Appoximate Cohomology
David Kazhdan, Tamar Ziegler

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 be a field, be an abelian group and . Let be an infinite dimensional -vector space. For any we denote by the rank of . We define by the minimal such that for any map with , there exists a homomorphism such that for all . We show the finiteness of for the case when is a finite field, is a -vector space of countable dimension. We actually prove a generalization of this result. In addition we introduce a notion of {\it Approximate Cohomology} groups (which is a purely algebraic analogue of the notion of -representation (\cite{ep})) and interperate our result as a computation of the groupβ¦
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Taxonomy
TopicsLimits and Structures in Graph Theory Β· Advanced Topology and Set Theory Β· Algebraic Geometry and Number Theory
Approximate cohomology
David Kazhdan
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givaat Ram The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
Β andΒ
Tamar Ziegler
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givaat Ram The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
Abstract.
Let be a field, be an abelian group and Let be an infinite dimensional -vector space. For any we denote by the rank of . We define by the minimal such that for any map with , there exists a homomorphism such that for all .
We show the finiteness of for the case when is a finite field, is a -vector space of countable dimension. We actually prove a generalization of this result.
In addition we introduce a notion of Approximate Cohomology groups (which is a purely algebraic analogue of the notion of -representation ([5])) and interperate our result as a computation of the group for some -modules .
The second author is supported by ERC grant ErgComNum 682150
1. Introduction
Let be a field, be an abelian group and Let be an infinite dimensional -vector space, and be the subspace of operators of finite rank. For any we denote by the rank of . We define by (correspondingly the minimal number such that for any map (correspondingly a map , with , there exists a homomorphism such that for all .
It is easy to see that in the case when and we have . We sketch the proof: one studies the rank operators . Since of rank , we replace by and can then assume that . Under this condition one must show that is a coboundary of rank one operators. The operators satisfy the equation . From this one can deduce that either there is a subspace of codimension in the kernel of all three operators, or a subspace of dimension containing the image of all three. One shows inductively that this property holds for all operators . Unfortunately we donβt know whether .
In this paper we first show that in the case when and is a - vector space of countable dimension and then show that .
Actually we prove the analogous bound in a more general case when is replaced by the space of tensors . To simplify the exposition we assume that and that is contained in the subset of symmetric tensors. In other words we consider map where is isomorphic to the space of homogeneous polynomials of degree on . We denote by the subspace of multilinear polynomials.
Now some formal definitions.
Definition 1.1** (Filtration).**
Let be an abelian group. A filtration on is an increasing sequence of subsets of such that for each there exists such that and . Two filtrations on are equivalent if there exist functions such that and .
Definition 1.2** (Algebraic rank filtration).**
Let be a field. Fix and consider the -vector space of homogeneous polynomials of degree in variables , . For a non-zero homogeneous polynomial on a -vector space , of degree we define the rank of as ,where is the minimal number such that it is possible to write in the form
[TABLE]
*where are homogeneous polynomials of positive degrees (in [9] this is called the -invariant). We denote by the filtration on such that is the subset of polynomials with . For the -space of non-homogeneous polynomials of degree define the rank similarly, and denote by the corresponding filtration.
Definition 1.3** (Finite rank homomorphisms).**
Let be a countable vector space over . We say that a linear map is of finite rank if we can write as a sum where each is a finite sum where and is a linear map from to . Denote the subspace of of finite rank maps.
Now we can formulate our main result.
Theorem 1.4**.**
For any finite field there exists such that for any map
[TABLE]
there exists a homomorphism such that . Moreover the homomorphism is unique up to an addition of a homomorphism of a finite rank.
Question 1.5**.**
* Is there a bound on independent of ? Moreover, does there exist such that ?*
* Could we drop the condition if ?*
Remark 1.6**.**
Theorem 1.4 does not hold for , see [8] for a function from to the space of quadratic forms over such that is of rank but for sufficiently large 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 is a finite cyclic group, any field, , one can show that .
We can reformulate Theorem 1.4 as an example of a computation of *approximate cohomology * groups.
Definition 1.7** (Approximate cohomology).**
- (1)
Let be an abelian group, and let be a filtration on . 2. (2)
Let be a discrete group acting on preserving the subsets . A cochain is an approximate -cocycle if for some . It is clear that the set of approximate -cocycles is a subgroup of the group of -chains which depends only on the equivalence class of a filtration . 3. (3)
A cochain is an approximate -coboundary if there exists an -cochain such that for some . It is clear that the set of approximate -coboundaries is a subgroup of . 4. (4)
We define . 5. (5)
Since any cocycle is an approximate cocycle and any coboundary is an approximate coboundary we have a morphism .
In this paper we consider the case when the group acts trivially on . So the group of -cocycles coincides with the group of linear maps, the subgroup of coboundaries is equal to and therefore . In this case we can reformulate the Theorem 1.4 in terms of a computation of the map .
Corollary 1.8**.**
Let be a prime finite field of characteristic , be a countable vector space over acting trivially on and assume that . Then the map is surjective, and .
Question 1.9**.**
How to describe for ?
Corollary 1.10**.**
Let be a prime finite field of characteristic , and let be a countable vector space over . Consider the filtration with and with -acting by translations and assume that . Then the map is surjective.
Proof.
Let where is the space of polynomial of degree . . We assume
[TABLE]
is of rank for any . Let be the homogeneous degree term of . Then since is of degree we have
[TABLE]
is of rank .
β
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 on a finite set we define
[TABLE]
We use to denote the estimate , where the constant depends only on . We fix a prime finite field of order and degree and suppress the dependence of all bounds on . We also fix a non-trivial additive character on .
Β
For a function a function between abelian groups we denote . if then for we write shorthand for .
Β
Let be a finite vector space over . Let . The -th Gowers norm of is defined by
[TABLE]
These were introduced by Gowers in [2], and were shown to be norms for .
Β
For a homogeneous polynomial on of degree we define
[TABLE]
This is a multilinear homogeneous form in such that
[TABLE]
Proposition 2.1**.**
There exists a function such that for any finite dimensional -vector space and a map such that for there exists a linear map such that .
Proof.
The proof is based on an argument of [4], [6]. Our aim is to show that if is of rank for all then there exists a homogeneous polynomial of degree such that and .
Lemma 2.2**.**
Let be a multilinear homogeneous polynomial of degree and rank . Then for some positive constant depending only on .
Proof.
We prove this by induction on . For quadratics: . If then , thus on a subspace of codimension at most we have , so that
[TABLE]
If is outside then the inner sum is nonnegative.
Β
Suppose now that . Let be multilinear homogeneous polynomial of degree and rank . Write
[TABLE]
with , are linear and is homogenous multilinear in such that the degrees of and are , and .
If for all we have , then with the degree of . For write . Then is of rank and degree for all and we obtain the claim by induction.
Β
Otherwise there is a , such that , without loss of generality . Let , then is of codimension at most . For let . Consider the sum
[TABLE]
For we have is of rank and homogeneous of degree . By the induction hypothesis the above sum is , so that
[TABLE]
For any , is of degree , and of rank thus by the induction hypothesis,
[TABLE]
Thus
[TABLE]
and we obtain the claim. β
Let be a map such that for all :
[TABLE]
We define a function on by
[TABLE]
Lemma 2.3**.**
.
Proof.
We expand
[TABLE]
with and . Since is of degree the above is equal
[TABLE]
Since , for any we have
[TABLE]
that for we have is of rank . For fixed , the above polynomial can be expresses as a multilinear homogeneous polynomial of degree in with
[TABLE]
which is of rank that is bounded in terms of . Now apply previous lemma. β
By the inverse theorem for the Gowers norm [1, 10] there is a polynomial on of degree on with 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 is homogeneous multilinear form and thus
[TABLE]
Applying Cauchy-Schwarz more times we obtain
[TABLE]
One more application of Cauchy-Schwarz gives
[TABLE]
Since is a polynomial of degree , is independent of so we obtain
[TABLE]
with a multilinear homogeneous form in . Denote by the function on given by . Then
[TABLE]
By [3] (Proposition 6.1) and [BL] (Lemma 4.17) it follows that for at least values of we have is of rank , where where is linear in . Recall now that is of rank , so that we get a set of size of for which
[TABLE]
with of rank . Since , by the Bogolyubov lemma (see e.g. [12])) contains a subspace of codimension in . For , define
[TABLE]
Let be any extension of linear in . Then is of rank , and is a cocycle. β
Remark 2.4**.**
In the case where , by the inverse theorem for the Gowers norms over finite fields the polynomial in the above argument on 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 polynomials - these are functions satisfying .
Proof of Theorem 1.4. Let , and let be an countable vector space over . Denote , then . Let be the subspace of homogeneous polynomials of degree . For any we denote by the subset of polynomials of in and denote by the projection defined by for . Observe that does not increase the rank. By Proposition 2.1 there is a constant depending only on (and ) such that for any there exists a linear map such that rank , for .
We now show that the existence of such linear maps implies the existence of a linear map such that , for .
Lemma 2.5**.**
Let , be finite not empty sets and be maps. Then one can find such that .
Proof.
This result is standard, but for the convenience of a reader we provide a proof. The claim is obviously true if the maps are surjective. For any we define the subset as the image of
[TABLE]
It is clear that for a fixed we have
[TABLE]
We define as the intersection . Since the set is finite, the sets stabilize as m grows and hence is not empty. Let be the restriction of on . Now the maps are surjective, thus the lemma follows. β
Lemma 2.6**.**
Let be a map such that for any there exists a linear map such that rank , . Then there exists a linear map such that , for all .
Proof.
Let be such that and . Since does not increase the rank, and since we have
[TABLE]
We apply Lemma 2.5 to the case when is the set of linear maps satisfying and are the restriction from onto we find the existence of linear maps satisfying the condition and such that the restriction of onto is equal to The system defines a linear map .
β
We now prove the result stated in the abstract by proving the equality . Let be an abelian group, where are finitely generated groups. Let be a finite field, and let be -vector spaces with bases . For let be the spans of . We denote by the natural imbedding and by the natural projection. Denote the finite rank homomorphisms from .
Proposition 2.7**.**
Suppose there exists such that for any map such that
[TABLE]
there exists a homomorphism such that . Then for any map such that
[TABLE]
there exists a homomorphism such that .
Proof.
We will use the following fact that is an immediate consequence of KΓΆnigβs lemma : Let be a locally finite tree, . If for any there exists a branch starting at of length then there exists an infinite branch starting at .
Β
Let be a map such that
[TABLE]
We define . Let and be given by
[TABLE]
where is the restriction of on .
Β
We denote by the subset of homomorphisms of such that
[TABLE]
Let be the disjoint union of and we connect with if .
Β
By the assumption are finite not empty sets and for any there exists a branch from to (any defines such a branch). Now the Lemma 2.5 implies the existence a character such that . β
To conclude the proof of Theorem 1.4 we calculate the kennel of the map :
Proposition 2.8**.**
The kernel of consists of maps of finite rank.
Proof.
Suppose are of dimension respectively. All the bounds below are independent of . Suppose is a linear map with . Let be the multilinear version of as in (2.1). Let . Now is a multilinear polynomial on of degree .
Β
For any fixed we have
[TABLE]
and thus
[TABLE]
It follows that is of bounded rank and thus of the form
[TABLE]
with of degree ,for any fixed also are of degree , and . For any fixed , is linear and thus either or are constant as a function of . Recall that , and let , similarly . Let be the set of in the sum such that is linear in and does not depend on . Let . The restriction of to has finite (that is by a constant which does not depend on ) rank. Since we see that has rank that is bounded by a constant which does not depend on and .
Β
Now let be infinite. Let be as in Lemma 2.6. Len . Now apply Lemma 2.5 for the collection of finite rank maps from , and the restriction as before. This finishes a proof of Theorem 1.4. β
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