3
votes
1answer
41 views

Computing the order of the first cohomology group $|H^1(S_n, \mathbb F_p^n)|$

Assume $n\geq 3$, $p$ is a prime, and that $S_n$ acts on $V=\mathbb F_p^n$ by permuting the basis vectors $v_1,\ldots, v_n$. I want to compute the order of the first cohomology group of this action. ...
0
votes
0answers
18 views

Exact sequence of linear spaces

While reading Nigel Higson's book Analytic K-homology i found the result (which was known to me earlier, but I never saw the proof) that the index of the product of two Fredholms operators is the sum ...
0
votes
0answers
94 views

Homology out of Smith normal form

Let $R$ be a PID and $A: R^m\rightarrow R^n$ and $B:R^n\rightarrow R^o$ with $BA=0$ and Smith normal forms $A=P\mathrm{diag}(a_1,\ldots,a_r,0,\ldots,0)Q^{-1}$ and ...
0
votes
0answers
21 views

Different definitions of connectedness of commutative cochain algebras

Let $(M,d)$ be a commutative cochain algebra over the rationals, that is a differential graded, graded commutative Algebra over $\mathbb{Q}$ concentrated in nonnegative degrees. In the literature, ...
1
vote
1answer
68 views

Matrices over a ring: does $PAQ=A'$ imply $\mathrm{Coker}A\cong\mathrm{Coker}A'$?

In A Singular Introduction to Commutative Algebra by Greuel & Pfister, there is written on p. 127: Let $R$ be a commutative unital ring and $A\in R^{n\times k}$, $P\in R^{n\times n}$, $Q\in ...
3
votes
1answer
101 views

Images in a short exact sequence

Suppose $$ 0\to V\to W\to X\to 0\\ \downarrow\quad\quad\downarrow\quad\quad\downarrow\\ 0\to V'\to W'\to X'\to 0\\ $$ is a commutative diagram of vector spaces, with the top and bottom rows short ...
4
votes
2answers
62 views

On chain complex morphisms

The following seems quite obvious to me. Nevertheless I would like to have another opinion. Suppose $(A_\bullet,d_A)$ and $(B_\bullet,d_B)$ are chain cmplexes, such that $d_A$ is the trivial ...
5
votes
1answer
431 views

Dimensions of vector spaces in an exact sequence

I've read the following formula in wikipedia: Given finite dimensional vector spaces $V_i$ and an exact sequence $\cdots\rightarrow V_i\rightarrow V_{i+1}\rightarrow\cdots$, we have $$ \sum_{n\in ...
2
votes
1answer
593 views

When is the pullback of a linear injection a surjection on dual space?

Due to the contravariance of the dual space functor on vector spaces, one might expect the pullback of an injection to be a surjection, and the pullback of a surjection to be an injection. Indeed, for ...
3
votes
2answers
277 views

limits of finite dimensional vector spaces

Let $A$ be a finite dimensional vector space, $\cdots \rightarrow A_{n+1}\rightarrow^{f_{n+1}} A_n \rightarrow^{f_{n}} A_{n-1}\rightarrow \cdots $ be an inverse system of finite dimensional vector ...