# 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### 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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### 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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### 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}\...

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### 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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### $\mathrm{Ext}^1$-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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### Verma module and vanishing of extension groups

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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### Bridgeland stability for restricted Kahler moduli?

Let $X$ be a simply-connected, smooth, projective Calabi-Yau threefold. To my understanding, Bridgeland introduced stability conditions on triangulated categories to give a proper mathematical ...

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### Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'

It appears to me (though I may be wrong) that the common opinion is that the main difference between two is that Rosenberg's version of noncommutative algebraic geometry mainly concerns as ...

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### Are hearts of all $t$-structures on smashing triangulated categories closed with respect to coproducts (also)?

Let $T$ be a triangulated category closed with respect to (small) coproducts, and $t$ be (an arbitrary!) a $t$-structure on $T$. I have noted that the heart $\underline{Ht}$ of $t$ is closed with ...