2
votes
0answers
37 views

Kernels of induced maps in cohomology

Let $k$ be a field, and let $A$ and $B$ be two commutative $k$-algebras. Suppose $\varphi,\psi:A\to B$ are maps of $k$-algebras such that there are algebra automorphisms $G:A\to A$ and $F:B\to B$ ...
6
votes
0answers
228 views

Another Algebraic de Rham Cohomology question…

NOTE: scroll down to read my latest edit first if you're reading this for the first time :) My aim is to calculate the de Rham cohomology of the variety $U = \text{Spec} \ A$, where: $$A = ...
7
votes
1answer
137 views

Does the rank of homology and cohomology groups always coincide?

Let $(C_i)_{i \in \mathbb{Z}}$ be a chain complex of free abelian groups. Does the rank of the homology and cohomology groups of $(C_i)_{i \in \mathbb{Z}}$ always coincide, i.e. is ...
3
votes
1answer
26 views

Generalisation of cochain complexes and “curvature”

Someone has mentioned to me that generalizations of co-chain complexes and their cohomology have been studied, where instead of $d^2 = 0$ we have something like $d^2 \alpha = q \alpha $, which is ...
2
votes
1answer
63 views

The cohomology ring of the nerve of a category associated to a vector space

Let $n\ge2$, and let $V$ be an $n$-dimensional vector space over a field $k$. Consider the category $\mathcal{C}$ whose objects are nonzero, proper subspaces of $V$, and whose morphisms are ...
1
vote
0answers
50 views

how is the jordan-holder theorem used in conjunction with short exact sequences to construct groups of certain order?

I am an undergraduate and have been asked to explain how simple groups can be used to construct groups of finite order. I started with reading about the extension problem in group theory and from ...
5
votes
1answer
101 views

(Co)homology of free symmetric algebra

Let $V$ be a (co)chain complex, and let $Sym(V)$ be the free differential graded-commutative algebra generated by $V$. Definition and examples below in case you don't know what I mean. Question: ...
3
votes
1answer
56 views

Why do we have that Hom is an exact functor in the situation described below?

We are given a finite $p$-group $G$ and a finite $G$-module $M$ such that $pM=0$ (therefore $M$ is in particular a $\mathbb{F}_p$-vector space). In addition we have an arbitrary $G$-module $N$ which ...
3
votes
0answers
51 views

For which categories do injections induce surjections in cohomology?

I'll ask a specific question first, but I believe my question might have a rather immediate abstraction, with which I'll finish. Let $H,G$ be finite affine group schemes over an algebraically closed ...
6
votes
1answer
57 views

Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
1
vote
1answer
41 views

Natural map of extension groups

Let $\Lambda$ be a cocommutative Hopf algebra over a commutative ring $R$. For two left $\Lambda$-modules $M$ and $N$, interpret $\mathrm{Ext}_{\Lambda}^n(M,N)$ as the set of equivalence classes of ...
2
votes
1answer
133 views

Definition/existence/uniqueness of a minimal projective resolution

I'm reading Dave Benson's book "Representations and Cohomology," Volume I, and I'm trying to understand the following discussion on page $32$ in which he introduces the notion of a minimal projective ...
2
votes
1answer
85 views

Understanding the Bockstein homomorphism in group cohomology

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a finite group. In group cohomology, the Bockstein homomorphism is the connecting homomorphism ...
0
votes
0answers
37 views

Uniqueness of the cohomological functor

This question is from the chapter 'Cohomology of Groups' by Atiyah and Wall in Cassels' and Frohlich's book 'Algebraic Number Theory'. Let $G$ be a group. Theorem 1 on page 95 says that there is a ...
1
vote
1answer
32 views

A G-isomorphism of certain Hom groups

This question is from 'Cohomology of Groups' by Atiyah and Wall, p.95 of Cassels' and Frohlich's book 'Algebraic Number Theory'. Let $G$ be a group and $A={\rm Hom}_{\mathbb Z}(\mathbb Z[G],X)$ where ...
2
votes
2answers
72 views

Cochains: terminology

Let a real, smooth manifold $M$ be given. Let $C_k(\mathbb Z, M$) denote the set of $k$-chains with integer coefficients, and let $C_k(\mathbb R, M)$ denote the set of $k$-chains with real ...
4
votes
1answer
61 views

Show that image of $res$ lies in $H^n(H,A)^{G/H}$

Let $G$ and $G^{\prime}$ be groups, $A$ and $A^{\prime}$ be $G$-module and $G^{\prime}$-module respectively, $C^n(G,A)$ be set of all maps from $G \times \cdots \times G$ ($n$ times) to $A$, $d_n ...
6
votes
0answers
141 views

Morita-invariance of Hochschild (co)homology.

Ok, I'm reading this paper by Christian Kassel on associative algebras and Hochschild (co)homology and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ ...
5
votes
0answers
66 views

Corestriction map in lie algebra cohomology

Given a lie algebra $\mathfrak{g}$ over a field $k$, we can define the cohomology groups of $\mathfrak{g}$ as follows: $$H^n(\mathfrak{g},k):=\mathrm{Ext}_{U(\mathfrak{g})}^n(k,k)$$ where ...
1
vote
1answer
80 views

Cohomological ($p$-)dimension of a pro-$p$ group

I have a question concerning the cohomological dimension and $p$-dimension of a pro-$p$-group. Let's first recall the definitions of that The cohomological dimension $cd \ G$ of a pro-finite group ...
7
votes
1answer
86 views

Finite generation of Tate cohomology groups

Let $G$ be a finite group, and let $F$ be a complete resolution for $G$. In other words, $F$ is an acyclic chain complex of projective $\mathbb{Z}G$-modules together with a map ...
3
votes
1answer
79 views

Extensions of $\mathbb{Z}_n$ by $\mathbb{Z}$

Given that $H^2(\mathbb{Z}_n,\mathbb{Z})=\mathbb{Z}_n$, it follows that up to equivalence there should be $n$ extensions of $\mathbb{Z}_n$ by $\mathbb{Z}$, one for each cohomology class. I'd like to ...
7
votes
1answer
132 views

Computing the action of $S_3$ on $H^n(\mathbb{Z}_3,\mathbb{Z})$

Let $G=S_3$ and let $H$ be the Sylow $3$-subgroup in $G$. If $\mathbb{Z}$ is the trivial module, then it can be shown that $$H^n(H,\mathbb{Z})=\begin{cases}\mathbb{Z}&n=0\\0&n\text{ ...
0
votes
1answer
73 views

Non abelian $H^1(G,A)$ problem.

Let $G,A$ be groups. We do not assume that $A$ is abelian. For $f,g\in Z^1(G,A)$, we write $f\backsim g$ if there is an $a\in A$ such that $g(x)=a^{-1}f(x)\ ^xa$ (we use the pre-exponential notation ...
0
votes
0answers
57 views

Exact sequence and Tor functor.

Say $M$ is an $R$-module and $\operatorname{gld}(R)=n$, i.e. global dimension of $R$ is n. Is then $\operatorname{Tor}_i^{R}(M,N)=0$ for any $i>n$ and $M,N$ any $R$-module? Is it possible to ...
5
votes
2answers
101 views

Question on $\mbox{Ext}^1$

I have 2 questions, one of them concerning the isomorphicity of quotient groups (rings), and the other is on $\mbox{Ext}^1$. It's pretty long, but somehow related to each other. So I just kinda put ...
1
vote
1answer
79 views

filtration on the (co)homology of a space from the filtration of a space

Fix $n\!\in\!\mathbb{N}$. Let $X$ be a topological space and $X_0\subseteq X_1\subseteq X_2\subseteq \ldots$ subspaces of $X$. Let $\iota_k:X_k\rightarrow X$ be the inclusion. Let ...
8
votes
1answer
126 views

“Inverse problem” for Brauer groups

This question is really just a curiosity, but I'm really interested in the answer. Given a field $K$, we can form the set$^*$ $Br(K)$ consisting of equivalence classes of finite-dimensional central ...
4
votes
1answer
96 views

Order of the first cohomology group and subgroups

Let $M$ be a $G$-module and $H$ a subgroup of $G$. Is $\# H^{1}(H, M) < \# H^{1}(G, M)$?
2
votes
1answer
276 views

Kunneth formula for cohomology

Why I can use Kunneth formula to say that $H^{*}(\mathbb{C}P^{\infty} \times \mathbb{C}P^{\infty})= \mathbb{C}[x_{1}] \otimes \mathbb{C}[x_{2}]$?
6
votes
0answers
124 views

Computing a specific direct limit

Suppose we have the following sequence of $\mathbb{Z}$-modules $$G\,\,\overset{M}{\longrightarrow}\,\, G\,\,\overset{M}{\longrightarrow}\,\, G\,\,\overset{M}{\longrightarrow}\,\, \cdots,$$ where each ...
1
vote
2answers
285 views

Conjugation action group cohomology

I have a question from Lang's Algebra (chapter twenty ex 6d). I think the vagueness confuses me as I am not even sure where to start If $H$ is a normal subgroup of $G$, we have the cohomology groups ...
41
votes
2answers
1k views

Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres

So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
1
vote
2answers
183 views

cohomology of a finite cyclic group

I apologize if this is a duplicate. I don't know enough about group cohomology to know if this is just a special case of an earlier post with the same title. Let $G=\langle\sigma\rangle$ where ...
10
votes
2answers
386 views

Does every Poisson bracket on a commutative algebra come from a second-order deformation?

Let $A$ be a commutative algebra over a field $k$ (of characteristic not equal to $2$ to be safe). Recall that $f : A \otimes A \to A$ is a Hochschild $2$-cocycle if it satisfies $$f(ab, c) + f(a, b) ...
10
votes
1answer
338 views

Finite groups with periodic cohomology

I'm trying to understand Chapter 12, Section 11 in Cartan + Eilenberg's Homological Algebra, which concerns finite groups with periodic cohomology. Unfortunately I am jumping right to this section in ...
3
votes
3answers
470 views

What is the connection between Grothendieck's Differential Operators and Hochschild Cohomology

For a given commutative algebra $A$ over a field $\mathbb{K}$(with char=0) the algebra of differential operators on $A$ is the set of endomorphism $D$ of $A$ such for some $n$ we have that for any ...
2
votes
3answers
438 views

Acyclic vs Exact

I have a question about the words "acyclic" and "exact." Why does Brown use the term "acyclic" instead of "exact" in his book Cohomology of Groups? It seems to me that these two terms exactly ...
6
votes
2answers
2k views

Tensors as matrices vs. Tensors as multi-linear maps

So I read the answers in this question, and don't feel that much closer to an answer about how tensors as multi-linear maps and tensors as "multi-dimensional" matrices are truly related. For ...