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 $F:B\to B$ ...
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 ...
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 ...
3
votes
1answer
38 views

Existence of Boundary Homomorphisms for Cohomology

I am just starting to learn the basics of cohomology and am confused about the construction of the cohomology groups. So given a group $G$, the idea is you take a projective resolution of $P_0 = ...
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 $R$ ...
1
vote
0answers
64 views

Easy characterization of Cohomology in an Abelian Category

It should be quite an easy question and probably there's also a certain degree of intrinsic silliness in it, but still... Let $\mathcal{C}$ be an abelian category and let $C(\mathcal{C})$ be the ...
3
votes
1answer
57 views

Proof that derived functors don't depend on choice of resolution.

Can somebody help me out with this? Let $X$ be an object in an abelian category $A$ with enough injectives, let $0 \rightarrow X \rightarrow M^{\bullet}$ be an injective resolution , let $0 ...
0
votes
1answer
81 views

Are the hom sets in the category of varieties abelian groups?

This is supposedly (though I know of the proof bud haven't read it) for the Hom sets of noetherian schemes. Since every variety can be thought of as a noetherian scheme then it seems right... when ...
2
votes
2answers
142 views

Equivalence of categories and derived functors.

Don't know if this kind of a dumb question but let $A$ and $B$ be abelian categories and suppose they're equivalent: there are two functors $P: A \rightarrow B$ and $Q: B \rightarrow A$ satisfying the ...
4
votes
1answer
69 views

What is the cohomological explanation for the Condorcets voting paradox?

according to the nlab entry on the Condorcet Paradox in social choice (that is voting preferences may be circular even if voters preferences are not) has a cohomological explanation - what is it?
15
votes
3answers
1k views

How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient Ker d / Im d where d as usual denotes the differentials, indexes skipped for simplicity. How ...