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.
3
votes
0answers
152 views
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)] ...
6
votes
0answers
93 views
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.
...
3
votes
0answers
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/...
4
votes
0answers
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 ...
4
votes
0answers
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 $A$-...
8
votes
1answer
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 ...
8
votes
0answers
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 ...
3
votes
0answers
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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votes
0answers
33 views
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?
6
votes
0answers
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 ...
7
votes
1answer
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}{...
3
votes
0answers
41 views
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 ...
1
vote
0answers
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 ...
4
votes
4answers
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 ...
3
votes
0answers
74 views
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-...
