# 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.

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### First homology group of the general linear group

The abelianization of the general linear group $GL(n,\mathbb{R})$ defined by $GL(n)^{ab} := GL(n)/[GL(n), GL(n)]$ is isomorphic to $\mathbb{R}^{\times}$. This follows from the fact that $[GL(n),GL(n)] ...

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### Hochschild cohomology of the $A_\infty$-category of paths

I would like to describe the Hochschild cohomology (in the sense of $A_\infty$-categories) of the following $A_\infty$-category associated to a topological space $X$:
It has points of $X$ as objects.
...

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28 views

### Cartan determinants of minimal Auslander-Gorenstein algebras

Iyama and Solberg introduced minimal Auslander-Gorenstein algebras as algebras having finite dominant dimension ($\geq 2$) equal to the Goreinstein dimension in https://www.sciencedirect.com/science/...

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54 views

### Stable equivalence and stable Auslander algebras

Let $A$ be a representation-finite finite dimensional quiver algebra and $M$ the basic direct sum of all indecomposable $A$-modules.
Recall that the Auslander algebra of $A$ is $End_A(M)$ and the ...

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54 views

### Derived equivalence between two exotic algebras

Let $A$ and $B$ be two connected finite dimensional quiver algebras having the same underlying quiver.
Question 1:
In case $A$ and $B$ have exactly one indecomposable projective non-injective $A$-...

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**1**answer

235 views

### Example of a ring where every module of finite projective dimension is free?

I'm interested in seeing an example of a ring which is not self-injective where every module admitting a finite projective resolution is free, or at least projective.
Note that self-injectivity says ...

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75 views

### Stability property for differential graded algebras

For a short exact sequence $1 \to A \stackrel{\iota}{\to} B \stackrel{\pi}{\to} C \to 1$ we have the situation that the square
is both a pushout in groups and also a pullback.
This would be wrong if ...

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32 views

### Graded commutative PBW bases

A Poincaré–Birkhoff–Witt (PBW) basis is a particularly nice basis of a quadratic algebra that can be used to prove that it is Koszul (see Priddy's 1970 paper "Koszul resolutions", Trans. Amer. Math. ...

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### Relation between maximal simplices and homology group [closed]

Can I say that only maximal simplices in a simplicial complex determine the homology group? If not what's wrong?

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153 views

### Homotopy quotient of groups

Suppose $0\to A \stackrel{\iota}{\to} B \stackrel{\pi}{\to} C \to 0$ is a short exact sequence of groups.
We have an induced map $k[\iota] : k[A] \to k[B]$ of group algebras over a field $k$.
What ...

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**1**answer

280 views

### Abstract proof that $\lvert H^2(G,A)\rvert$ counts group extensions

(This question is originally from Math.SE, where it didn't receive any answers.)
$\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\ext}{...

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### Bounds for the finitistic dimension

The finitistic dimension of an algebra is defined as the supremum of all projective dimensions of modules having finite projective dimension.
For finite dimensional algebras $A$ with radical cube ...

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53 views

### Two definitions of minimal models

Is there any relationship between both definitions of minimal models? (the couple of definitions I know are the one mentioned in Lefèvre's thesis, in the sense that the differential is zero, and the ...

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**4**answers

295 views

### Homology of solvable (nilpotent) Lie algebras

Let $\mathfrak{g}$ be a solvable Lie algebra over $\mathbb{C}$ and $\lambda\in(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be a character of $\mathfrak{g}$. I'm interested in calculating homology for ...

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### Decomposability of chain complexes

The following is stated in Luc Illusie, "Frobenius and Hodge degeneration", part 4.6.
Let $L$ be a bounded chain complex. There is a sequence of obstructions, first $c_i\in \mathrm{Ext}^2(H^iL, H^{i-...