# 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.

1,621 questions
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### Quadratic algebras and Koszul algebras

Let $A$ be a quadratic algebra and $B$ the Ext-algebra of $A$. In case $A$ is a Koszul algebra, we should have that the global dimension of $A$ plus one is equal to the Loewy length of $B$ (is there a ...
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### Decomposition of bordism groups for $BG$ where $G$ is a product of two groups

Let a group $G=G_1 \times G_2$, where $G_1$ is a discrete group (can be finite or infinite), $G_2$ be any compact Lie group or finite group. Question: Is there some simple result that we can ...
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### Do we have criteria of strict localization of a Grothendieck category?

Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under ...
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### How to show mapping cones are homotopy cofibers

In a dg-category $\mathcal{C}$, the $n$-translation of an object $C$ is an object $C[n]$ representing the functor $${\rm Hom}(-,C)[n].$$ The cone of a closed morphism $f\colon C \to D$ of degree ...
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### Length 2 modules over finite dimensional algebras

Given a finite dimensional algebra $A$ over an infinite field and two simple modules $S,T$. Question 1: Is there a useful (homological/computational) crtierion to decide when there are infinitely ...
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### Projective dimension of graded modules

Short version: Why is the projective dimension of a graded module the same as the projective dimension of its underlying ungraded module? Longer version: Let $G$ be a commutative group, let $R$ ...
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### $\Omega^2(S) \cong \tau(S)$ for simple modules

Let $A$ be an Artin algebra with $\Omega^2(S) \cong \tau(S)$ for each simple module $S$. Is $A$ a quasi-Frobenius (=selfinjective) algebra? Here $\tau$ denotes the Auslander-Reiten translate, which is ...
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### DG functors along which contractions can be lifted

For an object $X$ in a DG category, its contraction is $r \in Hom^{-1}(X,X)$ such that $d(r)=1_X$. Let us say that contractions lift along a DG functor $F: \mathcal{C} \to \mathcal{D}$, if for a ...
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### Reference for isomorphism $Ext_A^n(X,Y) \cong Ext_A^1(\Omega^{n-1}(X),Y)$

Let $A$ be a finite dimensional algebra and $X,Y$ indecomposable modules and $n \geq 2$. We know that $Ext_A^n(X,Y) \cong Ext_A^1(\Omega^{n-1}(X),Y)$. Now such an isomorphism should be given by ...
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### Hochschild homology with coefficients in a certain bimodule

Let $A$ be a finite-dimensional $k$-algebra and $U$ and $V$ two finite-dimensional projective $A$-modules (maybe neither the finiteness nor projectivity has to play a role, but these requirements are ...
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### Is there a homological interpretation for the cokernel of the kernel of a map between complexes induced by tensor product?

Let $A$ be a free abelian group of rank 2, and let $S = \mathbb{Z}[A]\cong\mathbb{Z}[a_1^{\pm1},a_2^{\pm1}]$ the group algebra for $A$. Let $t : S\times S\rightarrow S$ be the $S$-module map given by ...
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### Questions on group and Nakayama algebras from a book

Recall that a Nakayama algebra (also called serial algebras in the literature) is an algebra such that every indecomposable module has a unique composition series. In the book "Classical artinian ...
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### Generalized Postnikov square

Following Wikipedia (https://en.wikipedia.org/wiki/Postnikov_square), a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, ...
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### Künneth formulas/theorem for bordism groups and cobordisms?

We are familiar with Künneth theorem: The Kunneth formula is given by $R$ as a ring, $M,M'$ as the R-modules, $X,X'$ are some chain complex. The Kunneth formula shows the cohomology of a chain ...
Recall that an object $a$ in a symmetric monoidal category $(\mathcal{C}, \otimes, e)$ is dualizable if there exists an object $b$ and morphisms $\varepsilon\colon b \otimes a \to e$ and $\eta\colon e ... 2answers 296 views ### When is$\Omega^1$an equivalence? Let$C$be an abelian category with enough projectives and$\underline{C}$the stable category of$C$that is obtained by factoring out projective modules. When is the functor$\Omega^1 : \underline{...
Consider the Goodwillie calculus of a homotopy functor $F : \mathrm{Sp} \to \mathrm{Sp}$, where $\mathrm{Sp}$ denotes an appropriate model for spectra (... orthogonal spectra for instance). Then ...