## New answers tagged homological-algebra

2
votes

### Tensor product of a DGA and an $A_\infty$ algebra

To follow up on Dan's answer, a geometric construction of the Saneblidze--Umble diagonal is given by Masuda--Tonks--Thomas--Vallette in http://www.numdam.org/item/JEP_2021__8__121_0/, and an ...

4
votes

### If a bimodule is "generated" by single elements, must the elements be conjugate?

No.
Let $k = \mathbb{R}$ and $A = B = \mathbb{C}$. We can identify $A \otimes B \cong \mathbb{C} \times \mathbb{C}$. On pure tensors, this isomorphism takes $a \otimes b \mapsto (ab, a\bar{b})$. ...

5
votes

Accepted

### FI-homology of a spectral sequence of rational FI-modules

The work of Church-Ellenberg that you linked to bounds $t_k(M)$ in terms of $t_0(M)$ and $t_1(M)$.
This paper of Church-Reinhold-Miller-Nagpal introduces two invariants: the stable degree $\delta(M)$ ...

10
votes

Accepted

### Reference request: locally erasable delta-functor is universal

This is Proposition 4.2 in Buchsbaum’s Satellites and universal functors, Annals of Mathematics 71(2), pp. 199–209 (1960).
Well, to be precise, that is the dual result (for contravariant functors). ...

0
votes

### Arithmetic application: Complete group ring and group ring for infinite group

I am not sure precisely what your question is asking, but I have found that the old paper of Brumer, Pseudocompact algebras, profinite groups and class formations, J. Alg, 4, (1966), 442–470, link, ...

1
vote

### Is there a finite dimensional algebra with left finitistic dimension different from its right finitistic dimension?

Consider the algebras $A_n$ given by the following quiver:
$$
\circlearrowright 1 \to 2 \to \cdots \to n-1 \to n
$$
with the relation that the composition of any two arrows is zero, i.e., $r^2 = 0$.
...

6
votes

Accepted

### If the Hom-space of finite length modules is generated by single elements, must the elements be conjugate?

This isn't even true for vector spaces over a field. Let $A=k$ be a field. Let $X$ and $Y$ be two dimensional vector spaces, and let $f$ be the homomorphism $\left(\begin{smallmatrix}1&0\\0&0\...

2
votes

Accepted

### Does a crossed extension with trivial Postnikov invariant admit a section?

No. For $n>1$ you can always choose a representative with $E_n$ a free non-Abelian group, so there cannot be a section unless $G$ is free, and in that case the cohomology is trivial and all cross ...

2
votes

Accepted

### Could I get an interpretation for application of Euler characteristics in number theory？

If $\rho$ is a Galois representation of geometric origin and if you consider a cohomology complex $C$ computing Galois cohomology satisfying (supplementary cleverly) defined arithmetic conditions, ...

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