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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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      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 ...
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      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 ...
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      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
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      1answer
      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 ...
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      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 ...
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      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 $M$ ...
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      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 ...
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      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 ...
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      0answers
      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 ...
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      1answer
      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
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      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. ...
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      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 ...
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      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
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      0answers
      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 $\...
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      vote
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      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^+...

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