C}}/P_{\text{\bf C}},\text{\bf Z})$ [6, Proposition 11], [8]. (That makes sense because the classification of split reductive groups and their parabolic subgroups is the same over all fields.)

Proof.

(Proposition 8.1) We use that $X=G/P$ has a cell decomposition, the Bruhat decomposition. It follows that the Chow group of $X$ is free abelian on the set of cells. In fact, the Chow motive of $X$ is isomorphic to a direct sum of Tate motives $\text{\bf Z}(a)$, indexed by the cells [22, 2.6].

Next, Hodge cohomology is a functor on Chow motives over $k$. (That is, we have to show that a correspondence between smooth projective varieties gives a homomorphism on Hodge cohomology, which follows from Gros’s cycle map and proper pushforward for Hodge cohomology (Proposition 7.1).) As a result, the calculation follows from the Hodge cohomology of projective space, which implies that the Chow motive $M=\text{\bf Z}(a)$ has Hodge cohomology $H^{i}(M,\Omega^{j})$ isomorphic to $k$ if $i=j=a$ and zero otherwise. ∎

9 Invariant functions on the Lie algebra

Theorem 9.1**.**

Let $G$ be a reductive group over a field $k$, $T$ a maximal torus in $G$, $\mathfrak{g}$ and $\mathfrak{t}$ the Lie algebras. If $k$ has characteristic $p>0$, assume that no root of $G$ is divisible by $p$ in the weight lattice $\operatorname{Hom}(T,G_{m})$. Then the restriction $O(\mathfrak{g})^{G}\rightarrow O(\mathfrak{t})^{W}$ is an isomorphism.

Theorem 9.1 was proved by Springer and Steinberg for any adjoint group $G$, in which case the assumption on the roots always holds [24, II.3.17’]. If we do not assume that $G$ is adjoint, then the assumption on the roots is necessary, as shown by the example of the symplectic group $Sp(2n)$ in characteristic 2 (where some roots are divisible by 2 in the weight lattice, and the conclusion fails, as discussed in the proof of Theorem 10.2); but that is the only exception among simple groups.

In particular, Theorem 9.1 applies to cases such as the spin group $\operatorname{Spin}(n)$ in characteristic 2 with $n\geq 6$, which we study further in Theorem 13.1.

Proof.

It suffices to prove that the map is an isomorphism after enlarging $k$ to be algebraically closed. Define a morphism $\varphi\colon G/T\times\mathfrak{t}\rightarrow\mathfrak{g}$ by $(gT,x)\mapsto gxg^{-1}\in\mathfrak{g}$. Let the Weyl group $W=N_{G}(T)/T$ act on $G/T\times\mathfrak{t}$ by $w(gT,x)=(gw^{-1}T,wxw^{-1})$; then $\varphi$ factors through the quotient variety $W\backslash(G/T\times\mathfrak{t})$. Since we assume that no root of $G$ is divisible by $p=\operatorname{char}(k)$, each root of $G$ determines a nonzero linear map $\mathfrak{t}\rightarrow k$. So there is a *regular *element $x$ of $\mathfrak{t}$, meaning an element on which all roots are nonzero.

It follows that the derivative of $\varphi$ at $(1\cdot T,x)$ is bijective. (Indeed, the image of the derivative is at this point is $\mathfrak{t}$ plus the image of $\operatorname{ad}(x):\mathfrak{g}/\mathfrak{t}\rightarrow\mathfrak{g}$. The vector space $\mathfrak{g}/t$ is a direct sum of the 1-dimensional root spaces as a representation of $T$, and $x$ acts by a nonzero scalar on each space since $x$ is regular.) So $\varphi\colon G/T\times\mathfrak{t}\rightarrow\mathfrak{g}$ is a separable dominant map.

Next, I claim that $W\backslash(G/T\times\mathfrak{t})\rightarrow\mathfrak{g}$ is generically bijective; then it will follow that this map is birational. Note that the vector space $\mathfrak{t}$ is defined over $\text{\bf F}_{p}$, by the isomorphism $\mathfrak{t}\cong\operatorname{Hom}(G_{m},T)\otimes_{\text{\bf Z}}k$ (or over Z, if $k$ has characteristic 0). Let $x$ be a regular element of $\mathfrak{t}$ which is not in any hyperplane defined over $\text{\bf F}_{p}$ (or over Z, if $k$ has characteristic 0). Then our claim follows if the inverse image of $x$ in $G/T\times\mathfrak{t}$ is only the $W$-orbit of $(1\cdot T,x)$. Equivalently, we have to show that any element $g$ of $G(k)$ that conjugates $x$ into $\mathfrak{t}$ lies in the normalizer $N_{G}(T)$.

First suppose that $p>0$. Then, for any $g\in G(k)$, the intersection of $T$ with $gTg^{-1}$ has $p$-torsion subgroup scheme $H$ contained in $T[p]\cong(\mu_{p})^{l}$, where $l$ is the dimension of $T$. Here the Lie algebra of $T[p]$ is equal to the Lie algebra of $T$, and the Lie algebra of $H$ is defined over $\text{\bf F}_{p}$ in terms of the $\text{\bf F}_{p}$-structure above on $\mathfrak{t}$. So if $gxg^{-1}$ is in $\mathfrak{t}$, then $g\mathfrak{t}g^{-1}=\mathfrak{t}$, since $x$ is contained in no hyperplane of $\mathfrak{t}$ defined over $\text{\bf F}_{p}$. For $p=0$, the same conclusion holds, since the Lie algebra of $T\cap gTg^{-1}$ is a subspace of $\mathfrak{t}$ defined over Z. The rest of the argument works for any $p\geq 0$. Let $E$ be the normalizer of $\mathfrak{t}$ in $G$; then we have shown that $g\in E(k)$.

Clearly $E$ contains $T$. Also, the Lie algebra of $E$ is $\{y\in\mathfrak{g}:[y,\mathfrak{t}]\subset\mathfrak{t}\}$. Since $\mathfrak{t}$ acts nontrivially on each of the 1-dimensional root spaces which span $\mathfrak{g}/\mathfrak{t}$, the Lie algebra of $E$ is equal to $\mathfrak{t}$. Thus $E$ is smooth over $k$, with identity component equal to $T$. So $E$ is contained in $N_{G}(T)$. The reverse inclusion is clear, and so $E=N_{G}(T)$. Thus the element $g$ above is in $N_{G}(T)$, proving our claim.

As mentioned above, it follows that the morphism $\alpha\colon W\backslash(G/T\times\mathfrak{t})\rightarrow\mathfrak{g}$ is birational. This map is also $G$-equivariant, where $G$ acts on $G/T$ and by conjugation on $\mathfrak{g}$. Because $\alpha$ is dominant, the restriction $O(\mathfrak{g})^{G}\rightarrow O(\mathfrak{t})^{W}$ is injective. Because $\alpha$ is birational, every $W$-invariant polynomial $f$ on $\mathfrak{t}$ corresponds to a $G$-invariant rational function on $\mathfrak{g}$. We follow Springer-Steinberg’s argument: write $f=f_{1}/f_{2}$ with $f_{1}$ and $f_{2}$ relatively prime polynomials. The center $Z(G)$ acts trivially on $\mathfrak{g}$. Since $G/Z(G)$ equals its own commutator subgroup, every homomorphism $G/Z(G)\rightarrow G_{m}$ is trivial, and so both $f_{1}$ and $f_{2}$ are $G/Z(G)$-invariant. Thus $O(\mathfrak{t})^{W}$ is contained in the fraction field of $O(\mathfrak{g})^{G}$. Since the ring $O(\mathfrak{g})^{G}$ is normal, it follows that $O(\mathfrak{t})^{W}=O(\mathfrak{g})^{G}$, as we want. ∎

10 Hodge cohomology of $BG$ at non-torsion primes

Theorem 10.1**.**

Let $G$ be a reductive group over a field $k$ of characteristic $p\geq 0$. Then $H^{>0}(G,O(\mathfrak{g}))=0$ if and only if $p$ is not a torsion prime for $G$.

Theorem 10.2**.**

Let $G$ be a split reductive group over Z, and let $p$ be a non-torsion prime for $G$. Then $H^{j}(BG_{\text{\bf Z}},\Omega^{i})$ localized at $p$ is zero for $i\neq j$. Moreover, the Hodge cohomology ring $H^{*}(BG_{\text{\bf Z}},\Omega^{*})$ and the de Rham cohomology $H_{\operatorname{dR}}^{*}(BG/{\text{\bf Z}})$, localized at $p$, are polynomial rings on generators of degrees equal to 2 times the fundamental degrees of $G$. These rings are isomorphic to the cohomology of the topological space $BG_{\text{\bf C}}$ with $\text{\bf Z}_{(p)}$ coefficients.

We recall the definition of torsion primes for a reductive group $G$ over a field $k$. Let $B$ be a Borel subgroup of $G_{\overline{k}}$, and $T$ a maximal torus in $B$. Then there is a natural homomorphism from the character group $X^{*}(T)=\operatorname{Hom}(T,G_{m})$ (the weight lattice of $G$) to the Chow group $CH^{1}(G_{\overline{k}}/B)$. Therefore, for $N=\dim(G_{\overline{k}}/B)$, there is a homomorphism from the symmetric power $S^{N}(X^{*}(T))$ to $CH^{N}(G_{\overline{k}}/B)$; taking the degree of a zero-cycle on $G_{\overline{k}}/B$ gives a homomorphism (in fact, an isomorphism) $CH^{N}(G_{\overline{k}}/B)\rightarrow\text{\bf Z}$. A prime number $p$ is said to be a torsion prime for $G$ if the image of $S^{N}(X^{*}(T))\rightarrow\text{\bf Z}$ is zero modulo $p$. Borel showed that $p$ is a torsion prime for $G$ if and only if the cohomology $H^{*}(BG_{\text{\bf C}},\text{\bf Z})$ has $p$-torsion, where $G_{\text{\bf C}}$ is the corresponding complex reductive group. Various other characterizations of the torsion primes for $G$ are summarized in [27, section 1].

In most cases, Theorem 10.1 follows from Theorem 4.1. Explicitly, a prime number $p$ is non-torsion for a simply connected simple group $G$ if $p\neq 2$ if $G$ has a simple factor not of type $A_{n}$ or $C_{n}$, $p\neq 2,3$ if $G$ has a simple factor of type $F_{4}$, $E_{6}$, $E_{7}$, or $E_{8}$, and $p\neq 2,3,5$ if $G$ has an $E_{8}$ factor. So the main new cases in Theorem 10.1 are the symplectic groups $Sp(2n)$ in characteristic 2 and $G_{2}$ in characteristic 3. (These are non-torsion primes, but not good primes in the sense of Theorem 4.1.) In these cases, the representation-theoretic result that $H^{>0}(G,O(\mathfrak{g}))=0$ seems to be new. Does $O(\mathfrak{g})$ have a good filtration in these cases?

The following spectral sequence, modeled on the Leray-Serre spectral sequence in topology, will be important for the rest of the paper.

Proposition 10.3**.**

Let $P$ be a parabolic subgroup of a split reductive group $G$ over a field $k$. Let $L$ be the quotient of $P$ by its unipotent radical. Then there is a spectral sequence of algebras

[TABLE]

Proof.

Consider $\Omega^{*}=\oplus\Omega^{i}$ as a presheaf of commutative dgas on smooth $k$-schemes, with zero differential.

For a smooth morphism $f\colon X\rightarrow Y$ of smooth $k$-schemes, consider the object $Rf_{*}(\Omega^{*}_{X})$ in the derived category $D(Y)$ of etale sheaves on $Y$. Here the sheaf $\Omega^{*}_{X}$ on $X$ has an increasing filtration, compatible with its ring structure, with 0th step the subsheaf $f^{*}(\Omega^{*}_{Y})$ and $j$th graded piece $f^{*}(\Omega^{*}_{Y})\otimes\Omega^{j}_{X/Y}$. So $Rf_{*}(\Omega^{*}_{X})$ has a corresponding filtration in $D(Y)$, with $j$th graded piece $Rf_{*}(f^{*}(\Omega^{*}_{Y})\otimes\Omega^{j}_{X/Y})\cong\Omega^{*}_{Y}\otimes Rf_{*}\Omega^{j}_{X/Y}$. This gives a spectral sequence

[TABLE]

Now specialize to the case where $f\colon X\rightarrow Y$ is the $G/P$-bundle associated to a principal $G$-bundle over $Y$. The Hodge cohomology of $G/P$ is essentially independent of the base field, by the isomorphism $H^{*}_{\operatorname{H}}((G/P)/k)\cong CH^{*}(G/P)\otimes_{\text{\bf Z}}k$ (Proposition 8.1). Therefore, each object $Rf_{*}(\Omega^{j}_{X/Y})$ is a trivial vector bundle on $EG/P$, with fiber $H^{j}(G/P,\Omega^{j})$, viewed as a complex in degree $j$. So we can rewrite the spectral sequence as

[TABLE]

All differentials in the spectral sequence above preserve the degree in the grading of $\Omega^{*}$. Therefore, we can renumber the spectral sequence so that it is graded by total degree:

[TABLE]

Finally, we consider the analogous spectral sequence for the morphism $f\colon EG/P\rightarrow BG$ of simplicial schemes:

[TABLE]

By Lemma 1.2, the output of the spectral sequence is isomorphic to $H^{*}_{\operatorname{H}}(BP/k)$, or equivalently (by Theorem 6.1) to $H^{*}_{\operatorname{H}}(BL/k)$. This is a spectral sequence of algebras. All differentials preserve the degree in the grading of $\Omega^{*}$. ∎

Proof.

(Theorem 10.1) First, suppose that $H^{>0}(G,O(\mathfrak{g}))=0$; then we want to show that $\operatorname{char}(k)$ is not a torsion prime for $G$. By Theorem 3.1, the assumption implies that $H^{j}(BG,\Omega^{i})=0$ for all $i\neq j$. Apply Proposition 10.3 when $P$ is a Borel subgroup $B$ in $G$; this gives a spectral sequence

[TABLE]

where $T$ is a maximal torus in $B$. Under our assumption, this spectral sequence degenerates at $E_{2}$, because the differential $d_{r}$ (for $r\geq 2$) takes $H^{i}(BG,\Omega^{i})\otimes H^{j}(G/B,\Omega^{j})$ into $H^{i+r}(BG,\Omega^{i+r-1})\otimes H^{j-r+1}(G/B,\Omega^{j-r+1})$, which is zero. It follows that $H^{*}_{\operatorname{H}}(BT/k)\rightarrow H^{*}_{\operatorname{H}}((G/B)/k)$ is surjective. Here $H^{*}_{\operatorname{H}}(BT/k)$ is the polynomial ring $S(X^{*}(T)\otimes k)$ by Theorem 4.1, and $H^{*}_{\operatorname{H}}((G/B)/k)=CH^{*}(G/B)\otimes k$ by Proposition 8.1. It follows that the ring $CH^{*}(G/B)\otimes k$ is generated as a $k$-algebra by the image of $X^{*}(T)\rightarrow CH^{1}(G/B)$. Equivalently, $p$ is not a torsion prime for $G$.

Conversely, suppose that $p$ is not a torsion prime for $G$. That is, the homomorphism $S(X^{*}(T)\otimes k)\rightarrow CH^{*}(G/B)\otimes k$ is surjective. Equivalently, $H^{*}_{\operatorname{H}}(BT/k)\rightarrow H^{*}_{\operatorname{H}}((G/B)/k)$ is surjective. By the product structure on the spectral sequence above, it follows that the spectral sequence degenerates at $E_{2}$. Since $H^{j}(BT,\Omega^{i})=0$ for $i\neq j$, it follows that $H^{j}(BG,\Omega^{i})=0$ for $i\neq j$. Equivalently, $H^{>0}(G,O(\mathfrak{g}))=0$. ∎

Proof.

(Theorem 10.2) Let $G$ be a split reductive group over Z, and let $p$ be a non-torsion prime for $G$. We have a short exact sequence

[TABLE]

By Theorem 10.1, the Hodge cohomology ring $H^{*}(BG_{\text{\bf Z}},\Omega^{*})$ localized at $p$ is concentrated in bidegrees $H^{i,i}$ and is torsion-free. This ring tensored with Q is the ring of invariants $O({\mathfrak{g}}_{\text{\bf Q}})^{G}$, which is a polynomial ring on generators of degrees equal to the fundamental degrees of $G$.

To show that the Hodge cohomology ring over $\text{\bf Z}_{(p)}$ is a polynomial ring on generators in $H^{i.i}$ for $i$ running through the fundamental degrees of $G$, it suffices to show that the Hodge cohomology ring $H^{*}_{\operatorname{H}}(BG/\text{\bf F}_{p})$ is a polynomial ring in the same degrees. Given that, the other statements of the theorem will follow. Indeed, the statement on Hodge cohomology implies that the de Rham cohomology ring $H^{*}_{\operatorname{dR}}(BG/\text{\bf Z})$ localized at $p$ is also a polynomial ring, on generators in 2 times the fundamental degrees of $G$. The cohomology of the topological space $BG_{\text{\bf C}}$ localized at $p$ is known to be a polynomial ring on generators in the same degrees, by Borel [27, section 1].

From here on, let $k=\text{\bf F}_{p}$, and write $G$ for $G_{k}$. By definition of the Weyl group $W$ as $W=N_{G}(T)/T$, the image of $H^{*}_{\operatorname{H}}(BG/k)$ in $H^{*}_{\operatorname{H}}(BT/k)=S(X(T)\otimes k)$ is contained in the subring of $W$-invariants. We now use that $p$ is not a torsion prime for $G$. By Demazure, except in the case where $p=2$ and $G$ has an $Sp(2n)$ factor, the ring of $W$-invariants in $S(X(T)\otimes k)$ is a polynomial algebra over $k$, with the degrees of generators equal to the fundamental degrees of $G$ [8, Théorème].

By Theorem 9.1, for any simple group $G$ over a field $k$ of characteristic $p$ with $p$ not a torsion prime, except for $G=Sp(2n)$ with $p=2$, the restriction $O(\mathfrak{g})^{G}\rightarrow O(\mathfrak{t})^{W}$ is an isomorphism. In particular, for $G=SL(n)$ with $n\geq 3$ over any field $k$, it follows that $O(\mathfrak{g})^{G}$ is a polynomial ring with generators in the fundamental degrees of $G$, that is, $2,3,\ldots,n$.

The case of $Sp(2n)$ in characteristic 2 (including $SL(2)=Sp(2)$) is a genuine exception: here $O(\mathfrak{g})^{G}$ is a subring of $O(\mathfrak{t})^{W}$, not equal to it. However, it is still true in this case that $O(\mathfrak{g})^{G}$ is a polynomial ring with generators in the fundamental degrees of $G$, that is, $2,4,\ldots,2n$. One way to check this is first to compute that, for $k$ of characteristic 2, $O(\mathfrak{sl}(2))^{SL(2)}$ is the subring $k[c_{2}]$ of $O(\mathfrak{t})^{W}=k[x_{1}]$, where $x_{1}$ is in degree 1, $c_{2}$ is in degree 2, and $c_{2}\mapsto x_{1}^{2}$. (Note that $W\cong\text{\bf Z}/2$ acts trivially on $\mathfrak{t}\cong k$ since the characteristic is 2.) Here $c_{2}$ is the determinant on the space $\mathfrak{sl}(2)$ of matrices of trace zero, and $O(\mathfrak{sl}(2))^{SL(2)}$ is only $k[c_{2}]$ (not $k[x_{1}]$) because the determinant

[TABLE]

is visibly not a square in $O(\mathfrak{sl}(2))^{SL(2)}$. To handle $G=Sp(2n)$ for any $n$, note that the inclusion of $O(\mathfrak{sp}(2n))^{Sp(2n)}$ into $O(\mathfrak{t})^{W}$ factors through $((O(\mathfrak{sl}(2))^{SL(2)})^{n})^{S_{n}}$, because of the subgroup $S_{n}\ltimes SL(2)$ in $Sp(2n)$. By the calculation for $SL(2)$, $((O(\mathfrak{sl}(2))^{SL(2)})^{n})^{S_{n}}$ is isomorphic to $k[c_{2},c_{4},\ldots,c_{2n}]$ where $c_{2i}$ is in degree $2i$; so $O(\mathfrak{sp}(2n))^{Sp(2n)}$ is a subring of that polynomial ring. Conversely, the even coefficients of the characteristic polynomial for a matrix in $\mathfrak{sp}(2n)\subset\mathfrak{gl}(2n)$ restrict to these classes $c_{2i}$, and so $O(\mathfrak{sp}(2n))^{Sp(2n)}$ is isomorphic to the polynomial ring $k[c_{2},c_{4},\ldots,c_{2n}]$, as we want. ∎

11 $\mu_{p}$

Proposition 11.1**.**

Let $k$ be a field of characteristic $p>0$. Let $G$ be the group scheme $\mu_{p}$ of $p$th roots of unity over $k$. Then

[TABLE]

where $c_{1}$ is in $H^{1}(B\mu_{p},\Omega^{1})$ and $v_{1}$ is in $H^{0}(B\mu_{p},\Omega^{1})$. Likewise $H^{*}_{\operatorname{dR}}(B\mu_{p}/k)\cong k[c_{1}]\langle v_{1}\rangle$ with $|v_{1}|=1$ and $|c_{1}|=2$.

Here $R\langle v\rangle$ denotes the exterior algebra over a graded-commutative ring $R$ with generator $v$; that is, $R\langle v\rangle=R\oplus R\cdot v$, with product $v^{2}=0$. See section 1 for the definitions of Hodge and de Rham cohomology we are using for a non-smooth group scheme such as $\mu_{p}$. Proposition 11.1 can help to compute Hodge cohomology of $BG$ for smooth group schemes $G$, as we will see in the proof of Theorem 12.1 for $G=SO(n)$.

Proposition 11.1 is roughly what the topological analogy would suggest. Indeed, for $k$ of characteristic $p$, the ring $H^{*}((B\mu_{p})_{\text{\bf C}},k)$ is a polynomial ring $k[x]$ with $|x|=1$ if $p=2$, or a free graded-commutative algebra $k\langle x,y\rangle$ with $|x|=1$ and $|y|=2$ if $p$ is odd. So $H^{*}_{\operatorname{dR}}(B\mu_{p}/k)$ is isomorphic to $H^{*}((B\mu_{p})_{\text{\bf C}},k)$ additively for any prime $p$, and as a graded ring if $p>2$.

Proof.

Let $G=\mu_{p}$ over $k$. The co-Lie complex $l_{G}$ in the derived category of $G$-modules, discussed in section 3, has $H^{0}(l_{G})\cong\mathfrak{g}^{*}\cong k$ and also $H^{-1}(l_{G})\cong k$, with other cohomology groups being zero. (In short, this is because $G$ is a complete intersection in the affine line, defined by the one equation $x^{p}=1$.)

Since representations of $G$ are completely reducible, we have $\operatorname{Ext}_{G}^{>0}(M,N)=0$ for all $G$-modules $M$ and $N$ [15, Lemma I.4.3]. The isomorphism class of $l_{G}$ is described by an element of $\operatorname{Ext}^{2}_{G}(k,k)$, which is zero. So $l_{G}\cong k\oplus k[1]$ in the derived category of $G$-modules.

By Theorem 3.1, we have

[TABLE]

Here

[TABLE]

which is isomorphic to $k\oplus k[1]$ if $j\geq 1$ and to $k$ if $j=0$. Therefore, $H^{i}(BG,\Omega^{j})$ is isomorphic to $k$ if $0\leq i=j$ or if $0\leq i=j-1$, and is otherwise zero.

Write $c_{1}$ for the generator of $H^{1}(BG,\Omega^{1})$, which is pulled back from the Chern class $c_{1}$ in $H^{1}(BG_{m},\Omega^{1})$ via the inclusion $G\hookrightarrow G_{m}$. Write $v_{1}$ for the generator of $H^{0}(BG,\Omega^{1})$. We have $v_{1}^{2}=0$ because $H^{0}(BG,\Omega^{2})=0$. By the proof, Theorem 3.1 also describes the ring structure on the Hodge cohomology of $BG$. In particular, $\oplus_{i}H^{i}(BG,\Omega^{i})$ is the ring of invariants of $G$ acting on $O(\mathfrak{g})$, which is the polynomial ring $k[c_{1}]$. Finally, the description of $S^{j}(l_{G})$ also shows that $\oplus_{i}H^{i}(BG,\Omega^{i+1})$ is the free module over $k[c_{1}]$ on the generator $v_{1}$. This completes the proof that

[TABLE]

Finally, consider the Hodge spectral sequence for $BG$. The element $v_{1}$ is a permanent cycle because $H^{0}(BG,\Omega^{2})=0$, and $c_{1}$ is a permanent cycle because it is pulled back from a permanent cycle on $BG_{m}$. Therefore, the Hodge spectral sequence degenerates at $E_{1}$. We have $v_{1}^{2}=0$ in de Rham cohomology as in Hodge cohomology, because $\oplus_{i}H^{0}(BG,\Omega^{i})$ is a subring of de Rham cohomology, using degeneration of the Hodge spectral sequence. Therefore, the de Rham cohomology of $BG$ is isomorphic to $k[c_{1}]\langle v_{1}\rangle$ as a graded ring. ∎

Lemma 11.2**.**

Let $G$ be a discrete group, considered as a group scheme over a field $k$. Then the Hodge cohomology of the algebraic stack $BG$ is the group cohomology of $G$:

[TABLE]

It follows that $H^{*}_{\operatorname{dR}}(BG/k)\cong H^{*}(G,k)$.

Proof.

Since $G$ is smooth over $k$, we can compute the Hodge cohomology of the stack $BG$ as the etale cohomology of the simplicial scheme $BG$ with coefficients in $\Omega^{j}$. Since $G$ is discrete, the sheaf $\Omega^{j}$ is zero for $j>0$. For $j=0$, the spectral sequence

[TABLE]

reduces to a single row, since $H^{b}(G^{a},O)=0$ for $b>0$. That is, $H^{*}(BG,O)$ is the cohomology of the standard complex that computes the cohomology of the group $G$ with coefficients in $k$. ∎

More generally, we have the following “Hochschild-Serre” spectral sequence for the Hodge cohomology of a non-connected group scheme:

Lemma 11.3**.**

Let $G$ be an affine group scheme of finite type over a field $k$. Let $G^{0}$ be the identity component of $G$, and suppose that the finite group scheme $G/G^{0}$ is the $k$-group scheme associated to a finite group $Q$. Then there is a spectral sequence

[TABLE]

for any $a\geq 0$.

Proof.

By Theorem 3.1, $H^{r}(BG,\Omega^{a})$ is isomorphic to $H^{r-a}(G,S^{a}(l_{G}))$. The lemma then follows from the Hochschild-Serre spectral sequence for the cohomology of $G$ as an algebraic group, Theorem 6.4. ∎

12 The orthogonal groups

Theorem 12.1**.**

Let $G$ be the split group $SO(n)$ (also called $O^{+}(n)$) over a field $k$ of characteristic 2. Then the Hodge cohomology ring of $BG$ is a polynomial ring $k[u_{2},u_{3},\ldots,u_{n}]$, where $u_{2a}$ is in $H^{a}(BG,\Omega^{a})$ and $u_{2a+1}$ is in $H^{a+1}(BG,\Omega^{a})$. Also, the Hodge spectral sequence degenerates at $E_{1}$, and so $H^{*}_{\operatorname{dR}}(BG/k)$ is also isomorphic to $k[u_{2},u_{3},\ldots,u_{n}]$.

Likewise, the Hodge and de Rham cohomology rings of $BO(2r)$ are isomorphic to the polynomial ring $k[u_{1},u_{2},\ldots,u_{2r}]$. Finally, the Hodge and de Rham cohomology rings of $BO(2r+1)$ are isomorphic to $k[v_{1},c_{1},u_{2},\ldots,u_{2r+1}]/(v_{1}^{2})$, where $v_{1}$ is in $H^{0}(BO(2r+1),\Omega^{1})$ and $c_{1}$ is in $H^{1}(BO(2r+1),\Omega^{1})$.

Thus the de Rham cohomology ring of $BSO(n)_{\text{\bf F}_{2}}$ is isomorphic to the mod 2 cohomology ring of the topological space $BSO(n)_{\text{\bf C}}$ as a graded ring:

[TABLE]

where the classes $w_{i}$ are the Stiefel-Whitney classes. Theorem 12.1 gives a new analog of the Stiefel-Whitney classes for quadratic bundles in characteristic 2. (Note that the $k$-group scheme $O(2r+1)$ is not smooth in characteristic 2. Indeed, it is isomorphic to $SO(2r+1)\times\mu_{2}$.)

The proof is inspired by topology. In particular, it involves some hard work with spectral sequences, related to Borel’s transgression theorem and Zeeman’s comparison theorem. The method should be useful for other reductive groups.

The formula for the classes $u_{i}$ of a direct sum of two quadratic bundles is not the same as for the Stiefel-Whitney classes in topology. To state this, define a quadratic form $(q,V)$ over a field $k$ to be *nondegenerate *if the radical $V^{\perp}$ of the associated bilinear form is zero, and *nonsingular *if $V^{\perp}$ has dimension at most 1 and $q$ is nonzero on any nonzero element of $V^{\perp}$. (In characteristic not 2, nonsingular and nondegenerate are the same.) The orthogonal group is defined as the automorphism group scheme of a nonsingular quadratic form [16, section VI.23]. For example, over a field $k$ of characteristic 2, the quadratic form

[TABLE]

is nonsingular of even dimension $2r$, while the form

[TABLE]

is nonsingular of odd dimension $2r+1$, with $V^{\perp}$ of dimension 1. Let $u_{0}=1$.

Proposition 12.2**.**

Let $X$ be a scheme of finite type over a field $k$ of characteristic 2. Let $E$ and $F$ be vector bundles with nondegenerate quadratic forms over $X$ (hence of even rank). Then, for any $a\geq 0$, in either Hodge cohomology or de Rham cohomology,

[TABLE]

and

[TABLE]

Thus the even $u$-classes of $E\oplus F$ depend only on the even $u$-classes of $E$ and $F$. By contrast, Stiefel-Whitney classes in topology satisfy

[TABLE]

for all $m$ [19, Theorem III.5.11].

Theorem 13.1 gives an example of a reductive group $G$ for which the de Rham cohomology of $BG_{\text{\bf F}_{p}}$ and the mod $p$ cohomology of $BG_{\text{\bf C}}$ are not isomorphic. It is a challenge to find out how close these rings are, in other examples.

Via Theorem 3.1, Theorem 12.1 can be viewed as a calculation in the representation theory of the algebraic group $G=SO(n)$ for any $n$, over a field $k$ of characteristic 2. For example, when $G=SO(3)=PGL(2)$ over $k$ of characteristic 2, we find (what seems to be new):

[TABLE]

Proof.

(Theorem 12.1) We will assume that $k=\text{\bf F}_{2}$. This implies the theorem for any field of characteristic 2.

We begin by computing the ring $\oplus_{i}H^{i}(BG,\Omega^{i})$ for $G=SO(n)$. By Theorem 3.1, this is equal to the ring of $G$-invariant polynomial functions on the Lie algebra $\mathfrak{g}$ over $k$. By Theorem 9.1, since no roots of $G$ are divisible by 2 in the weight lattice for $G$, the restriction $O(\mathfrak{g})^{G}\rightarrow O(\mathfrak{t})^{W}$ is an isomorphism.

Let $r=\lfloor n/2\rfloor$. For $n=2r+1$, the Weyl group $W$ is the semidirect product $S_{r}\ltimes(\text{\bf Z}/2)^{r}$. There is a basis $e_{1},\ldots,e_{r}$ for $\mathfrak{t}$ on which $(\text{\bf Z}/2)^{r}$ acts by changing the signs, and so that action is trivial since $k$ has characteristic 2. The group $S_{r}$ has its standard permutation action on $e_{1},\ldots,e_{r}$. Therefore, the ring of invariants $O(\mathfrak{t})^{W}$ is the ring of symmetric functions in $r$ variables. Let $u_{2},u_{4},\ldots,u_{2r}$ denote the elementary symmetric functions. By the isomorphisms mentioned, we can view $u_{2a}$ as an element of $H^{a}(BSO(2r+1),\Omega^{a})$ for $1\leq a\leq r$, and $\oplus_{i}H^{i}(BSO(2r+1),\Omega^{i})$ is the polynomial ring $k[u_{2},u_{4},\ldots,u_{2r}]$.

For $n=2r$, the Weyl group $W$ of $SO(2r)$ is the semidirect product $S_{r}\ltimes(\text{\bf Z}/2)^{r-1}$. Again, the subgroup $(\text{\bf Z}/2)^{r-1}$ acts trivially on $\mathfrak{t}$, and $S_{r}$ acts by permutations as usual. So $\oplus_{i}H^{i}(BSO(2r),\Omega^{i})$ is also the polynomial ring $k[u_{2},u_{4},\ldots,u_{2r}]$, with $u_{2a}$ in $H^{a}(BSO(2r),\Omega^{a})$ for $1\leq a\leq r$.

For the smooth $k$-group $G=O(2r)$, we can also compute the ring $\oplus_{i}H^{i}(BG,\Omega^{i})$. By Theorem 3.1, this is the ring of $G$-invariant polynomial functions on the Lie algebra $\mathfrak{g}=\mathfrak{so}(2r)$. This is contained in the ring of $SO(2r)$-invariant functions on $\mathfrak{g}$, and I claim that the two rings are equal. It suffices to show that an $SO(2r)$-invariant function on $\mathfrak{g}$ is also invariant under the normalizer $N$ in $O(2r)$ of a maximal torus $T$ in $SO(2r)$, since that normalizer meets both connected components of $O(2r)$. Here $N=S_{r}\ltimes(\text{\bf Z}/2)^{r}$, which acts on $\mathfrak{t}$ in the obvious way; in particular, $(\text{\bf Z}/2)^{r}$ acts trivially on $\mathfrak{t}$. Therefore, an $SO(2r)$-invariant function on $\mathfrak{g}$ (corresponding to an $S_{r}$-invariant function on $\mathfrak{t}$) is also $O(2r)$-invariant. Thus we have $\oplus_{i}H^{i}(BO(2r),\Omega^{i})=k[u_{2},u_{4},\ldots,u_{2r}]$.

For a smooth group scheme $G$ over $R=\text{\bf Z}/4$, define the Bockstein

[TABLE]

on the Hodge cohomology of $BG_{k}$ (where $k=\text{\bf Z}/2$) to be the boundary homomorphism associated to the short exact sequence of sheaves

[TABLE]

on $BG_{R}$. The Bockstein on Hodge cohomology is not defined for group schemes such as $\mu_{2}$ which are flat but not smooth over $R=\text{\bf Z}/4$, because the sequence of sheaves above need not be exact.

Next, define elements $u_{1},u_{3},\ldots,u_{2r-1}$ of $H^{*}_{\operatorname{H}}(BO(2r)/k)$ as follows. First, let $u_{1}\in H^{1}(BO(2r),\Omega^{0})$ be the pullback of the generator of $H^{1}(\text{\bf Z}/2,k)=k$ via the surjection $O(2r)\rightarrow\text{\bf Z}/2$ (Lemma 11.2). Next, use that the split group $O(2r)$ over $k=\text{\bf F}_{2}$ lifts to a smooth group $O(2r)$ over Z. As a result, we have a Bockstein homomorphism on the Hodge cohomology of $BO(2r)$. For $0\leq a\leq r-1$, let $u_{2a+1}=\beta u_{2a}+u_{1}u_{2a}\in H^{a+1}(BO(2r),\Omega^{a})$. This agrees with the previous formula for $u_{1}$, if we make the convention that $u_{0}=1$. (The definition of $u_{2a+1}$ is suggested by the formula for odd Stiefel-Whitney classes in topology: $w_{2a+1}=\beta w_{2a}+w_{1}w_{2a}$ [19, Theorem III.5.12].)

I claim that the homomorphism

[TABLE]

is an isomorphism. To see this, consider the Hochschild-Serre spectral sequence of Lemma 11.3,

[TABLE]

Here $SO(2)$ is isomorphic to $G_{m}$, and so we know the Hodge cohomology of $BSO(2)$ by Theorem 4.1: $H^{*}_{\operatorname{H}}(BSO(2)/k)\cong k[c_{1}]$ with $c_{1}$ in $H^{1}(BSO(2),\Omega^{1})$. We read off that the $E_{2}$ page of the spectral sequence is the polynomial ring $k[u_{1},u_{2}]$, with $u_{1}$ in $H^{1}(\text{\bf Z}/2,H^{0}(BSO(2),\Omega^{0}))$ and $u_{2}$ in $H^{0}(\text{\bf Z}/2,H^{1}(BSO(2),\Omega^{1}))$. Here $u_{1}$ is a permanent cycle, because all differentials send $u_{1}$ to zero groups. Also, because the surjection $O(2)\rightarrow\text{\bf Z}/2$ of $k$-groups is split, there are no differentials into the bottom row of the spectral sequence; so $u_{2}$ is also a permanent cycle. It follows that the spectral sequence degenerates at $E_{2}$, and hence that $H^{*}_{\operatorname{H}}(BO(2)/k)\cong k[u_{1},u_{2}]$.

We also need to compute the Bockstein on the Hodge cohomology of $BO(2)$, which is defined because $O(2)$ lifts to a smooth group scheme over $R:=\text{\bf Z}/4$. The Bockstein is related to the Hodge cohomology of $BO(2)_{R}$ by the exact sequence

[TABLE]

Consider the Hochschild-Serre spectral sequence of Lemma 11.3 for $BO(2)_{R}$:

[TABLE]

Here $H^{1}(BO(2)_{R},\Omega^{1})$ is isomorphic to $H^{0}(\text{\bf Z}/2,H^{1}(BSO(2)_{R},\Omega^{1}))$, where $\text{\bf Z}/2$ acts by $-1$ on $H^{1}(BSO(2)_{R},\Omega^{1})\cong\text{\bf Z}/4$. So the generator of $H^{1}(BO(2)_{R},\Omega^{1})\cong\text{\bf Z}/2$ maps to zero in $H^{1}(BO(2)_{k},\Omega^{1})=k\cdot u_{2}$. Therefore, $\beta(u_{2})\neq 0$. Since $k=\text{\bf F}_{2}$, the element $\beta(u_{2})$ in $H^{2}(BO(2)_{k},\Omega^{1})=k\cdot u_{1}u_{2}$ must be equal to $u_{1}u_{2}$. A similar analysis shows that $\beta(u_{1})=u_{1}^{2}$.

Finally, think of $O(2)$ as the isometry group of the quadratic form $q(x,y)=xy$ on $V=A^{2}_{k}$. There is an inclusion $H=\text{\bf Z}/2\times\mu_{2}\subset O(2)$, where $\text{\bf Z}/2$ switches $x$ and $y$ and $\mu_{2}$ acts by scalars on $V$. For later use, it is convenient to say something about the restriction from $BO(2)$ to $BH$ on Hodge cohomology. By Lemma 11.2, the Hodge cohomology of $B(\text{\bf Z}/2)$ over $k$ is the cohomology of $\text{\bf Z}/2$ as a group, namely the polynomial ring $k[s]$ with $s\in H^{1}(B(\text{\bf Z}/2),O)$. Also, by Proposition 11.1, the Hodge cohomology of $B\mu_{2}$ is $k[t,v]/(v^{2})$ with $t\in H^{1}(B\mu_{2},\Omega^{1})$ and $v\in H^{0}(B\mu_{2},\Omega^{1})$. Thus we have a homomorphism from $H^{*}_{\operatorname{H}}(BO(2)/k)=k[u_{1},u_{2}]$ to $H^{*}_{\operatorname{H}}(BH/k)\cong k[s,t,v]/(v^{2})$ (by the Künneth theorem, Proposition 5.1). Here $u_{1}$ restricts to $s$, since both elements are pulled back from the generator of $H^{1}(B\text{\bf Z}/2,O)$. Also, $u_{2}$ restricts to either $t$ or $t+sv$, because $u_{2}$ restricts to the generator $c_{1}$ of $H^{1}(BG_{m},\Omega^{1})$ and hence to $t$ in $H^{1}(B\mu_{2},\Omega^{1})$. Thus the homomorphism from $H^{*}_{\operatorname{H}}(BO(2)/k)$ to $H^{*}_{\operatorname{H}}(BH/k)/\text{rad}=k[s,t]$ is an isomorphism. (Here the *radical *of a commutative ring means the ideal of nilpotent elements.) A direct cocycle computation shows that $u_{2}$ restricts to $t+sv$ in $H^{1}(BH,\Omega^{1})$, but we do not need that fact in this paper.

We now return to the group $O(2r)$ over $k=\text{\bf F}_{2}$ for any $r$. I claim that the homomorphism

[TABLE]

is injective. The idea is to compose this homomorphism with restriction to the Hodge cohomology of $BO(2)^{r}$. Let $s_{1},\ldots,s_{r}\in H^{1}(BO(2)^{r},\Omega^{0})$ be the pullbacks of $u_{1}$ from the $r$ $BO(2)$ factors, and let $t_{1},\ldots,t_{r}$ be the pullbacks of $u_{2}$ from those $r$ factors. By the Künneth theorem (Proposition 5.1), the Hodge cohomology of $BO(2)^{r}$ is the polynomial ring $k[s_{1},\ldots,s_{r},t_{1},\ldots,t_{r}]$. The elements $u_{2},u_{4},\ldots,u_{2r}$ restrict to the elementary symmetric functions in $t_{1},\ldots,t_{r}$:

[TABLE]

Also,

[TABLE]

The inclusion $O(2)^{2}\subset O(2r)$ lifts to an inclusion of smooth groups over Z, and so the restriction homomorphism commutes with the Bockstein. Therefore, for $0\leq a\leq r-1$,

[TABLE]

We want to show that this homomorphism $k[u_{1},\ldots,u_{2r}]\rightarrow k[s_{1},\ldots,s_{r},t_{1},\ldots,t_{r}]$ is injective. We can factor this homomorphism through $k[u_{1},u_{3},\ldots,u_{2r-1},t_{1},\ldots,t_{r}]$, by the homomorphism $\rho$ sending $u_{2},u_{4},\ldots,u_{2r}$ to the elementary symmetric polynomials in $t_{1},\ldots,t_{r}$. Since $\rho$ is injective, it remains to show that

[TABLE]

is injective.

More strongly, we will show that $\sigma$ is generically etale; that is, its Jacobian determinant is not identically zero. Because $\sigma$ is the identity on the $t_{i}$ coordinates, it suffices to show that the matrix of derivatives of $u_{1},u_{3},\ldots,u_{2r-1}$ with respect to $s_{1},\ldots,s_{r}$ is nonzero for $s_{1},\ldots,s_{r},t_{1},\ldots,t_{r}$ generic. This matrix of derivatives in fact only involves $t_{1},\ldots,t_{r}$, because $u_{1},u_{3},\ldots,u_{2r-1}$ have degree 1 in $s_{1},\ldots,s_{r}$. For example, for $r=3$, this matrix of derivatives is

[TABLE]

where the $a$th column gives the derivatives of $u_{2a-1}$ with respect to $s_{1},\ldots,s_{r}$. For any $r$, column 1 consists of 1s, while entry $(j,a)$ for $a\geq 2$ is

[TABLE]

This determinant is equal to the Vandermonde determinant $\delta:=\prod_{i<j}(t_{i}-t_{j})$, and in particular it is not identically zero [10, Theorem 1]. (The reference works over C, but it amounts to an identity of polynomials over Z, which therefore holds over any field.)

Thus we have shown that the composition $k[u_{1},\ldots,u_{2r}]\rightarrow H^{*}_{\operatorname{H}}(BO(2r)/k)$ is injective, because the composition to $H^{*}_{\operatorname{H}}(BO(2)^{r}/k)$ is injective. Analogously, let us show that $k[u_{2},\ldots,u_{n}]\rightarrow H^{*}_{\operatorname{H}}(BSO(n)/k)$ is injective for every $n\geq 1$.

For $n=2r+1$, this is easy, using the inclusions $O(2)^{r}\subset O(2r)\subset SO(2r+1)$. Write $u_{2},u_{3},\ldots,u_{2r+1}$ for the elements of the Hodge cohomology of $BSO(2r+1)$ defined by the same formulas as used above for $BO(2r)$ (which simplify to $u_{2a+1}=\beta u_{2a}$, since there is no element $u_{1}$ for $BSO(2r+1)$). Also, let $v_{1},\ldots,v_{2r}$ be the elements of the Hodge cohomology of $BO(2r)$ that were called $u_{1},\ldots,u_{2r}$ above. Then restricting from $BSO(2r+1)$ to $BO(2r)$ sends $u_{2a}\mapsto v_{2a}$ and $u_{2a+1}=\beta u_{2a}\mapsto\beta v_{2a}=v_{2a+1}+v_{1}v_{2a}$ for $1\leq a\leq r-1$. It is not immediate how to compute the restriction of the remaining element $u_{2r+1}$ to $BO(2r)$, but we can compute its restriction to $BO(2)^{r}$:

[TABLE]

Thus, the restriction from $BSO(2r+1)$ to $BO(2)^{r}$ sends $k[u_{2},\ldots,u_{2r+1}]$ into the subring

[TABLE]

by $u_{2a}\mapsto v_{2a}$ for $1\leq a\leq r$, $u_{2a+1}\mapsto v_{2a+1}+v_{1}v_{2a}$ for $1\leq a\leq r-1$, and $u_{2r+1}\mapsto v_{1}v_{2r}$. This homomorphism is injective, because the corresponding morphism $A^{2r}\rightarrow A^{2r}$ is birational (for $u_{2r}\neq 0$, one can solve for $v_{1},\ldots,v_{2r}$ in terms of $u_{2},\ldots,u_{2r+1}$). So the homomorphism $k[u_{2},\ldots,u_{2r+1}]\rightarrow H^{*}_{\operatorname{H}}(BSO(2r+1)/k)$ is injective (because its composition to $H^{*}_{\operatorname{H}}(BO(2)^{r}/k)$ is injective).

For $SO(2r)$, we argue a bit differently. As discussed above, there is a subgroup $\text{\bf Z}/2\times\mu_{2}\subset O(2)$. Therefore, we have a $k$-subgroup scheme $(\text{\bf Z}/2\times\mu_{2})^{r}\subset O(2)^{r}\subset O(2r)$. Since $SO(2r)$ is the kernel of a homomorphism from $O(2r)$ onto $\text{\bf Z}/2$, $SO(2r)$ contains a $k$-subgroup scheme $H\cong(\text{\bf Z}/2)^{r-1}\times(\mu_{2})^{r}$. By Lemma 11.2, the Hodge cohomology of $B(\text{\bf Z}/2)$ over $k$ is the cohomology of $\text{\bf Z}/2$ as a group, namely the polynomial ring $k[x]$ with $x\in H^{1}(B(\text{\bf Z}/2),\Omega^{0})$. Also, by Proposition 11.1, the Hodge cohomology of $B\mu_{2}$ is $k[t,v]/(v^{2})$ with $t\in H^{1}(B\mu_{2},\Omega^{1})$ and $v\in H^{0}(B\mu_{2},\Omega^{1})$. Thus we have a homomorphism from $k[u_{2},u_{3},\ldots,u_{2r}]$ to $H^{*}_{\operatorname{H}}(BSO(2r)/k)$ and from there to $H^{*}_{\operatorname{H}}(BH/k)\cong k[x_{1},\ldots,x_{r-1},t_{1},\ldots,t_{r},v_{1},\ldots,v_{r}]/(v_{i}^{2})$ (by the Künneth theorem, Proposition 5.1). We want to show that this composition is injective. For convenience, we will prove the stronger statement that the composition to

[TABLE]

is injective.

We compare the restriction from $O(2r)$ to $(\text{\bf Z}/2)^{r}\times(\mu_{2})^{r}$ with that from $SO(2r)$ to $H$:

[TABLE]

The bottom homomorphism is given (for a suitable choice of generators $x_{1},\ldots,x_{r-1}$) by $s_{i}\mapsto x_{i}$ for $1\leq i\leq r-1$ and $s_{r}\mapsto x_{1}+\cdots+x_{r-1}$ (agreeing with the fact that $u_{1}\mapsto s_{1}+\cdots+s_{r}\mapsto 0$ in the Hodge cohomology of $BH$). By the formulas for $O(2r)$, we know how the elements $u_{2},\ldots,u_{2r}$ restrict to $k[s_{1},\ldots,s_{r},t_{1},\ldots,t_{r}]$, and hence to $k[x_{1},\ldots,x_{r-1},t_{1},\ldots,t_{r}]$. Namely,

[TABLE]

and, for $1\leq a\leq r-1$,

[TABLE]

We want to show that this homomorphism $k[u_{2},u_{3},\ldots,u_{2r}]\rightarrow k[x_{1},\ldots,x_{r-1},t_{1},\ldots,t_{r}]$ is injective. It can be factored through $k[u_{3},u_{5},\ldots,u_{2r-1},t_{1},\ldots,t_{r}]$, by the homomorphism $\rho$ sending $u_{2},u_{4},\ldots,u_{2r}$ to the elementary symmetric polynomials in $t_{1},\ldots,t_{r}$. Since $\rho$ is injective, it remains to show that $\sigma\colon k[u_{3},u_{5},\ldots,u_{2r-1},t_{1},\ldots,t_{r}]\rightarrow k[x_{1},\ldots,x_{r-1},t_{1},\ldots,t_{r}]$ is injective.

As in the argument for $O(2r)$, we will show (more strongly) that $\sigma$ is generically etale; that is, its Jacobian determinant is not identically zero. Because $\sigma$ is the identity on the $t_{i}$ coordinates, it suffices to show that the matrix of derivatives of $u_{3},u_{5},\ldots,u_{2r-1}$ with respect to $x_{1},\ldots,x_{r-1}$ is nonzero for $x_{1},\ldots,x_{r-1},t_{1},\ldots,t_{r}$ generic. This matrix of derivatives in fact only involves $t_{1},\ldots,t_{r}$, because $u_{3},u_{5},\ldots,u_{2r-1}$ have degree 1 as polynomials in $x_{1},\ldots,x_{r-1}$. For example, for $r=3$, this $(r-1)\times(r-1)$ matrix of derivatives is

[TABLE]

where the $a$th column gives the derivatives of $u_{2a+1}$ with respect to $x_{1},\ldots,x_{r-1}$. For any $r$, the entry $(j,a)$ of the matrix (with $j,a\in\{1,\ldots,r-1\}$) is $(t_{j}+t_{r})e_{ja}$, where

[TABLE]

Since row $j$ is a multiple of $(t_{j}+t_{r})$ for each $r$, the determinant is $(t_{1}+t_{r})(t_{2}+t_{r})\cdots(t_{r-1}+t_{r})$ times the determinant of the $(r-1)\times(r-1)$ matrix $E=(e_{ja})$. So it suffices to show that the determinant of $E$ is not identically zero. Indeed, the determinant of $E$ is the same determinant shown to be nonzero in the calculation above for $O(2r)$, but with $r$ replaced by $r-1$.

Thus we have shown that $k[u_{2},\ldots,u_{n}]\rightarrow H^{*}_{\operatorname{H}}(BSO(n)/k)$ is injective for $n$ even as well as for $n$ odd. We now show that this is an isomorphism.

Let $r=\lfloor n/2\rfloor$ and $s=\lfloor(n-1)/2\rfloor$. Let $P$ be the parabolic subgroup of $G=SO(n)$ that stabilizes a maximal isotropic subspace (that is, an isotropic subspace of dimension $r$). Then the quotient of $P$ by its unipotent radical is isomorphic to $GL(r)$. By Proposition 10.3, we have a spectral sequence

[TABLE]

The Chow ring of $G/P$ is isomorphic to

[TABLE]

where $e_{i}\in CH^{i}(G/P)$ is understood to mean zero if $i>s$ [19, III.6.11]. (This uses Chevalley’s theorem that the Chow ring of $G/P$ for a split group $G$ is independent of the characteristic of $k$, and is isomorphic to the integral cohomology ring of $G_{\bf C}/P_{\bf C}$.) By Proposition 8.1, it follows that the Hodge cohomology ring of $G/P$ is isomorphic to

[TABLE]

where $e_{i}$ is in $H^{i}(G/P,\Omega^{i})$. For any list of variables $x_{1},\ldots,x_{m}$, write $\Delta(x_{1},\ldots,x_{m})$ for the $k$-vector space with basis consisting of all products $x_{i_{1}}\ldots x_{i_{j}}$ with $1\leq i_{1}<\cdots<i_{j}\leq m$ and $0\leq j\leq m$. Then we can say that

[TABLE]

The spectral sequence converges to $H^{*}_{\operatorname{H}}(BGL(r)/k)=k[c_{1},\ldots,c_{r}]$, by Theorem 10.2. The elements $u_{2},u_{4},\ldots,u_{2r}$ (where $u_{2i}$ is in $H^{i}(BG,\Omega^{i})$) restrict to $c_{1},c_{2},\ldots,c_{r}$. So the $E_{\infty}$ term of the spectral sequence is concentrated on the [math]th row and consists of the polynomial ring $k[u_{2},u_{4},\ldots,u_{2r}]$.

To analyze the structure of the spectral sequence further, we use Zeeman’s comparison theorem, which he used to simplify the proof of the Borel transgression theorem [19, Theorem VII.2.9]. The key point is to show that the elements $e_{i}$ (possibly after adding decomposable elements) are transgressive. (By definition, an element $u$ of $E_{2}^{0,q}$ in a first-quadrant spectral sequence is *transgressive *if $d_{2}=\cdots=d_{q}=0$ on $u$; then $u$ determines an element $\tau(u):=d_{q+1}(u)$ of $E_{q+1}^{q+1,0}$, called the transgression of $u$.)

In order to apply Zeeman’s comparison theorem, we define a model spectral sequence that maps to the spectral sequence we want to analyze. (To be precise, we consider spectral sequences of $k$-vector spaces, not of $k$-algebras.) As above, let $k=\text{\bf F}_{2}$. For a positive integer $q$, define a spectral sequence $G_{*}$ with $E_{2}$ page given by $G_{2}=\Delta(y)\otimes k[u]$, $y$ in bidegree $(0,q)$, $u$ in bidegree $(q+1,0)$, and $d_{q+1}(yu^{j})=u^{j+1}$.

[TABLE]

Suppose that, for some positive integer $a$, we have found elements $y_{i}$ of $H^{2i}_{\operatorname{H}}((G/P)/k)$ for $1\leq i\leq a$ which are transgressive in the spectral sequence $E_{*}$ above. Because $y_{i}$ is transgressive, there is a map of spectral sequences $G_{*}\rightarrow E_{*}$ that takes the element $y$ (in degree $q=2i$) to $y_{i}$. Since $E_{*}$ is a spectral sequence of algebras, tensoring these maps gives a map of spectral sequences

[TABLE]

(Here we are using that the elements $u_{2},u_{4},\ldots,u_{2r}$ are in $H^{*}_{\operatorname{H}}(BG/k)$, which is row 0 of the $E_{2}$ page on the right, and so they are permanent cycles.) Although we do not view the domain as a spectral sequence of algebras, its $E_{2}$ page is the tensor product of row 0 and column 0, and the map $\alpha\colon F_{2}\rightarrow E_{2}$ of $E_{2}$ pages is the tensor product of the maps on row 0 and column 0.

Using these properties, we have the following version of Zeeman’s comparison theorem, as sharpened by Hilton and Roitberg [19, Theorem VII.2.4]:

Theorem 12.3**.**

Let $N$ be a natural number. Suppose that the homomorphism $\alpha\colon F_{*}\rightarrow E_{*}$ of spectral sequences is bijective on $E_{\infty}^{i,j}$ for $i+j\leq N$ and injective for $i+j=N+1$, and that $\alpha$ is bijective on row 0 of the $E_{2}$ page in degrees $\leq N+1$ and injective in degree $N+2$. Then $\alpha$ is bijective on column 0 of the $E_{2}$ page in degree $\leq N$ and injective in degree $N+1$.

The inductive step for computing the Hodge cohomology of $BSO(n)$ is as follows.

Lemma 12.4**.**

Let $G$ be $SO(n)$ over $k=\text{\bf F}_{2}$, $P$ the parabolic subgroup above, $r=\lfloor n/2\rfloor$, $s=\lfloor(n-1)/2\rfloor$. Let $N$ be a natural number, and let $a=\min(s,\lfloor N/2\rfloor)$. Then, for each $1\leq i\leq a$, there is an element $y_{i}$ in $H^{i}(G/P,\Omega^{i})$ with the following properties. First, $y_{i}$ is equal to $e_{i}$ modulo polynomials in $e_{1},\ldots,e_{i-1}$ with exponents $\leq 1$. Also, each element $y_{i}$ is transgressive, and any lift $v_{2i+1}$ to $H^{i+1}(BG,\Omega^{i})$ of the element $\tau(y_{i})$ has the property that

[TABLE]

is bijective in degree $\leq N+1$ and injective in degree $N+2$. Finally, each element $v_{2i+1}$ is equal to $u_{2i+1}$ modulo polynomials in $u_{2},u_{3},\ldots,u_{2i}$.

More precisely, if this statement holds for $N-1$, then it holds for $N$ with the same elements $y_{i}$, possibly with one added.

We will apply Lemma 12.4 with $N=\infty$, but the formulation with $N$ arbitrary is convenient for the proof.

Proof.

As discussed earlier, the $E_{\infty}$ page of the spectral sequence

[TABLE]

is isomorphic to $k[u_{2},u_{4},\ldots,u_{2r}]$, concentrated on row 0.

We prove the lemma by induction on $N$. For $N=0$, it is true, using that $H^{0}_{\operatorname{H}}(BG/k)=k$ and $H^{1}_{\operatorname{H}}(BG/k)=0$, as one checks using our knowledge of the $E_{\infty}$ term.

We now assume the result for $N-1$, and prove it for $N$. By the inductive assumption, for $b:=\min(s,\lfloor(N-1)/2\rfloor)$, we can choose $y_{1},\ldots,y_{b}$ such that $y_{i}\in H^{i}(G/P,\Omega^{i})$ is equal to $e_{i}$ modulo polynomials in $e_{1},\ldots,e_{i-1}$ with exponents $\leq 1$, $y_{i}$ is transgressive for the spectral sequence, and, if we define $v_{2i+1}\in H^{i+1}(BG,\Omega^{i})$ to be any lift (from the $E_{2i+1}$ page to the $E_{2}$ page) of the transgression $\tau(y_{i})$ for $1\leq i\leq b$, the homomorphism

[TABLE]

is bijective in degree $\leq N$ and injective in degree $N+1$. Finally, the element $v_{2i+1}$ for $1\leq i\leq b$ is equal to $u_{2i+1}$ modulo polynomials in $u_{2},u_{3},\ldots,u_{2i}$.

Also, by the injectivity in degree $N+1$ (above), it follows that there is a set (possibly empty) of elements $z_{i}$ in $H^{N+1}_{\operatorname{H}}(BG/k)$ such that

[TABLE]

is bijective in degrees at most $N+1$. (Recall that $b=\min(s,\lfloor(N-1)/2\rfloor)$.) The elements $z_{i}$ do not affect the domain of $\varphi$ in degree $N+2$ (because that ring is zero in degree 1). Therefore, $\varphi$ is injective in degree $N+2$, because

[TABLE]

is injective. (This uses that $v_{2i+1}$ is equal to $u_{2i+1}$ modulo polynomials in $u_{2},u_{3},\ldots,u_{2i}$, together with the injectivity of $k[u_{2},u_{3},\ldots,u_{n}]\rightarrow H^{*}_{\operatorname{H}}(BG/k)$, shown earlier.)

The elements $z_{i}$ can be chosen to become zero in the $E_{\infty}$ page, because the $E_{\infty}$ page is just $k[u_{2},u_{4},\ldots,u_{2r}]$ on row 0. Therefore, there are transgressive elements $w_{i}\in H^{N}_{\operatorname{H}}((G/P)/k)$ with $z_{i}=\tau(w_{i})$ in the $E_{N+1}$ page. (If $z_{i}$ is killed before $E_{N+1}$, we can simply take $w_{i}=0$.) By Zeeman’s comparison theorem (Theorem 12.3), the homomorphism

[TABLE]

is bijective in degrees $\leq N$ and injective in degree $N+1$.

Let $a=\min(s,\lfloor N/2\rfloor)$. We know that $\Delta(e_{1},\ldots,e_{a})\rightarrow H^{*}_{\operatorname{H}}((G/P)/k)$ is bijective in degrees $\leq N$. Since the elements $w_{i}$ are in degree $N$, while $b=\min(s,\lfloor(N-1)/2\rfloor)$, we deduce that there is no element $w_{i}$ if $N$ is odd or $N>2s$, and there is exactly one $w_{i}$ if $N$ is even and $N\leq 2s$. In the latter case, we have $a=N/2$; in that case, let $y_{a}$ denote the single element $w_{i}$. Since we know that $H^{*}_{\operatorname{H}}((G/P)/k)=\Delta(e_{1},\ldots,e_{s})$, $y_{a}$ must be equal to $e_{a}$ modulo polynomials in $e_{1},\ldots,e_{a-1}$ with exponents $\leq 1$. By construction, $y_{a}$ is transgressive. Also, in the case where $N$ is even and $N\leq 2s$, let $v_{2a+1}$ in $H^{a+1}(BG,\Omega^{a})$ be a lift to the $E_{2}$ page of the element $\tau(y_{a})$ (formerly called $z_{i}$). Then we know that

[TABLE]

is bijective in degree $\leq N+1$. In the case where $N$ is even and $N\leq 2s$ (where we have added one element $v_{2a+1}$ to those constructed before), this bijectivity in degree $N+1=2a+1$ together with the injectivity of $k[u_{2},u_{3},\ldots,u_{n}]\rightarrow H^{*}_{\operatorname{H}}(BG/k)$ in all degrees implies that $v_{2a+1}$ must be equal to $u_{2a+1}$ modulo polynomials in $u_{2},u_{3},\ldots,u_{2a}$. By the same injectivity, it follows that $\varphi$ is injective in degree $N+2$. ∎

We can take $N=\infty$ in Lemma 12.4, because the elements $y_{1},\ldots,y_{s}$ do not change as we increase $N$. This gives that $k[u_{2},u_{3},\ldots,u_{n}]\rightarrow H^{*}_{\operatorname{H}}(BSO(n)/k)$ is an isomorphism. (The element $v_{2i+1}$ produced by Lemma 12.4 need not be the element $u_{2i+1}$ defined earlier, but $v_{2i+1}$ is equal to $u_{2i+1}$ modulo decomposable elements, which gives this conclusion.)

Using the Hodge cohomology of $BSO(2r)$, we can compute the Hodge cohomology of $BO(2r)$ over $k$ using the Hochschild-Serre spectral sequence of Lemma 11.3:

[TABLE]

We have a homomorphism $k[u_{1},u_{2},\ldots,u_{2r}]\rightarrow BO(2r)$ whose composition to $BSO(2r)$ is surjective. Therefore, $\text{\bf Z}/2$ acts trivially on the Hodge cohomology of $BSO(2r)$, and all differentials are zero on column 0 of this spectral sequence. It follows that the spectral sequence degenerates at $E_{2}$, and hence

[TABLE]

Finally, we show that the Hodge spectral sequence

[TABLE]

degenerates for $G=SO(n)$ over $k$. Indeed, by restricting to a maximal torus $T=(G_{m})^{r}$ of $G$, the elements $u_{2},u_{4},\ldots,u_{2r}$ restrict to the elementary symmetric polynomials in the generators of $H^{*}_{\operatorname{dR}}(BT/k)=k[t_{1},\ldots,t_{r}]$. Therefore, the ring $k[u_{2},u_{4},\ldots,u_{2r}]$ injects into $H^{*}_{\operatorname{dR}}(BG/k)$. So all differentials into the main diagonal $\oplus_{i}H^{i,i}$ of the Hodge spectral sequence for $BG$ are zero.

[TABLE]

It follows that all differentials are zero on the elements $u_{2i+1}\in H^{i+1}(BG,\Omega^{i})$: only $d_{1}$ maps $u_{2i+1}$ into a nonzero group, and that is on the main diagonal. Also, all differentials are zero on the elements $u_{2i}$ in the main diagonal (since they map into zero groups). This proves the degeneration of the Hodge spectral sequence. Therefore, $H^{*}_{\operatorname{dR}}(BSO(n)/k)$ is isomorphic to $k[u_{2},u_{3},\ldots,u_{n}]$.

The same argument proves the degeneration of the Hodge spectral sequence for $BO(2r)$. Therefore, $H^{*}_{\operatorname{dR}}(BO(2r)/k)$ is isomorphic to $k[u_{1},u_{2},\ldots,u_{2r}]$.

Finally, $O(2r+1)$ is isomorphic to $SO(2r+1)\times\mu_{2}$, and so the calculation for $BO(2r+1)$ follows from those for $BSO(2r+1)$ (above) and $B\mu_{2}$ (Proposition 11.1), by the Künneth theorem (Proposition 5.1). Theorem 12.1 is proved. ∎

Proof.

(Proposition 12.2) Let $2r$ and $2s$ be the ranks of the quadratic bundles $E$ and $F$. The problem amounts to computing the restriction from $BO(2r+2s)$ to $BO(2r)\times BO(2s)$ on Hodge cohomology or de Rham cohomology. We first compute $u(E\oplus F)$ in Hodge cohomology. The formula for $u_{2a}(E\oplus F)$ follows from the definition of $u_{2a}$ in $H^{a}(BO(2r+2s),\Omega^{a})$. (Since $u_{2a}$ is in $H^{a}(BO(2r+2s),\Omega^{a})$, its restriction to the Hodge cohomology of $BO(2r)\times BO(2s)$ must be in $H^{a}(BO(2r)\times BO(2s),\Omega^{a})$, which explains why only the even $u$-classes of $E$ and $F$ appear in the formula.) The formula for $u_{2a+1}(E\oplus F)$ follows from the formula for $u_{2a}(E\oplus F)$, using that $u_{2a+1}=\beta u_{2a}+u_{1}u_{2a}$.

In de Rham cohomology, the same formulas hold for $u(E\oplus F)$. This uses that for any affine $k$-group scheme $G$, since $H^{i}(BG,\Omega^{j})=0$ for $i<j$ by Theorem 3.1, the subring $\oplus_{i}H^{i}(BG,\Omega^{i})$ of Hodge cohomology canonically maps into de Rham cohomology. ∎

13 The spin groups

In contrast to the other calculations in this paper, we now exhibit a reductive group $G$ such that the mod 2 cohomology of the topological space $BG_{\text{\bf C}}$ is not isomorphic to the de Rham cohomology of the algebraic stack $BG_{\text{\bf F}_{2}}$, even additively. The example was suggested by the observation of Feshbach, Benson, and Wood that the restriction $H^{*}(BG_{\text{\bf C}},\text{\bf Z})\rightarrow H^{*}(BT_{\text{\bf C}},\text{\bf Z})^{W}$ fails to be surjective for $G=\operatorname{Spin}(n)$ if $n\geq 11$ and $n\equiv 3,4,5\pmod{8}$ [2]. For simplicity, we work out the case of $\operatorname{Spin}(11)$. It would be interesting to make a full computation of the de Rham cohomology of $B\operatorname{Spin}(n)$ in characteristic 2.

Theorem 13.1**.**

[TABLE]

Proof.

Let $k=\text{\bf F}_{2}$. Let $n$ be an integer at least 6; eventually, we will restrict to the case $n=11$. Let $G$ be the split group $\operatorname{Spin}(n)$ over $k$, and let $T$ be a maximal torus in $G$. Let $r=\lfloor n/2\rfloor$. The Weyl group $W$ of $G$ is $S_{r}\ltimes(\text{\bf Z}/2)^{r}$ for $n=2r+1$, and the subgroup $S_{r}\ltimes(\text{\bf Z}/2)^{r-1}$ for $n=2r$. We start by computing the ring $O(\mathfrak{t})^{W}$ of $W$-invariant functions on the Lie algebra $\mathfrak{t}$ of $T$.

First consider the easier case where $n$ is odd, $n=2r+1$. The element $-1$ in $(\text{\bf Z}/2)^{r}\subset W$ acts as the identity on $\mathfrak{t}$, since we are in characteristic 2. The ring $O(\mathfrak{t})^{W}$ can also be viewed as $S(X^{*}(T)\otimes k)^{W}$. Computing this ring is similar to, but simpler than, Benson and Wood’s calculation of $S(X^{*}(T))^{W}=H^{*}(BT_{\text{\bf C}},\text{\bf Z})^{W}$ [2]. We follow their notation.

We have

[TABLE]

by thinking of $T$ as the double cover of a maximal torus in $SO(2r+1)$. The symmetric group $S_{r}$ in $W$ permutes $x_{1},\ldots,x_{r}$ and fixes $A$. The elementary abelian group $E_{r}=(\text{\bf Z}/2)^{r}$ in $W$, with generators $\epsilon_{1},\ldots,\epsilon_{r}$, acts by: $\epsilon_{i}$ changes the sign of $x_{i}$ and fixes $x_{j}$ for $j\neq i$, and $\epsilon_{i}(A)=A-x_{i}$. So

[TABLE]

Note that $-1:=\epsilon_{1}\cdots\epsilon_{r}$ in $W$ acts as the identity on $S^{*}(X^{*}(T)\otimes k)$.

We first compute the invariants of the subgroup $E_{r}$ on $S(X^{*}(T)\otimes k)$, using the following lemma.

Lemma 13.2**.**

Let $R$ be an $\text{\bf F}_{2}$-algebra which is a domain, $S$ the polynomial ring $R[x]$, and $a$ a nonzero element of $R$. Let $G=\text{\bf Z}/2$ act on $S$ by fixing $R$ and sending $x$ to $x+a$. Then the ring of invariants is

[TABLE]

where $u=x(x+a)$.

Proof.

Clearly $u=x(x+a)$ in $S$ is $G$-invariant. Since $u$ is a monic polynomial of degree 2 in $x$, we have $S=R[u]\oplus x\cdot R[u]$. Let $\sigma$ be the generator of $G=\text{\bf Z}/2$. Any element of $S$ can be written as $f+xg$ for some (unique) elements $f,g\in R[u]$. If $f+xg$ is $G$-invariant, then $0=\sigma(f+xg)-(f+xg)=(x+a)g-xg=ag$. Since $a$ is a non-zero-divisor in $R$, it is a non-zero-divisor in $R[u]$; so $g=0$. Thus $S^{G}=R[u]$. ∎

Let $E_{j}\cong(\text{\bf Z}/2)^{j}$ be the subgroup of $W$ generated by $\epsilon_{1},\ldots,\epsilon_{j}$. Let

[TABLE]

which is $E_{j}$-invariant. Here $\eta_{j}$ has degree $2^{j}$ in $S^{*}(X^{*}(T)\otimes k)$. By Lemma 13.2 (with $R=k[x_{1},\ldots,x_{r}]/(x_{1}+\cdots+x_{r})$) and induction on $j$, we have

[TABLE]

for $1\leq j\leq r-1$. Since $-1=\epsilon_{1}\cdots\epsilon_{r}$ acts as the identity on these rings, we also have

[TABLE]

The symmetric group $S_{r}$ permutes $x_{1},\ldots,x_{r}$, and it fixes $\eta_{r-1}$. Therefore, computing the invariants of the Weyl group on $S^{*}(X^{*}(T)\otimes k)$ reduces to computing the invariants of the symmetric group $S_{r}$ on $R=k[x_{1},\ldots,x_{r}]/(x_{1}+\cdots+x_{r})$. Write $c_{1},\ldots,c_{r}$ for the elementary symmetric polynomials in $k[x_{1},\ldots,x_{r}]$. For $r\geq 3$, the ring of invariants $R^{S_{r}}$ is equal to $k[c_{1},\ldots,c_{r}]/(c_{1})=k[c_{2},\ldots,c_{r}]$ [20, Proposition 4.1].

The answer is different for $r=2$: then $S_{2}$ acts trivially on $R=k[x_{1},x_{2}]/(x_{1}+x_{2})$, and so $R^{S_{2}}=R=k[x_{1}]$.

Combining these calculations with the earlier ones, we have found the invariants for the Weyl group $W$ of $G=\operatorname{Spin}(2r+1)$: for $r\geq 1$,

[TABLE]

Here $|c_{i}|=i$ for $2\leq i\leq r$, $|x_{1}|=2$, and $|\eta_{r-1}|=2^{r-1}$.

We now compute $S^{*}(X^{*}(T)\otimes k)^{W}$ for $G=\operatorname{Spin}(2r)$. Note that a maximal torus in $\operatorname{Spin}(2r)$ is also a maximal torus in $\operatorname{Spin}(2r+1)$. So we have again

[TABLE]

The Weyl group $W=S_{r}\ltimes(\text{\bf Z}/2)^{r-1}$ acts on this ring by: $S_{r}$ permutes $x_{1},\ldots,x_{r}$, and fixed $A$, and $(\text{\bf Z}/2)^{r-1}$ is the subgroup $\langle\epsilon_{1}\epsilon_{2},\ldots,\epsilon_{1}\epsilon_{r}\rangle$ in the notation above. Thus $\epsilon_{1}\epsilon_{j}$ fixes each $x_{j}$ (since we are working modulo 2) and sends $A$ to $A-x_{1}-x_{j}$.

For $1\leq j\leq r$, let $F_{j}$ be the subgroup $\langle\epsilon_{1}\epsilon_{2},\ldots,\epsilon_{1}\epsilon_{j}\rangle\cong(\text{\bf Z}/2)^{j-1}\subset W$. Let

[TABLE]

Then $|\mu_{j}|=2^{j-1}$ and $\mu_{1}=A$. Clearly $\mu_{j}$ is $F_{j}$-invariant. Benson and Wood observed (or one can check directly) that if $r$ is even and $r\geq 4$, then $\mu_{r-1}$ is in fact $W$-invariant, while if $r$ is odd and $r\geq 3$, then $\mu_{r}$ is $W$-invariant [2, Proposition 4.1].

For $1\leq j\leq r-1$, an induction on $j$ using Lemma 13.2 gives that

[TABLE]

If $r$ is even, then $-1:=\epsilon_{1}\cdots\epsilon_{r}$ is in $F_{r}\subset W$, and it acts trivially on $S^{*}(X^{*}(T)\otimes k)$. Therefore, for $r$ even, we have

[TABLE]

If $r$ is odd, then we can apply Lemma 13.2 one more time, yielding that

[TABLE]

The subgroup $S_{r}\subset W$ permutes $x_{1},\ldots,x_{r}$, and fixes $\mu_{r-1}$, resp. $\mu_{r}$. We showed above that

[TABLE]

Therefore, for $G=\operatorname{Spin}(2r)$, we have

[TABLE]

Here $|c_{i}|=i$ for $2\leq i\leq r$ and $|\mu_{r-1}|=2^{r-2}$, resp. $|\mu_{r}|=2^{r-1}$.

Thus we have determined $S^{*}(X^{*}(T)\otimes k)^{W}$ for $G=\operatorname{Spin}(n)$ for all $n$, even or odd. Now think of $G=\operatorname{Spin}(n)$ as a split reductive group over $k$. By Theorem 9.1, the ring $S^{*}(X^{*}(T)\otimes k)^{W}=O(\mathfrak{t})^{W}$ can be identified with $O(\mathfrak{g})^{G}$ for all $n\geq 6$. (The exceptional cases $\operatorname{Spin}(3),\operatorname{Spin}(4),\operatorname{Spin}(5)$ are the spin groups that have a factor isomorphic to a symplectic group: $\operatorname{Spin}(3)\cong Sp(2)$, $\operatorname{Spin}(4)\cong Sp(2)\times Sp(2)$, and $\operatorname{Spin}(5)\cong Sp(4)$.) We deduce that for $n\geq 6$,

[TABLE]

For $G=\operatorname{Spin}(n)$ and any $n\geq 6$, we have homomorphisms

[TABLE]

whose composition is the obvious inclusion. (The first homomorphism comes from the isomorphism of $O(\mathfrak{g})^{G}$ with $\oplus_{i}H^{i}(BG_{k},\Omega^{i})$, using that $H^{i}(BG_{k},\Omega^{j})=0$ for $i<j$.) In this case, the restriction $O(\mathfrak{g})^{G}\rightarrow O(\mathfrak{t})^{W}$ is a bijection. So $H^{*}_{\operatorname{dR}}(BG/k)$ contains the ring computed above (with degrees multiplied by 2), and retracts onto it. It follows that for all $n\geq 6$, $H^{*}_{\operatorname{dR}}(BG/k)$ has an indecomposable generator in degree $2^{r}$ if $n=2r+1$, in degree $2^{r-1}$ if $n=2r$ and $r$ is even, and in degree $2^{r}$ if $n=2r$ and $r$ is odd. (For this argument, we do not need to find all the indecomposable generators of $H^{*}_{\operatorname{dR}}(BG/k)$.)

Compare this with Quillen’s calculation of the cohomology of the classifying space of the complex reductive group $\operatorname{Spin}(n)_{\text{\bf C}}$, or equivalently of the compact Lie group $\operatorname{Spin}(n)$ [21, Theorem 6.5]:

[TABLE]

Here $\Delta_{\theta}$ is a faithful orthogonal representation of $\operatorname{Spin}(n)_{\text{\bf C}}$ of minimal dimension, and $J$ is the ideal generated by the regular sequence

[TABLE]

in the polynomial ring $H^{*}(BSO(n)_{\text{\bf C}},k)=k[w_{2},w_{3},\ldots,w_{n}]$, where $|w_{i}|=i$. Finally, the number $h$ is given by the following table:

[TABLE]

The Steenrod operations on the mod 2 cohomology of $BSO(n)_{\text{\bf C}}$, as used in the formula above, are known, by Wu’s formula [19, Theorem III.5.12]:

[TABLE]

for $0\leq i\leq j$, where by convention $\binom{-1}{0}=1$.

Write $r=\lfloor n/2\rfloor$. If $n=2r+1$, then the generator $w_{2^{h}}(\Delta_{\theta})$ is in degree $2^{r}$ if $r\equiv 0,3\pmod{4}$ and in degree $2^{r+1}$ if $r\equiv 1,2\pmod{4}$. If $n=2r$, then the generator $w_{2^{h}}(\Delta_{\theta})$ is in degree $2^{r-1}$ if $r\equiv 0\pmod{4}$ and in degree $2^{r}$ if $r\equiv 1,2,3\pmod{4}$. Therefore, for $n\geq 11$, $H^{*}(B\operatorname{Spin}(n)_{\text{\bf C}},k)$ has no indecomposable generator in degree $2^{r}$ if $n\equiv 3,5\pmod{8}$, and no indecomposable generator in degree $2^{r-1}$ if $n\equiv 4\pmod{8}$. But $H^{*}_{\operatorname{dR}}(BG/k)$ does have an indecomposable generator in the indicated degree $2^{a}$, as shown above. Thus, for $G=\operatorname{Spin}(n)$, $H^{*}(BG_{\text{\bf C}},k)$ is not isomorphic to $H^{*}_{\operatorname{dR}}(BG/k)$ as a graded ring when $n\geq 11$ and $n\equiv 3,4,5\pmod{8}$.

We want to show, more precisely, that for $n=11$, $H^{32}_{\operatorname{dR}}(BG/k)$ is bigger than $H^{32}(BG_{\text{\bf C}},k)$. We know the cohomology of $BG_{\text{\bf C}}$ by Quillen (above), and so it remains to give a lower bound for the de Rham cohomology of $BG$ over $k$.

We do this by restricting to a suitable abelian $k$-subgroup scheme of $G=\operatorname{Spin}(n)$. Assume that $n\not\equiv 2\pmod{4}$; this includes the case $\operatorname{Spin}(11)$ that we are aiming for. Then the Weyl group $W$ of $\operatorname{Spin}(n)$ contains $-1$. So $\operatorname{Spin}(n)$ contains an extension of $\text{\bf Z}/2$ by a split maximal torus $T\cong(G_{m})^{r}$, where $\text{\bf Z}/2$ acts by inversion on $T$. Let $L$ be the subgroup of the form $1\rightarrow T[2]\rightarrow L\rightarrow\text{\bf Z}/2\rightarrow 1$; then $L$ is abelian (because inversion is the identity on $T[2]\cong(\mu_{2})^{r}$). Since the field $k=\text{\bf F}_{2}$ is perfect, the reduced locus of $L$ is a $k$-subgroup scheme (isomorphic to $\text{\bf Z}/2$) [18, Corollary 1.39], and so the extension splits. That is, $L\cong(\mu_{2})^{r}\times\text{\bf Z}/2$.

Let us compute the pullbacks of the generators $u_{i}$ of $H^{*}_{\operatorname{dR}}(BSO(n)/k)$ (Theorem 12.1) to the subgroup $L$ of $G=\operatorname{Spin}(n)$. It suffices to compute the restrictions of the classes $u_{i}$ to the image $K$ of $L$ in $SO(n)$; clearly $K\cong(\mu_{2})^{r-1}\times\text{\bf Z}/2$. In notation similar to that used earlier in this proof, the ring of polynomial functions on the Lie algebra of the subgroup $(\mu_{2})^{r-1}$ here is

[TABLE]

This ring can be viewed as the Hodge cohomology ring of $B(\mu_{2})^{r-1}$ modulo its radical, with the generators $t_{i}$ in $H^{1}(B(\mu_{2})^{r-1},\Omega^{1})$ (by Propositions 11.1 and 5.1). Using Lemma 11.2, we conclude that

[TABLE]

where $s$ is pulled back from the generator of $H^{1}(B(\text{\bf Z}/2),O)$. The Hodge spectral sequence for $BK$ degenerates at $E_{1}$, since we know this degeneration for $B\text{\bf Z}/2$ and $B(\mu_{2})^{r-1}$. Therefore,

[TABLE]

Note that the surjection $L\rightarrow K$ is split. So if we compute that an element of $H^{*}_{\operatorname{dR}}(BSO(n)/k)$ has nonzero restriction to $K$, then it has nonzero restriction to $L$, hence a fortiori to $G=\operatorname{Spin}(n)$.

Now strengthen the assumption $n\not\equiv 2\pmod{4}$ to assume that $n$ is odd and $n\geq 7$. In the proof of Theorem 12.1, we computed the restriction of $u_{2},u_{3},\ldots,u_{2r+1}$ from $SO(2r+1)$ to its subgroup $O(2)^{r}$, and hence to its subgroup $(\mu_{2})^{r}\times(\text{\bf Z}/2)^{r}$. (We worked there in Hodge cohomology, but the formulas remain true in de Rham cohomology.) We now want to restrict to the smaller subgroup $K=(\mu_{2})^{r-1}\times\text{\bf Z}/2$. This last step sends $H^{*}_{\operatorname{dR}}(B((\mu_{2})^{r}\times(\text{\bf Z}/2)^{r})/k)/\text{rad}=k[s_{1},\ldots,s_{r},t_{1},\ldots,t_{r}]$ to $H^{*}_{\operatorname{dR}}(BK/k)/\text{rad}=k[s,t_{1},\ldots,t_{r}]/(t_{1}+\cdots+t_{r})$ by $s_{i}\mapsto s$ for all $i$ and $t_{i}\mapsto t_{i}$. By the formulas from the proof of Theorem 12.1, the element $u_{2a}$ (for $1\leq a\leq r$) restricts to the elementary symmetric polynomial

[TABLE]

Thus $u_{2}$ restricts to 0 on $K$, but $u_{4},u_{6},\ldots,u_{2r}$ restrict to generators of the polynomial ring

[TABLE]

using that $r\geq 3$, as discussed earlier in this section.

Next, using notation from the proof of Theorem 12.1, for $1\leq a\leq r$, the restriction of $u_{2a+1}$ to $H^{*}_{\operatorname{dR}}(BK/k)/\text{rad}$ is (first restricting from $SO(2r+1)$ to its subgroup $(\mu_{2})^{r}\times(\text{\bf Z}/2)^{r}$, and then to $K=(\mu_{2})^{r-1}\times\text{\bf Z}/2$):

[TABLE]

Thus, for all $1\leq a\leq r$, $u_{2a+1}$ restricts in $H^{*}_{\operatorname{dR}}(BK/k)/\text{rad}$ to $su_{2a}$ if $a$ is odd, and otherwise to zero. (But $u_{2}$ restricts to 0, and so this also means that $u_{3}$ restricts to 0.)

This gives a lower bound for the image of $H^{*}_{\operatorname{dR}}(BSO(n)/k)\rightarrow H^{*}_{\operatorname{dR}}(B\operatorname{Spin}(n)/k)$ for $n$ odd. In particular, for $n=11$, this image has Hilbert series at least that of the ring

[TABLE]

since the latter ring is isomorphic to the image of restriction from $SO(11)$ to $H^{*}_{\operatorname{dR}}(BL/k)/\text{rad}$, where $L\subset\operatorname{Spin}(11)$.

We now compare this to Quillen’s computation (above) in the case of $\operatorname{Spin}(11)$:

[TABLE]

Since the last generator $w_{64}(\Delta_{\theta})$ is in degree 64 and the last relation is in degree 33, the degree-32 component of this ring has the same dimension as the degree-32 component of the lower bound above for $H^{*}_{\operatorname{dR}}(B\operatorname{Spin}(11)/k)$. However, earlier in this section, we showed that $H^{*}_{\operatorname{dR}}(B\operatorname{Spin}(11)/k)$ has an extra generator $\mu_{5}$ in degree 32. This is linearly independent of the image of restriction from $SO(11)$, as we see by restricting to a maximal torus $T$ in $\operatorname{Spin}(11)$. Indeed, we showed earlier in this section that the image of $H^{*}_{\operatorname{dR}}(B\operatorname{Spin}(11)/k)\rightarrow H^{*}_{\operatorname{dR}}(BT/k)$ is the polynomial ring $k[c_{2},\ldots,c_{5},\mu_{5}]$, whereas the image of the pullback from $SO(11)$ to $T\subset\operatorname{Spin}(11)$ is just $k[c_{2},\ldots,c_{5}]$ ($=k[w_{4},w_{6},w_{8},w_{10}]$). Thus we have shown that

[TABLE]

∎