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        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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        votes
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        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)] ...
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        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/...
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        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 ...
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        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 $A$-...
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        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 ...
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        votes
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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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        votes
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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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        votes
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        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 ...
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        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}{...
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        votes
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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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        vote
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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 ...
        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 ...
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        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-...

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