Appoximate Cohomology

TL;DR

This paper introduces approximate cohomology groups for vector spaces over finite fields and proves the finiteness of a related minimal rank bound for approximate homomorphisms, extending classical cohomology concepts.

Contribution

It defines a new notion of approximate cohomology groups and establishes finiteness results for minimal rank bounds in the context of approximate homomorphisms over finite fields.

Findings

Finiteness of R(G,r,k) for finite fields and countable-dimensional vector spaces.

Introduction of approximate cohomology groups as algebraic analogues of epsilon-representations.

Computation of the first approximate cohomology group for certain modules.

Abstract

Let $k$ be a field, $G$ be an abelian group and $r\in \mathbb N$. Let $L$ be an infinite dimensional $k$-vector space. For any $m\in End_k(L)$ we denote by $r(m)\in [0,\infty ]$ the rank of $m$. We define by $R(G,r,k)\in [0,\infty]$ the minimal $R$ such that for any map $A:G \to End_k(L)$ with $r(A(g'+g'')-A(g')-A(g''))\leq r$, $g',g''\in G$ there exists a homomorphism $\chi :G\to End_k(L)$ such that $r(A(g)-\chi (g))\leq R(G, r, k)$ for all $g\in G$. We show the finiteness of $R(G,r,k)$ for the case when $k$ is a finite field, $G=V$ is a $k$-vector space $V$ of countable dimension. We actually prove a generalization of this result. In addition we introduce a notion of {\it Approximate Cohomology} groups $H^k_{\mathcal F} (V,M)$ (which is a purely algebraic analogue of the notion of $\epsilon$-representation (\cite{ep})) and interperate our result as a computation of the group $H^1_{\mathcal F} (V,M)$ for some $V$-modules $M$.

Equations

$$P=\sum _{i=1}^r{l}_i{R}_i,$$

$$A:V\to M=M^d,r(A(v^{\prime }+v^{\prime \prime })-A(v^{\prime })-A(v^{\prime \prime }))\le r,v^{\prime },v^{\prime \prime }\in V$$

$$\partial P(v,v^{\prime })(x)=P(v+v^{\prime })(x)-P(v)(x+v^{\prime })-P(v^{\prime })(x)$$

$$Q(v+v^{\prime })(x)-Q(v)(x)-Q(v^{\prime })(x)$$

$${\mathbb{E}}_{x\in X}f(x)=\frac{1}{|X|}\sum _{x\in X}f(x).$$

$$\Vert \psi (F){\Vert }_{U_m}^{2^m}={\mathbb{E}}_{v,v_1,\dots ,v_m\in V}\psi ({\Delta }_{v_m}\dots {\Delta }_{v_1}F(v)).$$

$$\tilde{P}(x_1,\dots ,x_d)={\Delta }_{x_d}\dots {\Delta }_{x_1}P(x).$$

$$P(x)=\frac{1}{d!}\tilde{P}(x,\dots ,x).$$

$$\sum _{x_1\in W}{\mathbb{E}}_{x_2\in V}\psi (P(\bar{x}))=|W|.$$

$$P(x_1,\dots ,x_d)=\sum _{j\le d}\sum _{i\le L_j}{l}_i^j(x_j)Q_i^j(x_1,\dots ,{\hat{x}}_j,\dots ,x_d)+\sum _{k\le M}{T}_k(\bar{x})R_k(\bar{x})$$

$${\mathbb{E}}_{x_2,\dots ,x_d\in V}\psi (P_{x_1}(x_2,\dots ,x_d)).$$

$$\sum _{x_1\in W}{\mathbb{E}}_{x_2,\dots ,x_d\in V}\psi (P_{x_1}(x_2,\dots ,x_d))\ge C_{L,d-1}|W|.$$

$${\mathbb{E}}_{x_2,\dots ,x_d\in V}\psi (P_{x_1}(x_2,\dots ,x_d))\ge 0.$$

$${\mathbb{E}}_{x_1\in V}{\mathbb{E}}_{x_2,\dots ,x_d\in V}\psi (P_{x_1}(x_2,\dots ,x_d))\ge C_{L,d-1}|W|/|V|,$$

$$rk(P(v)+P(u)-P(v+u))<L.$$

$$f(v,x_1,\dots ,x_d)=\psi (\tilde{P}(v)(x_1,\dots ,x_d))=\psi (\tilde{P}(v)(\bar{x})).$$

$${\Delta }_{(v_{d+2},{\bar{h}}_{d+2})}\dots {\Delta }_{(v_1,{\bar{h}}_1)}\tilde{P}(v)(\bar{x})=\sum _{k=0}^{d+2}{\Delta }_{{\bar{h}}_{d+2}}\dots {\Delta }_{{\bar{h}}_{k+1}}({\Delta }_{v_k}\dots {\Delta }_{v_1}(\tilde{P}(v)))(\bar{x}+{\bar{h}}_1+\dots +h_k)$$

$$\sum _{k=2}^{d+2}{\Delta }_{{\bar{h}}_{d+2}}\dots {\Delta }_{{\bar{h}}_{k+1}}({\Delta }_{v_k}\dots {\Delta }_{v_1}\tilde{P}(v))(\bar{x}+{\bar{h}}_1+\dots +{\bar{h}}_k)$$

$$rk(P(v_1)+P(u_1)-P(v_2)-P(u_2))<2L.$$

$$y_j=(x_j,h_1^j,\dots ,h_{d+2}^j).$$

$$|{\mathbb{E}}_{v,x_1,\dots ,x_d}\psi (\tilde{P}(v)(x_1,\dots ,x_d)-Q(v,x_1,\dots ,x_d))|{\gg }_{L}1.$$

$$|{\mathbb{E}}_{v,x_1,\dots ,x_d}\psi (\tilde{P}(v)(x_1,\dots ,x_d)-Q(v,x_1,\dots ,x_d)){|}^2$$

$$\le {\left|{\mathbb{E}}_{v,x_1,\dots ,x_{d-1}}\left|{\mathbb{E}}_{x_d}\psi (\tilde{P}(v)(x_1,\dots ,x_d)-Q(v,x_1,\dots ,x_d))\right|\right|}^2$$

$$\le {\mathbb{E}}_{v,x_1,\dots ,x_{d-1}}{\left|{\mathbb{E}}_{x_d}\psi (\tilde{P}(v)(x_1,\dots ,x_d)-Q(v,x_1,\dots ,x_d))\right|}^2$$

$$={\mathbb{E}}_{v,x_1,\dots ,x_{d-1}}{\mathbb{E}}_{x_d,x_d^{\prime }}\psi (\tilde{P}(v)(x_1,\dots ,x_d+x_d^{\prime })-\tilde{P}(v)(x_1,\dots ,x_d)-Q(v,x_1,\dots ,x_d+x_d^{\prime })+Q(v,x_1,\dots ,x_d))$$

$$={\mathbb{E}}_{v,x_1,\dots ,x_{d-1}}{\mathbb{E}}_{x_d,x_d^{\prime }}\psi (\tilde{P}(v)(x_1,\dots ,x_{d-1},x_d^{\prime })-{\Delta }_{x_d^{\prime }}Q(v,x_1,\dots ,x_d)).$$

$$\tilde{P}(v)(x_1,\dots ,x_d+x_d^{\prime })-\tilde{P}(v)(x_1,\dots ,x_d)=\tilde{P}(v)(x_1,\dots ,x_{d-1},x_d^{\prime }).$$

$${\mathbb{E}}_{v,x_1,x_1^{\prime },\dots ,x_d,x_d^{\prime }}\psi ({\tilde{P}}_v(x_1^{\prime },\dots ,x_d^{\prime })-{\Delta }_{x_1^{\prime }}\dots {\Delta }_{x_d^{\prime }}Q(v,x_1,\dots ,x_d)){\gg }_{L}1$$

$${\mathbb{E}}_{v,v^{\prime },x_1,x_1^{\prime },\dots x_d,x_d^{\prime }}\psi ((\tilde{P}(v+v^{\prime })-\tilde{P}(v))(x_1^{\prime },\dots ,x_d^{\prime })-{\Delta }_{v^{\prime }}{\Delta }_{x_1^{\prime }}\dots {\Delta }_{x_d^{\prime }}Q(v,x_1,\dots ,x_d)){\gg }_{L}1.$$

$$|{\mathbb{E}}_{v,v^{\prime },x_1^{\prime },\dots ,x_d^{\prime }}\psi ((\tilde{P}(v+v^{\prime })-\tilde{P}(v))(x_1^{\prime },\dots ,x_d^{\prime })-\tilde{Q}(v^{\prime },x_1^{\prime },\dots ,x_d^{\prime }))|{\gg }_{L}1$$

$${\mathbb{E}}_{v,v^{\prime }}|{\mathbb{E}}_{x_1^{\prime },\dots ,x_d^{\prime }}\psi ((\tilde{P}(v+v^{\prime })-\tilde{P}(v)-\tilde{Q}(v^{\prime }))(x_1^{\prime },\dots ,x_d^{\prime })|{\gg }_{L}1.$$

$$P(v)=Q^{\prime }(v)+R(v)$$

Full text

We show the finiteness of $R(G,r,k)$ for the case when $k$ is a finite field, $G=V$ is a $k$-vector space $V$ of countable dimension. We actually prove a generalization of this result.

In addition we introduce a notion of Approximate Cohomology groups $H^{k}_{\mathcal{F}}(V,M)$ (which is a purely algebraic analogue of the notion of $\epsilon$-representation ([5])) and interperate our result as a computation of the group $H^{1}_{\mathcal{F}}(V,M)$ for some $V$-modules $M$.

The second author is supported by ERC grant ErgComNum 682150

1. Introduction

Let $k$ be a field, $G$ be an abelian group and $r\in\mathbb{N}.$ Let $L$ be an infinite dimensional $k$-vector space, $End(L)=End_{k}(L)$ and $M\subset\operatorname{End}(L)$ be the subspace of operators of finite rank. For any $m\in\operatorname{End}(L)$ we denote by $r(m)\in[0,\infty]$ the rank of $m$. We define by $R(G,r,k)\in[0,\infty]$(correspondingly $R^{f}(G,r,k))$ the minimal number $R$ such that for any map $A:G\to\operatorname{End}(L)$(correspondingly a map $A:G\to M)$, $g\to m_{g}$ with $r(A(g^{\prime}+g^{\prime\prime})-A(g^{\prime})-A(g^{\prime\prime}))\leq r$, $g^{\prime},g^{\prime\prime}\in G$ there exists a homomorphism $\chi:G\to\operatorname{End}(L)$ such that $r(A(g)-\chi(g))\leq R(G,r,k)$ for all $g\in G$.

It is easy to see that in the case when $G={\mathbb{Z}}$ and $\operatorname{char}(k)\neq 2$ we have $R({\mathbb{Z}},1,\mathbb{F})\leq 2$. We sketch the proof: one studies the rank $\leq 1$ operators $r_{m,n}=A(m+n)-A(m)-A(n)$. Since $r_{0,0}$ of rank $\leq 1$, we replace $r_{m,n}$ by $r_{n,m}-r_{0,0}$ and can then assume that $r_{0,0}=0$. Under this condition one must show that $r_{n,m}$ is a coboundary of rank one operators. The operators $r_{n,m}$ satisfy the equation $r_{a,-a}=r_{a+c,-a}+r_{a,c}$. From this one can deduce that either there is a subspace of codimension $1$ in the kernel of all three operators, or a subspace of dimension $1$ containing the image of all three. One shows inductively that this property holds for all operators $r_{m,n}$. Unfortunately we don’t know whether $R({\mathbb{Z}},2,{\mathbb{C}})<\infty$.

In this paper we first show that $R^{f}(V,r,k)<\infty$ in the case when $k=\mathbb{F}_{p}$ and $G$ is a $k$- vector space $V$ of countable dimension and then show that $R(V,r,k)=R^{f}(V,r,k)$.

Actually we prove the analogous bound in a more general case when $M$ is replaced by the space of tensors $L^{1}\otimes L^{2}\otimes...\otimes L^{n}$. To simplify the exposition we assume that $L_{i}=L^{\vee}$ and that $Im(A)$ is contained in the subset $Sym^{d}(L^{\vee})\subset{L^{\vee}}^{\otimes n}$ of symmetric tensors. In other words we consider map $A:V\to M^{d}$ where $M^{d}$ is isomorphic to the space of homogeneous polynomials of degree $d$ on $L$. We denote by $N^{d}\subset M^{d}$ the subspace of multilinear polynomials.

Now some formal definitions.

Definition 1.1** (Filtration).**

Let $M$ be an abelian group. A filtration $\mathcal{F}$ on $M$ is an increasing sequence of subsets $M_{i},1\leq i\leq\infty$ of $M$ such that for each $i,j$ there exists $c(i,j)$ such that $M_{i}+M_{j}\subset M_{c(i,j)}$ and $M=\cup_{i}M_{i}$. Two filtrations $M_{i},M_{i}^{\prime}$ on $M$ are equivalent if there exist functions $a,b:\mathbb{Z}_{+}\to\mathbb{Z}_{+}$ such that $M_{i}\subset M_{a(i)}^{\prime}$ and $M^{\prime}_{i}\subset M_{b(i)}$.

Definition 1.2** (Algebraic rank filtration).**

Let $k$ be a field. Fix $d\geq 2$ and consider the $k$-vector space $M=M^{d}$ of homogeneous polynomials $P$ of degree $d$ in variables $x_{j}$, $j\geq 1$. For a non-zero homogeneous polynomial $P$ on a $k$-vector space $W$, $P\in k[W^{\vee}]$ of degree $d\geq 2$ we define the rank $r(P)$ of $P$ as $r$ ,where $r$ is the minimal number $r$ such that it is possible to write $P$ in the form

[TABLE]

*where $l_{i},R_{i}\in\bar{k}[W^{\vee}]$ are homogeneous polynomials of positive degrees (in [9] this is called the $h$-invariant). We denote by $\mathcal{A}_{d}$ the filtration on $M$ such that $M_{n}$ is the subset of polynomials $P$ with $r(P)<n$. For the $k$-space $\mathcal{P}^{d}$ of non-homogeneous polynomials of degree $\leq d$ define the rank similarly, and denote by $\mathcal{B}_{d}$ the corresponding filtration.

Definition 1.3** (Finite rank homomorphisms).**

Let $V$ be a countable vector space over $k$. We say that a linear map $P:V\to M^{d}$ is of finite rank if we can write $P$ as a sum $\sum_{k=1}^{d-1}P_{k}$ where each $P_{k}$ is a finite sum $P_{k}=\sum_{j}Q_{j}R_{j}$ where $R_{j}\in M^{k}$ and $Q_{j}$ is a linear map from $V$ to $M^{d-k}$. Denote $\operatorname{Hom}_{f}(V,M^{d})$ the subspace of $\operatorname{Hom}(V,M^{d})$ of finite rank maps.

Now we can formulate our main result.

Theorem 1.4**.**

For any finite field $k=\mathbb{F}_{p},d<p-1,r\geq 0$ there exists $R=R(r,k,d)$ such that for any map

[TABLE]

there exists a homomorphism $\chi_{A}:V\to M$ such that $r(A(v)-\chi_{A}(v))\leq R$. Moreover the homomorphism $\chi_{A}$ is unique up to an addition of a homomorphism of a finite rank.

Question 1.5**.**

$a)$* Is there a bound on $R$ independent of $k$ ? Moreover, does there exist $c(d)$ such that $R(V,r,k,d)\leq c(d)r$?*

$b)$* Could we drop the condition $d<p-1$ if $Im(A)\subset N^{d}$?*

Remark 1.6**.**

Theorem 1.4 does not hold for $p=d=2$, see [8] for a function from $\mathbb{F}_{2}^{n}$ to the space of quadratic forms over $\mathbb{F}_{2}$ such that $f(u+v)-f(u)-f(v)$ is of rank $\leq 3$ but for $n$ sufficiently large $f$ does not differ from a linear function by a function taking values in bounded rank quadratics. In the low characteristic case the same proof shows that obstructions come from non-classical polynomials see Remark 2.4. In the case when $G$ is a finite cyclic group, $k$ any field, $d=2$, one can show that $C(1,k)\leq 2$.

We can reformulate Theorem 1.4 as an example of a computation of *approximate cohomology * groups.

Definition 1.7** (Approximate cohomology).**

1. (1)

Let $M$ be an abelian group, and let $\mathcal{F}=\{M_{i}\}$ be a filtration on $M$. 2. (2)

Let $G$ be a discrete group acting on $M$ preserving the subsets $M_{i}$. A cochain $r:G^{n}\to M$ is an approximate $n$-cocycle if $Im(\partial r)\subset M_{i}$ for some $i\in{\mathbb{Z}}_{+}$. It is clear that the set $Z^{n}_{\mathcal{F}}$ of approximate $n$-cocycles is a subgroup of the group $C^{n}$ of $n$-chains which depends only on the equivalence class of a filtration $\mathcal{F}$. 3. (3)

A cochain $r:G^{n}\to M$ is an approximate $n$-coboundary if there exists an $n-1$-cochain $t\in C^{n-1}$ such that $Im(r-\partial t)\subset M_{i}$ for some $i\in{\mathbb{Z}}_{+}$. It is clear that the set $B^{n}_{\mathcal{F}}$ of approximate $n$-coboundaries is a subgroup of $\tilde{Z}^{n}_{\mathcal{F}}$. 4. (4)

We define $H^{n}_{\mathcal{F}}=Z^{n}_{\mathcal{F}}/B^{n}_{\mathcal{F}}$. 5. (5)

Since any cocycle is an approximate cocycle and any coboundary is an approximate coboundary we have a morphism $a^{n}_{\mathcal{F}}:H^{n}(G,M)\to H^{n}_{\mathcal{F}}(G,M)$.

In this paper we consider the case when the group $V$ acts trivially on $M$. So the group $Z^{1}(V,M)$ of $1$-cocycles coincides with the group $\operatorname{Hom}(V,M)$ of linear maps, the subgroup of coboundaries $B^{1}(V,M)\subset Z^{1}(V,M)$ is equal to $\{0\}$ and therefore $H^{1}(V,M)=\operatorname{Hom}(V,M)$. In this case we can reformulate the Theorem 1.4 in terms of a computation of the map $a^{1}_{\mathcal{F}}$.

Corollary 1.8**.**

Let $k$ be a prime finite field of characteristic $p$, $V$ be a countable vector space over $k$ acting trivially on $(M,\mathcal{F})=(M^{d},\mathcal{A}_{d})$ and assume that $p>d+1$. Then the map $a^{1}:H^{1}(V,M)=\operatorname{Hom}(V,M)\to\tilde{H}_{\mathcal{F}}^{1}(V,M)$ is surjective, and $\operatorname{Ker}(a^{1})=\operatorname{Hom}_{f}(V,M_{d})$.

Question 1.9**.**

How to describe $H_{\mathcal{F}}^{n}(V,M)$ for $n>1$ ?

Corollary 1.10**.**

Let $k$ be a prime finite field of characteristic $p$, and let $V$ be a countable vector space over $k$. Consider the filtration $(\mathcal{P}^{d},\mathcal{B}_{d})$ with $W=V$ and with $V$-acting by translations $(v.P)(x)=P(x+v)$ and assume that $p>d+1$. Then the map $a^{1}_{\mathcal{B}}$ is surjective.

Proof.

Let $P:V\to\mathcal{P}^{d}$ where $\mathcal{P}^{d}$ is the space of polynomial of degree $\leq d$. . We assume

[TABLE]

is of rank $\leq i$ for any $v,v^{\prime}\in V$. Let $Q(v)$ be the homogeneous degree $d$ term of $P(v)$. Then since $P(v)(x+v^{\prime})-P(v)(x)$ is of degree $<d$ we have

[TABLE]

is of rank $\leq i+1$.

∎

Remark 1.11**.**

The proof of Theorem 1.4 uses the inverse theorem for the Gowers norms [1, 10]. One can use also prove the reverse implication modifying the arguments in [7], and thus an independent proof of Theorem 1.4 could lead to a new proof of the inverse conjecture for the Gowers norms.

2. Proof of Theorem 1.4

For a function $f$ on a finite set $X$ we define

[TABLE]

We use $X\ll_{L}Y$ to denote the estimate $|X|\leq C(L)|Y|$, where the constant $C$ depends only on $L$. We fix a prime finite field $k$ of order $p$ and degree $d<p-1$ and suppress the dependence of all bounds on $k,d$. We also fix a non-trivial additive character $\psi$ on $k$.

 

For a function $f:G\to H$ a function between abelian groups we denote $\Delta_{h}f(x)=f(x+h)-f(x)$. if $f:G_{1}\times G_{2}\to H$ then for $g_{1}\in G_{1}$ we write $\Delta_{g_{1}}f$ shorthand for $\Delta_{(g_{1},0)}f$.

 

Let $V$ be a finite vector space over $k$. Let $F:V\to k$. The $m$-th Gowers norm of $F$ is defined by

[TABLE]

These were introduced by Gowers in [2], and were shown to be norms for $m>1$.

 

For a homogeneous polynomial $P$ on $V$ of degree $d$ we define

[TABLE]

This is a multilinear homogeneous form in $x_{1},\ldots,x_{d}\in V$ such that

[TABLE]

Proposition 2.1**.**

There exists a function $R(L)$ such that for any finite dimensional $k$-vector space $V$ and a map $P:V\to M=M_{d}$ such that $r(P(v+v^{\prime})-P(v)-P(v^{\prime}))\leq L$ for $v,v^{\prime}\in V$ there exists a linear map $Q:V\to M$ such that $r(P(v)-Q(v))\leq R(L)$.

Proof.

The proof is based on an argument of [4], [6]. Our aim is to show that if $\partial P(v,v^{\prime})(x)$ is of rank $\leq L$ for all $v,v^{\prime}\in V$ then there exists a homogeneous polynomial $Q(v)(x)$ of degree $\leq d$ such that $\partial P(v,v^{\prime})=0$ and $P(v)-Q(v)\ll_{L}1$.

Lemma 2.2**.**

Let $P:V^{d}\to k$ be a multilinear homogeneous polynomial of degree $d\geq 2$ and rank $L$. Then ${\mathbb{E}}_{\bar{x}\in V}\psi(P(x))\geq C_{L,d}$ for some positive constant $C_{L,d}$ depending only on $L,d$.

Proof.

We prove this by induction on $d$. For quadratics: $P(x_{1},x_{2})=\sum_{i=1}^{L}l^{1}_{i}(x_{1})l^{2}_{i}(x_{2})$. If $x_{1}\in\bigcap_{i}\operatorname{Ker}(l_{i}^{1})$ then $\psi(P({\bar{x}}))\equiv 1$, thus on a subspace $W$ of codimension at most $L$ we have $\psi(P({\bar{x}}))\equiv 1$, so that

[TABLE]

If $x_{1}$ is outside $W$ then the inner sum is nonnegative.

 

Suppose now that $d>2$. Let $P(x_{1},\ldots,x_{d})$ be multilinear homogeneous polynomial of degree $d$ and rank $L$. Write

[TABLE]

with ${\bar{x}}=(x_{1},\ldots,x_{d})$, $l_{i}^{j}$ are linear and $T_{k}({\bar{x}})R_{k}({\bar{x}})$ is homogenous multilinear in $x_{1},\ldots x_{d}$ such that the degrees of $T_{k}({\bar{x}})$ and $R_{k}({\bar{x}})$ are $\geq 2$, and $\sum_{j}L_{j}+M=L$.

If for all $j$ we have $L_{j}=0$, then $P({\bar{x}})=\sum_{k\leq L}T_{k}({\bar{x}})R_{k}({\bar{x}})$ with the degree of $T_{k}({\bar{x}}),R_{k}({\bar{x}})\geq 2$. For $x_{1}\in V$ write $P_{x_{1}}(x_{2},\ldots,x_{d})=P(x_{1},x_{2},\ldots,x_{d})$. Then $P_{x_{1}}$ is of rank $L$ and degree $d-1$ for all $x_{1}$ and we obtain the claim by induction.

 

Otherwise there is a $j$, such that $L_{j}>0$, without loss of generality $j=1$. Let $W=\bigcap_{i}\operatorname{Ker}(l_{i}^{1})$, then $W$ is of codimension at most $L_{1}$. For $x_{1}\in W$ let $P_{x_{1}}(x_{2},\ldots,x_{d})=P(x_{1},x_{2},\ldots,x_{d})$. Consider the sum

[TABLE]

For $x_{1}\in W$ we have $P_{x_{1}}$ is of rank $\leq L-L_{1}$ and homogeneous of degree $d-1$. By the induction hypothesis the above sum is $\geq C_{L,d-1}$, so that

[TABLE]

For any $x_{1}\notin W$, $P_{x_{1}}$ is of degree $d-1$, and of rank $<\infty$ thus by the induction hypothesis,

[TABLE]

Thus

[TABLE]

and we obtain the claim. ∎

Let $P:V\to M_{d}$ be a map such that for all $u,v$:

[TABLE]

We define a function on $V\times W^{d}$ by

[TABLE]

Lemma 2.3**.**

$\|f\|_{U^{d+2}}\geq c_{L}$.

Proof.

We expand

[TABLE]

with $v_{i}\in V$ and ${\bar{h}_{i}}\in V^{d}$. Since $P_{v}$ is of degree $d$ the above is equal

[TABLE]

Since $rk(P(v)+P(u)-P(v+u))<L$, for any $v_{1}+u_{1}=v_{2}+u_{2}$ we have

[TABLE]

that for $k\geq 2$ we have $\Delta_{v_{k}}\ldots\Delta_{v_{1}}\tilde{P}(v)$ is of rank $\ll_{L}1$. For fixed $v,v_{1},\ldots,v_{k}$, the above polynomial can be expresses as a multilinear homogeneous polynomial of degree $d$ in $y_{1},\ldots,y_{d}$ with

[TABLE]

which is of rank that is bounded in terms of $L,d$. Now apply previous lemma. ∎

By the inverse theorem for the Gowers norm [1, 10] there is a polynomial $Q$ on $V^{d+1}$ of degree $d+1$ on with $Q:V\times V^{d}\to k$ s.t. such that

[TABLE]

By an application of the triangle and Cauchy-Schwarz inequalities we obtain

[TABLE]

Where the last equality follows from the fact that $\tilde{P}$ is homogeneous multilinear form and thus

[TABLE]

Applying Cauchy-Schwarz $d-1$ more times we obtain

[TABLE]

One more application of Cauchy-Schwarz gives

[TABLE]

Since $Q$ is a polynomial of degree $d+1$, $\Delta_{v^{\prime}}\Delta_{x_{1}^{\prime}}\ldots\Delta_{x_{d}^{\prime}}Q$ is independent of $v,x_{1},\ldots,x_{d}$ so we obtain

[TABLE]

with $\tilde{Q}$ a multilinear homogeneous form in $v^{\prime},x^{\prime}_{1},\ldots,x^{\prime}_{d}$. Denote by $\tilde{Q}(v)$ the function on $W^{d}$ given by $\tilde{Q}(v,\bar{x})=Q(v,\bar{x})$. Then

[TABLE]

By [3] (Proposition 6.1) and [BL] (Lemma 4.17) it follows that for at least $\gg_{L}|V|^{2}$ values of $v,v^{\prime}$ we have $P(v+v^{\prime})-P(v)-Q^{\prime}(v^{\prime})$ is of rank $\ll_{L}1$, where $Q^{\prime}(v)(x)=\tilde{Q}(v)(x,\ldots,x)/d!$ where $Q^{\prime}(v)$ is linear in $v$. Recall now that $P(v+v^{\prime})-P(v)-P(v^{\prime})$ is of rank $\leq L$, so that we get a set $E$ of size $\gg_{L}|V|$ of $v$ for which

[TABLE]

with $R(v)$ of rank $\leq L$. Since $E\gg_{L}|V|$, by the Bogolyubov lemma (see e.g. [12])) $2E$ contains a subspace $E^{\prime}$ of codimension $K\ll_{L}1$ in $V$. For $v\in E^{\prime}$, define

[TABLE]

Let $P^{\prime}:V\to M_{d}$ be any extension of $P^{\prime}$ linear in $v\in V$. Then $P^{\prime}(v)-P(v)$ is of rank $\ll_{L}1$, and $P^{\prime}(v)$ is a cocycle. ∎

Remark 2.4**.**

In the case where $p\leq d+2$, by the inverse theorem for the Gowers norms over finite fields the polynomial $Q$ in the above argument on $V\times V^{d}$ would be replaced by a nonclassical polynomial see [11], and the same argument would give that the approximate cohomology obstructions lie in the nonclassical degree $d$ polynomials - these are functions $P:V\to\mathbb{T}$ satisfying $\Delta_{h_{d+1}}\ldots\Delta_{h_{1}}P\equiv 0$.

Proof of Theorem 1.4. Let $k=\mathbb{F}_{q}$, and let $V$ be an countable vector space over $k$. Denote $V_{n}=k^{n}$, then $V=\cup V_{n}$. Let $M\subset k[x_{1},...x_{n},...]$ be the subspace of homogeneous polynomials of degree $d$. For any $l\geq 1$ we denote by $N_{l}\subset M$ the subset of polynomials of in $x_{1},...,x_{l}$ and denote by $p_{l}:M\to N_{l}$ the projection defined by $x_{i}\to 0$ for $i>l$. Observe that $p_{n}$ does not increase the rank. By Proposition 2.1 there is a constant $C$ depending only on $L,d$ (and $k$) such that for any $n$ there exists a linear map $\phi_{n}:V_{n}\to M$ such that rank $(P(v)-\phi_{n}(v))\leq C$, for $v\in V_{n}$.

We now show that the existence of such linear maps $\phi_{n}$ implies the existence of a linear map $\psi:V\to M$ such that $rank(R(v)-\psi(v))\leq C$, for $v\in V$.

Lemma 2.5**.**

Let $X_{n}$, $n\geq 0$ be finite not empty sets and $f_{n}:X_{n+1}\to X_{n}$ be maps. Then one can find $x_{n}\in X_{n}$ such that $f_{n}(x_{n+1})=x_{n}$.

Proof.

This result is standard, but for the convenience of a reader we provide a proof. The claim is obviously true if the maps $f_{n}$ are surjective. For any $m>n$ we define the subset $X_{m,n}\subset X_{n}$ as the image of

[TABLE]

It is clear that for a fixed $n$ we have

[TABLE]

We define $Y_{n}$ as the intersection $\cap_{m>n}X_{m,n}$. Since the set $X_{n}$ is finite, the sets $X_{m,n}$ stabilize as m grows and hence $Y_{n}$ is not empty. Let $\tilde{f}_{n}$ be the restriction of $f_{n}$ on $Y_{n+1}$. Now the maps $\tilde{f}_{n}:Y_{n+1}\to Y_{n}$ are surjective, thus the lemma follows. ∎

Lemma 2.6**.**

Let $P:V\to M$ be a map such that for any $n$ there exists a linear map $\phi_{n}:V_{n}\to M$ such that rank $(P(v)-\phi_{n}(v))\leq C$, $v\in V_{n}$. Then there exists a linear map $\psi:V\to M$ such that $rank(P(v)-\psi(v))\leq C$, for all $v\in V$.

Proof.

Let $l(n)$ be such that $P(V_{n})\subset N_{l(n)}$ and $\psi_{n}=p_{l(n)}\circ\phi_{n}$. Since $p_{n}$ does not increase the rank, and since $(p_{l(n)}\circ P)(V_{n})=R(V_{n})$ we have

[TABLE]

We apply Lemma 2.5 to the case when $X_{n}$ is the set of linear maps $\psi_{n}:V_{n}\to N_{l(n)}$ satisfying $(\star_{n})$ and $f_{n}$ are the restriction from $V_{n+1}$ onto $V_{n}$ we find the existence of linear maps $\psi_{n}:V_{n}\to N_{l(n)}$ satisfying the condition $(\star_{n})$ and such that the restriction of $\psi_{n+1}$ onto $V_{n}$ is equal to $\psi_{n}.$ The system $\{\psi_{n}\}$ defines a linear map $\psi:V\to M$.

∎

We now prove the result stated in the abstract by proving the equality $R(V,r,k)=R^{f}(V,r,k)$. Let $\Gamma$ be an abelian group, $\Gamma=\cup\Gamma_{n}$ where $\Gamma_{n}$ are finitely generated groups. Let $k$ be a finite field, and let $V,W$ be $k$-vector spaces with bases $v_{j},w_{j}$. For $n\geq 1$ let $V_{n},W_{n}$ be the spans of $v_{j},w_{j},1\leq j\leq n$. We denote by $i_{n}:V_{n}\to V_{n+1}$ the natural imbedding and by $\beta_{n}:W_{n+1}\to W_{n}$ the natural projection. Denote $\operatorname{Hom}^{f}(V,W)$ the finite rank homomorphisms from $V\to W$.

Proposition 2.7**.**

Suppose there exists $C=C(c)$ such that for any map $a^{f}:\Gamma\to\operatorname{Hom}^{f}(V,W)$ such that

[TABLE]

there exists a homomorphism $\chi^{f}:\Gamma\to\operatorname{Hom}^{f}(V,W)$ such that $r(a^{f}(\gamma)-\chi^{f}(\gamma))\leq C$. Then for any map $a:\Gamma\to\operatorname{Hom}(V,W)$ such that

[TABLE]

there exists a homomorphism $\chi:\Gamma\to\operatorname{Hom}(V,W)$ such that $r(a(\gamma)-\chi(\gamma))\leq C$.

Proof.

We will use the following fact that is an immediate consequence of König’s lemma : Let $X$ be a locally finite tree, $x\in X$. If for any $N$ there exists a branch starting at $x$ of length $N$ then there exists an infinite branch starting at $x$.

 

Let $a:\Gamma\to\operatorname{Hom}(V,W)$ be a map such that

[TABLE]

We define $F_{n}=\operatorname{Hom}(V_{n},W_{n})$. Let $Y_{n}=\operatorname{Hom}(\Gamma_{n},F_{n})$ and $q_{n}:Y_{n+1}\to Y_{n}$ be given by

[TABLE]

where $\chi^{\prime}_{n+1}$ is the restriction of $\chi_{n+1}$ on $\Gamma_{n}$.

 

We denote by $X_{n}\subset Y_{n}$ the subset of homomorphisms $\chi_{n}$ of $\Gamma_{n}$ such that

[TABLE]

Let $X$ be the disjoint union of $X_{n}$ and we connect $\chi_{n}\in X_{n}$ with $\chi_{n+1}\in X_{n+1}$ if $\chi_{n}=q_{n}(\chi_{n+1})$.

 

By the assumption $X_{n}$ are finite not empty sets and for any $n$ there exists a branch from $X_{0}$ to $X_{n}$ (any $\chi_{n}\in X_{n}$ defines such a branch). Now the Lemma 2.5 implies the existence a character $\chi:\Gamma\to\operatorname{Hom}(V,W)$ such that $r(a(\gamma)-\chi(\gamma))\leq C$. ∎

To conclude the proof of Theorem 1.4 we calculate the kennel of the map $a^{1}$:

Proposition 2.8**.**

The kernel of $a^{1}$ consists of maps $P$ of finite rank.

Proof.

Suppose $V,W$ are of dimension $n_{1},n_{2}$ respectively. All the bounds below are independent of $n_{1},n_{2}$. Suppose $P:V\to M_{d}$ is a linear map with $r(P)\leq L$. Let $\tilde{P}$ be the multilinear version of $P$ as in (2.1). Let $f(v,\bar{x})=\psi(\tilde{P}(v)(\bar{x}))$. Now $\tilde{P}(v)(\bar{x})$ is a multilinear polynomial on $V\times W^{d}$ of degree $d+1$.

 

For any fixed $v$ we have

[TABLE]

and thus

[TABLE]

It follows that $\tilde{P}(v)(\bar{x})$ is of bounded rank $\ll_{L}1$ and thus of the form

[TABLE]

with $\tilde{Q}_{j},\tilde{R}_{j}$ of degree $\geq 1$ ,for any fixed $v$ also $\tilde{Q}_{j},\tilde{R}_{j}$ are of degree $\geq 1$, and $K\ll_{L}1$. For any fixed $x$, $\tilde{P}(V)$ is linear and thus either $\tilde{Q}_{j}$ or $\tilde{R}_{j}$ are constant as a function of $v$. Recall that $P(v)(x)=\frac{1}{d!}\tilde{P}(v)(x,\ldots,x)$, and let $Q_{j}(v,x)=\frac{1}{d!}\tilde{Q}_{j}(v,x,\ldots,x)$, similarly $R_{j}$. Let $J$ be the set of $j$ in the sum $P=\sum_{j}Q_{j}R_{j}$ such that $Q_{j}$ is linear in $v$ and does not depend on $x$. Let $V^{\prime}=\bigcap_{j}\operatorname{Ker}Q_{j}$. The restriction of $P$ to $V^{\prime}$ has finite (that is by a constant which does not depend on $n_{1},n_{2}$ ) rank. Since $codim(V^{\prime})\leq|J|$ we see that $P$ has rank that is bounded by a constant which does not depend on $n_{1}$ and $n_{2}$.

 

Now let $V,W$ be infinite. Let $V_{n},l(n),p_{l(n)}$ be as in Lemma 2.6. Len $P_{n}(V_{n})=(p_{l(n)}\circ P)(V_{n})$. Now apply Lemma 2.5 for $X_{n}$ the collection of finite rank maps from $V_{n}\to N_{l(n)}$, and $f_{n}$ the restriction as before. This finishes a proof of Theorem 1.4. ∎