The affine part of the Picard scheme

TL;DR

This paper characterizes the structure of the Picard scheme by describing its maximal torus and unipotent subgroup for proper schemes over perfect fields, enhancing understanding of its algebraic group components.

Contribution

It provides a detailed description of the affine parts of the Picard scheme, specifically the maximal torus and unipotent subgroup, which was not previously explicitly characterized.

Findings

Identifies the maximal torus of the Picard scheme.

Describes the maximal unipotent subgroup of the Picard scheme.

Clarifies the structure of the Picard scheme over perfect fields.

Abstract

We describe the maximal torus and maximal unipotent subgroup of the Picard variety of a proper scheme over a perfect field.

Equations

$$0\to L_X\to {\mathrm{Pic}}_X^{0,\text{red}}\to A_X\to 0.$$

$$0\to K_X\to {\mathrm{Pic}}_X^{0,\text{red}}\to {\mathrm{Pic}}_{X^{+}}^{0,\text{red}}\to 0,$$

$$U_X\subseteq K_X\subseteq p_{\ast }({\mathbb{G}}_{m,X^{+}}/{\mathbb{G}}_{m,X})$$

$$T\mapsto \{{\mathcal{O}}_{X\times T}\text{-line bundles }\mathcal{L}\subseteq {\mathcal{O}}_{X^{+}\times T}\text{ which are invertible in }{\mathcal{O}}_{X^{+}\times T}\}.$$

$$p_{\ast }{\mathbb{G}}_{m,X}(T):=\Gamma (X\times T,{\mathcal{O}}_{X\times T}{)}^{\times }=(\Gamma (X,{\mathcal{O}}_X)\otimes \Gamma (T,{\mathcal{O}}_T){)}^{\times },$$

$$H_{\text{et}}^1(X,\mathbb{Z})\cong H_{\text{fl}}^1(X,\mathbb{Z})\cong {\mathrm{Ext}}_X^1({\mathbb{G}}_{m,X},{\mathbb{G}}_{m,X}).$$

$$E_2^{s,t}=H_{\text{fl}}^s(X,{\mathrm{Ext}}_X^t({\mathbb{G}}_{m,X},{\mathbb{G}}_{mX}))\Rightarrow {\mathrm{Ext}}_X^{s+t}({\mathbb{G}}_{m,X},{\mathbb{G}}_{m,X}).$$

$$E_2^{s,t}={\mathrm{Ext}}_k^s({\mathbb{G}}_{m,k},R^t{p}_{\ast }{\mathbb{G}}_{m,X})\Rightarrow {\mathrm{Ext}}_X^{s+t}({\mathbb{G}}_{m,X},{\mathbb{G}}_{m,X}).$$

$$0\to {\mathrm{Ext}}_k^1({\mathbb{G}}_{m,k},p_{\ast }{\mathbb{G}}_{mX})\to {\mathrm{Ext}}_X^1({\mathbb{G}}_{m,X},{\mathbb{G}}_{m,X})\to {\mathrm{Hom}}_k({\mathbb{G}}_{m,k},R^1{p}_{\ast }{\mathbb{G}}_{m,X})\stackrel{{\delta }_X}{⟶}{\mathrm{Ext}}_k^2({\mathbb{G}}_{m,k},p_{\ast }{\mathbb{G}}_{m,X}).$$

$$\begin{array}{ccc}{\mathrm{Hom}}_k({\mathbb{G}}_{m,k},R^1{p}_{\ast }{\mathbb{G}}_{m,X})& \to & {\mathrm{Hom}}_k({\mathbb{G}}_{m,k},R^1{p}_{\ast }{i}_{\ast }{\mathbb{G}}_{m,Z})=0\\ {\delta }_X\downarrow & & {\delta }_Z\downarrow \\ {\mathrm{Ext}}_k^2({\mathbb{G}}_{m,k},p_{\ast }{\mathbb{G}}_{m,X})& \stackrel{\sim }{\to }& {\mathrm{Ext}}_k^2({\mathbb{G}}_{m,k},{\mathbb{G}}_m).\end{array}$$

$$H_{\text{et}}^1(\bar{X},\mathbb{Z})\otimes \mathbb{Q}/\mathbb{Z}$$

$$\mathrm{Div}({}_{\text{tor}}{H}_{\text{et}}^2(\bar{X},\mathbb{Z}))$$

$${}_{\text{tor}}{H}_{\text{et}}^2(\bar{X},\mathbb{Z})/\mathrm{Div}$$

$$\begin{CD}\operatornamewithlimits{colim}\operatorname{Hom}_{\bar{k}}(\mu_{m},T_{X})@=H^{1}_{\text{\rm et}}(\bar{X},{{\mathbb{Z}}})\otimes{{\mathbb{Q}}}/{{\mathbb{Z}}}\\ @V{}V{}V@V{}V{}V\\ \operatornamewithlimits{colim}\operatorname{Hom}_{\bar{k}}(\mu_{m},\operatorname{Pic}^{0,{\text{\rm red}}}_{X})@>{}>{}>\operatornamewithlimits{colim}\operatorname{Hom}_{\bar{k}}(\mu_{m},\operatorname{Pic}_{X})@>{}>{}>\operatornamewithlimits{colim}\operatorname{Hom}_{\bar{k}}(\mu_{m},NS_{X})\\ @V{}V{}V@V{}V{}V\Big{\|}\\ \operatornamewithlimits{colim}\operatorname{Hom}_{\bar{k}}(\mu_{m},A_{X})@>{f}>{}>{}_{\text{\rm tor}}H^{2}_{\text{\rm et}}(\bar{X},{{\mathbb{Z}}})@>{}>{}>\operatorname{coker}f.\end{CD}$$

$$0\to {}_{\text{tor}}{A}_X^t(k)\to C{H}_0(X,1,\mathbb{Q}/\mathbb{Z})\to \chi (T_X)\otimes \mathbb{Q}/\mathbb{Z}\to 0,$$

$$0\to C{H}_0(X,1)\otimes \mathbb{Q}/\mathbb{Z}\to C{H}_0(X,1,\mathbb{Q}/\mathbb{Z})\to {}_{\text{tor}}C{H}_0(X)\to 0$$

$${\dim }_k{H}^1(X,{\mathcal{O}}_X)=\dim U_X+{\dim }_{k}Lie(N{S}_X^0)+\mathrm{rank}{H}_{\text{et}}^1(X,\mathbb{Z})+\frac{1}{2}{\mathrm{corank}}_l{H}_{\text{et}}^1(\bar{X},{\mathbb{Q}}_l/{\mathbb{Z}}_l).$$

$$\mathrm{Pic}(X[t,t^{-1}])\cong \mathrm{Pic}(X)\oplus N\mathrm{Pic}(X)\oplus N\mathrm{Pic}(X)\oplus H_{\text{et}}^1(X,\mathbb{Z}).$$

$$\begin{CD}0@>{}>{}>H^{1}_{\text{\rm et}}({\mathbb{A}}_{k}^{1},p_{*}^{\prime}{\mathbb{G}}_{m})@>{}>{}>\operatorname{Pic}(X\times{\mathbb{A}}^{1}_{k})@>{}>{}>\operatorname{Pic}_{X}({\mathbb{A}}^{1}_{k})@>{}>{}>H^{2}_{\text{\rm et}}({\mathbb{A}}_{k}^{1},p^{\prime}_{*}{\mathbb{G}}_{m})\\ @I II@V{}V{}V@V{}V{}V@V{}V{}V@V{}V{}V\\ 0@>{}>{}>H^{1}_{\text{\rm et}}(k,p_{*}{\mathbb{G}}_{m})@>{}>{}>\operatorname{Pic}(X)@>{}>{}>\operatorname{Pic}_{X}(k)@>{}>{}>H^{2}_{\text{\rm et}}(k,p_{*}{\mathbb{G}}_{m}),\end{CD}$$

$$H_{\text{et}}^i({\mathbb{A}}_k^1,p_{\ast }^{\prime }{\mathbb{G}}_m)\cong H_{\text{et}}^i({\mathbb{A}}_L^1,g_{\ast }^{\prime }{\mathbb{G}}_m)\cong H_{\text{et}}^i({\mathbb{A}}_L^1,{\mathbb{G}}_m),$$

$$\sum _j{a}_j(nt{)}^j=n{f}^{\ast }(x_i)=f^{\ast }(n{x}_i)=n\sum _j{a}_j{t}^j$$

$$U_X=\ker ({\mathrm{Pic}}_X^{0,\text{red}}\to {\mathrm{Pic}}_{X^{+}}^{0,\text{red}}).$$

$$\begin{gathered} 0\to p_{\ast }{\mathbb{G}}_{m,X}\to p_{\ast }{\mathbb{G}}_{m,X^{+}}\to p_{\ast }({\mathbb{G}}_{m,X^{+}}/{\mathbb{G}}_{m,X}) \\ \to {\mathrm{Pic}}_X\to {\mathrm{Pic}}_{X^{+}} \end{gathered}$$

$$\begin{array}{ccccccccc}0& \to & K_X& \to & {\mathrm{Pic}}_X^{0,\text{red}}& \to & {\mathrm{Pic}}_{X^{+}}^{0,\text{red}}& \to & 0\\ u\downarrow & & v\downarrow & & \downarrow & & & & \\ 0& \to & p_{\ast }({\mathbb{G}}_{m,X^{+}}/{\mathbb{G}}_{m,X})& \to & {\mathrm{Pic}}_X& \to & {\mathrm{Pic}}_{X^{+}}& & \\ \downarrow & & \downarrow & & \downarrow & & & & \\ 0& \to & \mathrm{coker}u& \to & {\mathrm{NS}}_X& \to & {\mathrm{NS}}_{X^{+}}.& & \end{array}$$

$$\begin{CD}0@>{}>{}>H^{1}_{\text{\rm et}}(T,q_{*}{\mathbb{G}}_{m,X\times T})@>{}>{}>\operatorname{Pic}(X\times T)@>{}>{}>\operatorname{Pic}_{X/k}(T)@>{}>{}>H^{2}_{\text{\rm et}}(T,q_{*}{\mathbb{G}}_{m,X\times T})\\ @I II@V{}V{}V@V{r}V{}V@V{s}V{}V@V{}V{}V\\ 0@>{}>{}>H^{1}_{\text{\rm et}}(T,q_{*}{\mathbb{G}}_{m,X^{+}\times T})@>{}>{}>\operatorname{Pic}(X^{+}\times T)@>{}>{}>\operatorname{Pic}_{X^{+}/k}(T)@>{}>{}>H^{2}_{\text{\rm et}}(T,q_{*}{\mathbb{G}}_{m,X^{+}\times T}).\end{CD}$$

$$f:\{{\mathcal{O}}_Y\text{-line bundles }\mathcal{L}\subseteq {\mathcal{O}}_{Y^{\prime }}\text{ which are invertible in }{\mathcal{O}}_{Y^{\prime }}\}\to \mathrm{Pic}(Y).$$

Full text

Supported in part by NSF grant No.0556263

1. Introduction

For a proper scheme $p:X\to k$ over a perfect field, the Picard scheme $\operatorname{Pic}_{X}$ representing the functor $T\mapsto H^{0}(T_{\text{\rm et}},R^{1}p_{*}{\mathbb{G}}_{m})$ exists, and its connected component $\operatorname{Pic}^{0}_{X}$ is separated and of finite type [Mu64, II 15]. By Chevalley’s structure theorem [Chev60], the reduced connected component $\operatorname{Pic}_{X}^{0,{\text{\rm red}}}$ is an extension of an abelian variety $A_{X}$ by a linear algebraic group $L_{X}$:

[TABLE]

The commutative, smooth affine group scheme $L_{X}$ is the direct product of a torus $T_{X}$ and a unipotent group $U_{X}$. The following theorem completely characterizes $T_{X}$:

Theorem 1**.**

If $X$ is proper over a perfect field, then the cocharactermodule $\operatorname{Hom}_{\bar{k}}({\mathbb{G}}_{m},T_{X})$ of the maximal torus of $\operatorname{Pic}_{X}$ is isomorphic to $H^{1}_{\text{\rm et}}(\bar{X},{{\mathbb{Z}}})$ as a Galois-module.

To analyze the unipotent part, we let $\operatorname{Pic}(X[t])_{[1]}$ be the typical part, i.e. the subgroup of elements $x$ of $\operatorname{Pic}(X[t])$ such that the map $X[t]\to X[t]$, $t\mapsto nt$ sends $x$ to $nx$.

Theorem 2**.**

Let $X$ be proper over a perfect field. Then $\operatorname{Pic}(X[t])_{[1]}$ is isomorphic to the group of morphisms of schemes $f:{\mathbb{G}}_{a}\to U_{X}$ satisfying $f(nx)=nf(x)$ for every $n\in{{\mathbb{Z}}}$. In particular, $\operatorname{Hom}_{k}({\mathbb{G}}_{a},U_{X})\subseteq\operatorname{Pic}(X[t])_{[1]}$, and this is an equality in characteristic [math].

To get another description of $U_{X}$, we assume that $X$ is reduced (the map on the Picard scheme induced by the map $X^{\text{\rm red}}\to X$ is well understood by the work of Oort [Oort62]). The semi-normalization $X^{+}\to X$ is the largest scheme between $X$ and its normalization which is strongly universally homeomorphic to $X$ in the sense that the map $X^{+}\to X$ induces an isomorphism on all residue fields. A Theorem of Traverso [Tra70] implies that $\operatorname{Pic}(X[t])_{[1]}$, hence $U_{X}$, vanishes if $X$ is reduced and seminormal. We use this to show

Theorem 3**.**

Let $X$ be reduced and proper over a perfect field.

a) We have a short exact sequence

[TABLE]

and inclusions of unipotent group schemes

[TABLE]

with quotients finite $p$-primary group schemes.

b) The group scheme $p_{*}({\mathbb{G}}_{m,X^{+}}/{\mathbb{G}}_{m,X})$ represents the functor

[TABLE]

Notation: For a field $k$, we denote by $\bar{k}$ its algebraic closure, and for a scheme $X$ over $k$ we let $\bar{X}=X\times_{k}\bar{k}$. Unless specified otherwise, all extension and homomorphism groups are considered on the fpqc site. 111In [Gei09] we used the étale topology

Acknowledgements:

This (original) paper was written while the author was visiting T. Saito at the University of Tokyo, whom we thank for his hospitality. We are indebted to G. Faltings for pointing out a mistake in a previous version, and the referee, whose comments helped to improve the exposition and to give more concise proofs. O. Gabber pointed out mistakes in the original version and suggested improvements.

2. The torus

Proposition 4**.**

If $p:X\to k$ is reduced, geometrically connected, and proper over a perfect field, then ${\mathbb{G}}_{m,k}\to p_{*}{\mathbb{G}}_{m,X}$ is an isomorphism. Moreover, if $f:X^{\prime}\to X$ is a universal homeomorphism and $X^{\prime}$ is reduced as well, then $f$ induces an isomorphism $p_{*}{\mathbb{G}}_{m,X}\cong p_{*}{\mathbb{G}}_{m,X^{\prime}}$. 222This replaces [Gei09, Prop. 9 a)] which is incorrect as stated because the induction step in the proof does not preserve the hypothesis on reducedness.

*Proof. *Since any scheme $T$ over $k$ is flat, we have by flat base change $R^{j}q_{*}{\mathcal{O}}_{X_{T}}=H^{j}(X,{\mathcal{O}}_{X})\otimes_{k}{\mathcal{O}}_{T}$, where $q:X_{T}\to T$ is the projection. In particular,

[TABLE]

and it suffices to show that $\Gamma(X,{\mathcal{O}}_{X})\cong\Gamma(X^{\prime},{\mathcal{O}}_{X^{\prime}})\cong k$. Since $\Gamma(\bar{X},{\mathcal{O}}_{\bar{X}})^{Gal(\bar{k}/k)}=\Gamma(X,{\mathcal{O}}_{X})$, we can assume that $k$ is algebraically closed and that $X$ is connected, in which case the statement follows because $X$ and $X^{\prime}$ are reduced, proper, connected, and have a $k$-rational point. $\Box$

Lemma 5**.**

For any scheme $X$ we have isomorphisms

[TABLE]

*Proof. *The first isomorphism is [Mi80, III Rem. 3.11(b)]. To prove the second isomorphism, we note that $\operatorname{{\mathcal{H}{\mathfrak{om}}}}_{X}({\mathbb{G}}_{m,X},{\mathbb{G}}_{m,X})\cong{{\mathbb{Z}}}_{X}$ by [SGA3, VIII Cor. 1.5], and that $\operatorname{\mathcal{E}{\mathfrak{x}t}}^{1}_{X}({\mathbb{G}}_{m,X},{\mathbb{G}}_{m,X})$ is isomorphic to the group of extensions of group schemes [Oort66, Cor. 17.5], which vanishes by [SGA7, VIII Prop. 3.3.1].333This was claimed without proof in [Gei09]. Hence we obtain the isomorphism from the spectral sequence [Mi80, III Thm.1.22]

[TABLE]

$\Box$

*Proof. *(Theorem 1) Since the maps defined below are natural, we can assume that $k$ is algebraically closed and $X$ is connected. We can also assume that $X$ is reduced, because $H^{1}_{\text{\rm et}}(X,{{\mathbb{Z}}})\xrightarrow{\sim}H^{1}_{\text{\rm et}}(X^{\text{\rm red}},{{\mathbb{Z}}})$, and the map $\operatorname{Pic}_{X}\to\operatorname{Pic}_{X^{\text{\rm red}}}$ has unipotent kernel and cokernel [Oort62, Cor. page 9]. It suffices to calculate $\operatorname{Hom}_{k}({\mathbb{G}}_{m,k},\operatorname{Pic}_{X})$, because there are no homomorphisms from ${\mathbb{G}}_{m}$ to commutative group schemes other than tori [Oort66, p. 81]. By Yoneda’s Lemma, the latter group is isomorphic to the group of homomorphisms of sheaves on the fpqc site $\operatorname{Hom}_{k}({\mathbb{G}}_{m,k},R^{1}p_{*}{\mathbb{G}}_{m,X})$. The Leray spectral sequence

[TABLE]

gives an exact sequence

[TABLE]

By Proposition 4 the left term agrees with $\operatorname{Ext}^{1}_{k}({\mathbb{G}}_{m,k},{\mathbb{G}}_{m,k})$, and this vanishes by [Oort66, Cor. 17.5]. Thus it suffices to show that $\delta_{X}$ is the zero map 444The remainder of the proof is a simplification suggested by O. Gabber.. Choose a closed point of $X$ and let $i:Z\to X$ be the corresponding closed subscheme. Since $p_{*}\circ i_{*}=\operatorname{id}$ we have $R^{s}p_{*}i_{*}=R^{s}(p\circ i)_{*}=0$ for $s>0$. Hence we obtain a diagram

[TABLE]

By Proposition 4, the lower horizontal map is an isomorphism. $\Box$

Remark. The example in [Gei06, Prop. 8.2] shows that the map $H^{i}_{\text{\rm et}}(\bar{X},{{\mathbb{Z}}})\to\operatorname{Ext}^{i}_{\bar{X}}({\mathbb{G}}_{m},{\mathbb{G}}_{m})$ is not an isomorphism for $i\geq 2$. One can ask if it is an isomorphism if one replaces $H^{i}_{\text{\rm et}}(\bar{X},{{\mathbb{Z}}})$ by the eh-cohomology group $H^{i}_{eh}(\bar{X},{{\mathbb{Z}}})$ of [Gei06].

Example. If $X$ is the node over an algebraically closed field, then $H^{1}_{\text{\rm et}}(X,{{\mathbb{Z}}})\cong{{\mathbb{Z}}}$, and $T_{X}\cong{\mathbb{G}}_{m}$. Let $X$ be a node with non-rational tangent slopes at the singular point. Base changing to the algebraic closure, one sees that $H^{1}_{\text{\rm et}}(\bar{X},{{\mathbb{Z}}})\cong{{\mathbb{Z}}}$, with Galois group acting as multiplication by $-1$, hence $T_{X}$ is an anisotropic torus.

Using the theorem, we are able to recover the torsion of $T_{X}$, $A_{X}$ and the diagonalizable part of $NS_{X}:=\operatorname{Pic}_{X}/\operatorname{Pic}_{X}^{0,{\text{\rm red}}}$ in terms of etale cohomology:

Corollary 6**.**

Let $X$ be proper over a perfect field $k$. Then we have canonical isomorphisms

[TABLE]

*Proof. *Taking the colimit of the isomorphism $H^{1}_{\text{\rm et}}(\bar{X},{{\mathbb{Z}}}/m)\cong\operatorname{Hom}_{\bar{k}}(\mu_{m},\operatorname{Pic}_{X})$ of [Mi80, Prop.4.16] or [Ray70, §6.2], we obtain $H^{1}_{\text{\rm et}}(\bar{X},{{\mathbb{Q}}}/{{\mathbb{Z}}})\cong\operatornamewithlimits{colim}\operatorname{Hom}_{\bar{k}}(\mu_{m},\operatorname{Pic}_{X})$. Since $\operatorname{Ext}^{1}_{\bar{k}}({\mathbb{G}}_{m},T_{X})=0$, Theorem 1 implies that $\operatorname{Hom}_{\bar{k}}(\mu_{m},T_{X})\cong\operatorname{Hom}_{\bar{k}}({\mathbb{G}}_{m},T_{X})/m\cong H^{1}_{\text{\rm et}}(\bar{X},{{\mathbb{Z}}})/m$. Consider the commutative diagram:

[TABLE]

The middle column is the short exact coefficient sequence. The left column and middle row are short exact because $\operatorname{Ext}^{1}_{\bar{k}}(\mu_{m},T_{X})=\operatorname{Ext}^{1}_{\bar{k}}(\mu_{m},\operatorname{Pic}^{0,{\text{\rm red}}}_{X})=0$ by [Oort66, Cor. 17.5, II 14.2]. A diagram chase shows that $f$ is injective, and the right vertical map is an isomorphism. The Corollary follows because $\operatornamewithlimits{colim}\operatorname{Hom}_{\bar{k}}(\mu_{m},A_{X})$ is divisible and $\operatornamewithlimits{colim}\operatorname{Hom}_{\bar{k}}(\mu_{m},NS_{X})$ is finite. $\Box$

The above result should be compared to [Gei10, Prop.6.2], where we show that, for every proper scheme over an algebraically closed field, the higher Chow group of zero-cycles $CH_{0}(X,1,{{\mathbb{Z}}}/m)$ is the Pontrjagin dual of $H^{1}_{\text{\rm et}}(X,{{\mathbb{Z}}}/m)$. This implies a short exact sequence

[TABLE]

for $A_{X}^{t}$ the dual abelian variety of $A_{X}$, and $\chi(T_{X})$ the character module of $T_{X}$. However, in this case the contribution from the torus and from the abelian variety are not compatible with the coefficient sequence

[TABLE]

as in Corollary 6.

Looking at tangent spaces, the previous Corollary gives a dimension formula:

Corollary 7**.**

Let $l$ be a prime different from $\operatorname{char}k$. Then

[TABLE]

3. The unipotent part

Let N\operatorname{Pic}(X):=\ker\big{(}\operatorname{Pic}(X[t])\xrightarrow{0^{*}}\operatorname{Pic}(X)\big{)}. Since $t\mapsto 0t$ induces $x\mapsto 0x$ on the typical part, $\operatorname{Pic}(X[t])_{[1]}$ is a subgroup of $N\operatorname{Pic}(X)$. In [Wei91], Weibel shows that for every scheme there is a direct sum decomposition

[TABLE]

*Proof. *(Theorem 2). We show first that N\operatorname{Pic}(X)=\ker\big{(}U_{X}({\mathbb{A}}^{1})\to U_{X}(k)\big{)}. Since there are no non-trivial morphisms of schemes from ${\mathbb{A}}^{1}_{k}$ to an abelian variety, a torus, an infinitesimal group, or a discrete group, we see that the kernel of $U_{X}({\mathbb{A}}^{1}_{k})\to U_{X}(k)$ agrees with the kernel of $\operatorname{Pic}_{X}({\mathbb{A}}^{1}_{k})\to\operatorname{Pic}_{X}(k)$. Let $p:X\to k$ and $p^{\prime}:X\times{\mathbb{A}}^{1}_{k}\to{\mathbb{A}}^{1}_{k}$ be the structure morphisms. Then the Leray spectral sequence gives a commutative diagram

[TABLE]

and it suffices to show that the outer vertical maps are isomorphisms. Let $X\xrightarrow{g}L\to k$ be the Stein factorization of $p$, such that ${\mathcal{O}}_{L}\cong g_{*}{\mathcal{O}}_{X}$ and $L$ is the spectrum of an Artinian $k$-algebra. Since ${\mathbb{A}}^{1}_{k}\to k$ is flat, $p^{\prime}_{*}{\mathcal{O}}_{X\times{\mathbb{A}}^{1}_{k}}={\mathcal{O}}_{{\mathbb{A}}^{1}_{k}}\otimes_{k}p_{*}{\mathcal{O}}_{X}$, and $X\times{\mathbb{A}}^{1}_{k}\xrightarrow{g^{\prime}}{\mathbb{A}}^{1}_{L}\ \xrightarrow{}{\mathbb{A}}^{1}_{k}$ is the Stein factorization of $p^{\prime}$. We obtain

[TABLE]

and $H^{i}_{\text{\rm et}}(k,p_{*}{\mathbb{G}}_{m})\cong H^{i}_{\text{\rm et}}(L,{\mathbb{G}}_{m})$. Hence the terms on the left vanish because $\operatorname{Pic}(L)=\operatorname{Pic}({\mathbb{A}}_{L}^{1})=0$. To show that $H^{2}_{\text{\rm et}}({\mathbb{A}}_{L}^{1},{\mathbb{G}}_{m})\to H^{2}_{\text{\rm et}}(L,{\mathbb{G}}_{m})$ is an isomorphism, we can assume that $L$ is a local Artinian $k$-algebra with (perfect) residue field $k^{\prime}$. By [Mi80, III Rem.3.11] we are reduced to showing that $H^{2}_{\text{\rm et}}({\mathbb{A}}_{k^{\prime}}^{1},{\mathbb{G}}_{m})\to H^{2}_{\text{\rm et}}(k^{\prime},{\mathbb{G}}_{m})$ is an isomorphism, and this can be found in [Mi80, IV Ex.2.20].

Given an element $x$ of $N\operatorname{Pic}(X)$, the condition $x\in\operatorname{Pic}(X[t])_{[1]}$ implies that the corresponding $f\in\operatorname{Hom}_{Sch}({\mathbb{A}}^{1},U_{X})$ satisfies $f(nx)=nf(x)$ for all $n$. If $k$ has characteristic [math], then $U_{X}\cong{\mathbb{G}}_{a}^{r}$ for some $r$, and the map $f:{\mathbb{G}}_{a}\to U_{X}$ corresponds to a morphism of Hopf algebras $f^{*}:k[x_{1},\cdots,x_{r}]\to k[t]$. If $f^{*}(x_{i})=\sum_{j}a_{j}t^{j}$, then

[TABLE]

only if $n^{j}=n$ for all $n$, hence $j=1$. $\Box$

**Example. **If $k$ has characteristic $p$, then $t\mapsto t^{2p-1}$ induces a map ${\mathbb{G}}_{a}\to{\mathbb{G}}_{a}$ which is compatible with multiplication by $n$, but not a homomorphism of group schemes.

Corollary 8**.**

We have $U_{X}=0$ if and only if $N\operatorname{Pic}(X)=0$.

*Proof. *This follows from N\operatorname{Pic}(X)=\ker\big{(}U_{X}({\mathbb{A}}^{1})\to U_{X}(k)\big{)}, because any unipotent, connected, smooth affine group is an affine space as a scheme, hence admits a non-trivial morphism from ${\mathbb{A}}^{1}$ which sends [math] to [math] if it is non-trivial. $\Box$

The kernel and cokernel of $\operatorname{Pic}_{X}\to\operatorname{Pic}_{X^{\text{\rm red}}}$ has been described in [Oort62], hence we will from now assume that $X$ is reduced. If $X^{+}$ is the semi-normalization of $X$, then the map ${\mathcal{O}}_{X}\to{\mathcal{O}}_{X^{+}}$ is an injection of sheaves on the same topological space. For $X^{+}$ reduced and semi-normal, $N\operatorname{Pic}(X^{+})=0$ by Traverso’s theorem [Tra70] together with [Wei91, Thm. 4.7]. Hence the Corollary implies that $U_{X^{+}}=0$, and that

[TABLE]

(For curves, this recovers [BLR90, Prop.9.2/10].) Indeed, by Corollary 6, the map $\operatorname{Pic}_{X}^{0,{\text{\rm red}}}\to\operatorname{Pic}_{X^{+}}^{0,{\text{\rm red}}}$ induces an isomorphism on the torus and abelian variety part, because it induces an isomorphism on etale cohomology.

*Proof. *(Theorem 3) a) We have isomorphims $H^{i}_{\text{\rm et}}(X,{{\mathbb{Z}}})\cong H^{i}_{\text{\rm et}}(X^{+},{{\mathbb{Z}}})$, which combined with Corollary 6 shows that the canonical map $\operatorname{Pic}_{X}^{0,{\text{\rm red}}}\to\operatorname{Pic}_{X^{+}}^{0,{\text{\rm red}}}$ induces an isomorphism on the torus components, and is an isogeny with kernel a unipotent group scheme $P_{X}$ on the abelian variety parts. 555O.Gabber [Gab20] showed that, conversely, any finite unipotent commutative group scheme can appear as $P_{X}$. Hence the map is surjective and the kernel $K_{X}$ is an extension of $P_{X}$ by $U_{X}$. Applying the Proposition to the exact sequence of etale sheaves

[TABLE]

on $\operatorname{Spec}k$, we obtain the diagram with exact columns

[TABLE]

Since $v$ is injective, so is $u$. The Neron-Severi group schemes are extensions of finitely generated étale group schemes by a finite connected group scheme. The isomorphism $H^{2}_{\text{\rm et}}(X,\mu_{m})\cong H^{2}_{\text{\rm et}}(X^{\prime},\mu_{m})$ implies that $\operatorname{Pic}_{X}(\bar{k})/m\to\operatorname{Pic}_{X^{+}}(\bar{k})/m$ is injective for any $m$ prime to $p$, and since $\operatorname{Pic}_{X}^{0,{\text{\rm red}}}(\bar{k})$ and $\operatorname{Pic}_{X^{+}}^{0,{\text{\rm red}}}(\bar{k})$ are $m$-divisible, the same holds for $\operatorname{NS}_{X}(\bar{k})/m\to\operatorname{NS}_{X^{+}}(\bar{k})/m$, and consequently for $\operatorname{NS}_{X}[\frac{1}{p}]\to\operatorname{NS}_{X^{+}}[\frac{1}{p}]$. Thus $\operatorname{coker}u$ is contained in the extension of the $p$-primary torsion subgroup $\operatorname{NS}_{X}\{p\}$ by the finite connected group scheme $\operatorname{NS}_{X}^{0}$. Finally, the isomorphism $\operatorname{Hom}_{\bar{k}}(\mu_{m},\operatorname{Pic}_{X})\cong H^{1}_{\text{\rm et}}(\bar{X},{{\mathbb{Z}}}/m)$ from [Mi80, III Prop. 4.16] together with the isomorphism $H^{i}_{\text{\rm et}}(X,{{\mathbb{Z}}})\cong H^{i}_{\text{\rm et}}(X^{+},{{\mathbb{Z}}})$ and the result on $K_{X}$ shows that the three right maps in the diagram induce isomorphisms on $\operatorname{Hom}_{\bar{k}}(\mu_{m},-)$ for all $m$, hence the three groups on the left are unipotent.

b) Recall that $q:X_{T}\to T$, and consider the diagram

[TABLE]

Since $q_{*}{\mathbb{G}}_{a,X\times T}=H^{0}(X,{\mathcal{O}}_{X})\otimes{\mathcal{O}}_{T}=H^{0}(X^{+},{\mathcal{O}}_{X^{+}})\otimes{\mathcal{O}}_{T}=q_{*}{\mathbb{G}}_{a,X^{+}\times T}$ is an isomorphism as in Proposition 4a), the outer maps are isomorphisms, and it suffices to calculate $\ker r$. Let $Y=X\times T$ and $Y^{\prime}=X^{+}\times T$, and consider the tautological map

[TABLE]

It suffices to show the following statements:

1. a)

The image of $f$ is contained in \ker\big{(}\operatorname{Pic}(Y)\to\operatorname{Pic}(Y^{\prime})\big{)}. 2. b)

$f$ surjects onto \ker\big{(}\operatorname{Pic}(Y)\to\operatorname{Pic}(Y^{\prime})\big{)}. 3. c)

$f$ is injective.

a) We claim that the map ${\mathcal{L}}\otimes_{{\mathcal{O}}_{Y}}{\mathcal{O}}_{Y^{\prime}}\to{\mathcal{O}}_{Y^{\prime}}\otimes_{{\mathcal{O}}_{Y}}{\mathcal{O}}_{Y^{\prime}}\xrightarrow{\mu}{\mathcal{O}}_{Y^{\prime}}$ is an isomorphism. We can check this on an affine covering, and in this case it is proved in [RS93, Lemma 2.2(4)].

b) Let ${\mathcal{L}}\in\operatorname{Pic}(Y)$ with ${\mathcal{L}}\otimes_{{\mathcal{O}}_{Y}}{\mathcal{O}}_{Y^{\prime}}\cong{\mathcal{O}}_{Y^{\prime}}$. Since ${\mathcal{L}}$ is flat, we get an injection ${\mathcal{L}}={\mathcal{L}}\otimes_{{\mathcal{O}}_{Y}}{\mathcal{O}}_{Y}\to{\mathcal{L}}\otimes_{{\mathcal{O}}_{Y}}{\mathcal{O}}_{Y^{\prime}}\cong{\mathcal{O}}_{Y^{\prime}}$. We claim that the inverse of ${\mathcal{L}}$ in ${\mathcal{O}}_{Y^{\prime}}$ is the sheaf associated to the presheaf $U\mapsto\{x\in{\mathcal{O}}_{Y^{\prime}}(U)|x{\mathcal{L}}(U)\subseteq{\mathcal{O}}_{Y}(U)\}\subseteq{\mathcal{O}}_{Y^{\prime}}(U)$. This can be checked on an affine covering, and then it is [RS93, Lemma 2.2(2)].

c) Let ${\mathcal{L}}$ and ${\mathcal{L}}^{\prime}$ be subsheaves of ${\mathcal{O}}_{Y^{\prime}}$ which are invertible in ${\mathcal{O}}_{Y^{\prime}}$ and isomorphic as abstract invertible sheaves. Multiplying with the inverse of ${\mathcal{L}}^{\prime}$ inside ${\mathcal{O}}_{Y^{\prime}}$, it suffices to show that if ${\mathcal{L}}$ is a subsheaf of ${\mathcal{O}}_{Y^{\prime}}$, and $f:{\mathcal{O}}_{Y}\to{\mathcal{L}}$ an isomorphism, then ${\mathcal{L}}={\mathcal{O}}_{Y}\subseteq{\mathcal{O}}_{Y^{\prime}}$. But $f(1)$ is a global unit of ${\mathcal{O}}_{Y^{\prime}}(Y)$, and by Proposition 4a), ${\mathcal{O}}_{Y}(Y)^{\times}={\mathcal{O}}_{Y^{\prime}}(Y)^{\times}$. Hence ${\mathcal{L}}=f(1)^{-1}{\mathcal{L}}={\mathcal{O}}_{Y}$. $\Box$