## New answers tagged homological-algebra

5
votes

### Derived Hom without injectives nor projectives

Some construction of derived Hom complexes in an arbitrary $k$-linear Quillen exact category (for any commutative ring $k$) is worked out in the appendix to my paper "Artin-Tate motivic sheaves ...

4
votes

### Finding exceptional regular representations of $\tilde{D}_4$ efficiently

The AR quiver of the regular representations of an affine quiver consists of infinitely many "tubes". A tube of rank $r$ has $r$ modules on what you call the border. Let me number them $B_1, ...

1
vote

### Finiteness of $\ell^2$-Betti numbers

(This is more of a comment/sketch than a full answer but it seemed clearer to write things here.)
To remove some clutter let me write $A$ for ${\mathcal D}_{{\mathbb F}K}$. Then paraphrasing the ...

0
votes

### Explicitly calculating the homotopy fiber of a 3-cocycle in the category of simplicial sets

For anyone interested, I've found a solution. It is the central construction in https://arxiv.org/abs/2203.11392.

6
votes

Accepted

### Geometric interpretation of shuffle product

Perhaps this is more naïve than you are looking for, but here is one interpretation: one representation of an $n$ simplex is the following
$$\Delta_n=\{(t_1,t_2,\dotsc,t_n)\mid 0\leq t_1\leq t_2\leq\...

3
votes

Accepted

### Does the category of commutative and cocommutative Hopf algebras have enough injectives?

Over a field $k$, the answer is yes for injectives; I'm not sure about projectives. Over $\mathbb Z$ or other commutative rings, I really don't know -- the use of the fundamental theorem of coalgebra ...

4
votes

### Yoneda extensions in derived categories

There is such a sequence, but it's not very interesting.
Given an element of $\text{Hom}_{D^b(\mathcal{A})}(E,F[i])$, then in the same way you describe, this gives a distinguished triangle
$$F\to Z_{i-...

4
votes

### What are Koszul dualities?

I have finally found a source which puts together the pieces in a satisfactory way, at least in the stable setting, here:
Amabel, Araminta. "Poincaré/Koszul Duality for General Operads." ...

Community wiki

1
vote

### A three-line proof of global class field theory?

This is a late answer, but I would recommend that anyone interested in Dmitry's heuristic argument read "Notes on etale cohomology of number fields," by Barry Mazur.
Mazur's article is an ...

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