# Questions tagged [homological-algebra]

(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.

1,724
questions

**2**

votes

**0**answers

58 views

### Cohomology of sheaves on $X \cup_{Z} Y$

I am in the following situation, I have two schemes $X$, $Y$ and two closed immersions $Z \rightarrow Y$, $Z \rightarrow X$. Everything is smooth. I am interested in calculating morphisms in the ...

**3**

votes

**0**answers

48 views

### Auslander-Solberg algebras from non-rigid modules

Let $A$ be a Nakayama algebra and $M$ be the direct sum of all indecomposable $A$-modules $N$ with $Ext_A^1(N,N) \neq 0$.
The following is suggested by computer experiments with QPA:
Question: Is ...

**2**

votes

**2**answers

202 views

### Motivation/intuition behind the definition of delta-functors and related concepts

I originally posted this on Maths SE, but then realised that the question probably fits MO better, as my objective was to gain different perspectives regarding the matter.
Why are $\delta$-functors ...

**2**

votes

**1**answer

111 views

### Combinatorial problem on periodic dyck paths from homological algebra

edit: I added conjecture 2 that looks much more accessible.
Here is the elementary combinatorial translation of the problem (read below for the homological background):
Let $n \geq 2$.
A Nakayama ...

**1**

vote

**0**answers

64 views

### Eventually non vanishing tors

Let $A$ be a commutative $k$-algebra, for $k$ a field of characteristic $0$. Let $Perf_{A}$ denote the dg category of cohomologically graded $A$-modules and let $M\in Perf_{A}$ be a classical perfect ...

**2**

votes

**0**answers

72 views

### Free DGA given a map and cohomology groups

Why do Free DGAs on a morphism often give the same (co)homology as other (co)homologies?
Here is the example that comes to mind first:
Example: Let $R$ be a ring, let $A$ be an $R$-algebra, let $M$ ...

**2**

votes

**0**answers

41 views

### Selfinjective algebras with loops

Given a selfinjective finite dimensional algebra $A$ with an indecomposable module $M$ with $Ext_A^1(M,M) \neq 0$.
Question:
Is A derived equivalent to an algebra with a loop in the quiver in ...

**2**

votes

**0**answers

42 views

### Strong no loop conjecture for uniserial modules

Let $A$ be a an Artin algebra. The strong no loop conjecture states that a simple $A$-module with $Ext_A^1(S,S) \neq 0$ has infinite projective dimension.
This conjecture was recently proved for ...

**2**

votes

**0**answers

31 views

### On monomial and $\Omega^d$-finite algebras

Let $Q$ be a finite quiver and $I$ a monomial admissible ideal of the path algebra $KQ$ for a field $K$. Then an algebra $A=KQ/I$ is called a monomial algebra.
It is well known that monomial algebras ...

**6**

votes

**1**answer

187 views

### Why does every chain complex have a map into its cone?

In Weibel's An introduction to homological algebra he defines a cone as an explicit chain complex associated to the given one -i.e. for a chain $C=(C_i, d)$ he defines $Cone(C)=\left(C_{i-1} \oplus ...

**1**

vote

**0**answers

130 views

### What does “standard Koszul morphism” mean?

I'm reading a paper 'D'Andrea, Carlos(RA-UBA), Dickenstein, Alicia(RA-UBA)Explicit formulas for the multivariate resultant. (English summary)
Effective methods in algebraic geometry (Bath, 2000).
J. ...

**0**

votes

**0**answers

94 views

### Equivalence of the category of covariant functors and the category of contravariant functors

Let $\mathcal{C}$ be a category. Then we have the category $\mathcal{C}^{\vee}$ of contravariant functors from $\mathcal{C}$ to $\mathcal{Sets}$ which is the category of sets. In the textbook "Sheaves ...

**1**

vote

**0**answers

33 views

### Checking $\mathbb{K}_{U\times(a,b)}\ast\mathbb{K}_{[0,\infty)}\simeq \mathbb{K}_{U\times[a,\infty)}[-1]$ in derived category $D(X\times\mathbb{R})$

Let $D(X\times\mathbb{R})$ be the derived category of sheaves of $\mathbb{K}$-vector spaces on a smooth manifold $X\times\mathbb{R}$ where $\mathbb{K}$ is a ground field. Let $p_1:X\times\mathbb{R}\...

**1**

vote

**0**answers

33 views

### Computing injective resolution of some constant sheaves

I follow the notations on "Sheaves on manifolds" written by Kashiwara-Schapira. Let $\mathbb{K}$ be a ground field and $X$ be a smooth manifold. Let $D(X)$ be the derived category of sheaves of $\...

**1**

vote

**0**answers

49 views

### $\mathrm{Ext}^1$-ordering on ${}^IW^{\Sigma_\mu}$

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$.
Let $W$ be the associated Weyl group and let $\Phi$ be its root system.
We write $\Phi^+...