# 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.

1,947
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### On grades of torsion modules in noetherian rings

Let $A$ be a (not necessarily commutative) two-sided noetherian ring with minimal injective coresolution $(I_i)$ of the regular module $A$ as a right module.
Say that $A$ has dominant dimension $n$ ...

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67 views

### A cyclic-like chain complex

Let $V$ be a $\mathbb{Z}$-graded vector space over an algebraically closed field $k$ of characteristic zero.
Let $\overline{TV}= \bigoplus_{n=1}^{\infty} V^{\otimes n}$ be the reduced tensor algebra....

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116 views

### Reference for homotopy colimit = total complex

I'm looking for a reference for the following fact:
take a simplicial chain complex $ X:\Delta^{op}\to Ch_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough ...

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112 views

### Definition of the Yoneda Ext

Let $\mathcal{A}$ be an abelian category and let $X$ and $Y$ be objects in $\mathcal{A}$. The Yoneda $\text{Ext}^{n}(Y,X)$ is defined by the following:
First we consider the class $\text{E}^{n}(Y,X)$ ...

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102 views

### Does there always exists a locally free resolution of quasi-coherent sheaves on quasi-projective noetherian scheme?

We consider a quasi-projective noetherian scheme. It is well known that for a coherent sheaf we can construct a sheaf resolution of locally free of finite rank. It is introduced in Hartshorne chapter ...

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86 views

### Is the derived category of local systems equivalent to the derived category of sheaves of vector spaces with local system cohomology?

Let $k$ be a field and $X$ a topological space.
Write $\mathrm{Sh}(X)$ for the category of sheaves of vector spaces on $X$, and $\mathrm{Loc}(X)$ for the subcategory of local systems of finite ...

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110 views

### Action on group $\operatorname{Ext}^i(\mathcal{L}, \mathcal{M})$ by scalar multiplication

Let $X$ be a proper scheme over field $k$ and $\mathcal{L}, \mathcal{M}$ two invertible $\mathcal{O}_X$-modules. Then $Hom_{\mathcal{O}_X}(\mathcal{L}, \mathcal{M}) \cong Hom_{\mathcal{O}_X}(\mathcal{...

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98 views

### On inequality between number of generators of ideals

Let $(R, \mathfrak m,k )$ be a regular local ring of dimension $3$ with infinite residue field $k$. Let $I$ be an $\mathfrak m$-primary ideal such that for every ideal $J$ containing $I$, it holds ...

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107 views

### For an additive category $\mathcal{A}$, how does one show $K_0(\mathcal{A})\cong K_0(\mathcal{K}^b(\mathcal{A}))$?

This is an exercise in §3.13 Beilinson's notes on homological algebra. He doesn't specify but I'm pretty sure $K_0(\mathcal{A})$ is defined as the free group on the isomorphism classes of $\mathcal{A}$...

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119 views

### Bar-cobar for topological spaces

Let $(A, \{m_k\}_{k \geq 1} )$ be an $A_{\infty}$ algebra over a field $k$. Recall that this is the data of a $\mathbb{Z}$-graded $k$-vector space, along with a collection of $k$-nary operations which ...

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194 views

### Ideals generated by regular sequences

In Vasconcelos' paper (Ideals generated by R-sequences), he proved
If $R$ is a local ring, $I$ an ideal of finite projective dimension, and $I/I^2$ is a free $R/I$ module, then $I$ can be ...

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290 views

### Are projective modules over a certain localised Laurent polynomial ring free?

Let $R=\mathbb{Z}[t^{\pm 1}]$ be the ring of Laurent polynomials, and let $S \subset R$ be the multiplicative subset generated by the polynomial $t-1$. I am interested in the ring $S^{-1}R=\mathbb{Z}[...

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### Reflexive vs. pseudo-coherent abelian groups

Recall that a module M over some ring R is pseudo-coherent if it admits a resolution whose terms are finitely generated projective R-modules. Such a module is reflexive when regarded as an object in ...

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364 views

### When does QCoh have 'enough perfect complexes'?

Let $X$ be a derived fpqc stack on the $\infty$-category of connective spectral affine schemes $\mathbf{Aff}^{\mathrm{cn}}=(\mathbf{Ring}^{\mathrm{cn}}_{E_\infty})^{\mathrm{op}}$, that is to say, a ...

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120 views

### Splitting of short exact sequence in the category of finitely generated modules over a commutative Noetherian ring

In the category of finitely generated modules over a commutative Noetherian ring, the splitting of a short exact sequence can be checked locally at the maximal ideals of the ring. One reference for ...

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### Removing half of H_1 of a 2-manifold through a 3-dimensional bounding manifold

Let $M$ be a 2-manifold. If $X$ is a 3-manifold such that $\partial X = M$, we know that the image of the boundary map $H_2(X, M) \rightarrow H_1(M)$ has rank equal to one half the rank of $H_1(M)$.
...

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119 views

### Almost split sequences coming from bimodules

Let $A$ be a finite dimensional algebra with enveloping algebra $A^e$.
Auslander and Reiten proved in "On a theorem of E. Green on the dual of the transpose" that
$Hom_A(Tr_{A^e}(A),M) \cong \tau(M)$ ...

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44 views

### Bimodule isomorphism for representation-finite blocks of the Schur algebra

Let $A$ be a representation-finite block of a schur algebra with $n \geq 2$ simple modules. Then the global and dominant dimension of $A$ are equal to $g=2n-2$. You can find quiver and relations for ...

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151 views

### Categories of modules generated under coproducts by a small set?

Question 1: For which rings $R$ does there exist a small set $S \subseteq Mod_R$ such that every module $M \in Mod_R$ is a direct sum of modules in $S$?
Equivalenty, for which rings $R$ does there ...

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224 views

### Model categories and chain complexes

I'm fairly new to thinking about homological algebra and chain complexes in their own right, i.e outside of isolated examples such as for constructing simplicial homology, or for computing $Ext$ ...

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### Invariance under derived equivalence of a Gorenstein projective bimodule

A module $M$ over an finite dimensional algebra $A$ is called Gorenstein projective in case there exists an exact complex $(P_i)$ of projective $A$-modules such that the complex stays exact after ...

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66 views

### Higher analogue of the Auslander-Bridger transpose

Let $A$ be an Artin algebra and $M$ a module with $Ext^i(M,A)=0$ for $i=1,...,n-2$.
Then in case $P_{n-1} \rightarrow ... \rightarrow P_0 \rightarrow M \rightarrow 0$ is the beginning of a minimal ...

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134 views

### Convergence of a spectral sequence of a double complex

In Weibel's book, a spectral sequence $E^r_{p,q}$ is said to weakly converge to a graded object $H_{\ast}$ if for every $n$ there exists a filtration $\dots \subset F_{r}H_{n} \subset F_{r-1}H_{\ast} \...

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44 views

### intersection of left and right orthogonals of a module

$\DeclareMathOperator\Ext{Ext}$Let $(R, m)$ be a commutative Noetherian Gorenstein local ring and let $M$ be an $R$-module. Let
${^\perp M}$ and $M^\perp$ be respectively the left and the right ...

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912 views

### Has anyone seen this generalization of the snake lemma? Is it useful?

I originally posted this question on MSE (link), but was suggested to post here instead.
While learning about spectral sequences a friend of mine found a proof of the snake lemma using spectral ...

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624 views

### Is the tensor product of chain complexes a Day convolution?

Recently, Jade Master asked whether the tensor product of chain complexes could be viewed as a special case of Day convolution. Noting that chain complexes may be viewed as $\mathsf{Ab}$-functors from ...

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205 views

### Kan extensions between categories of monoid objects

Let $K\colon\mathcal{A}\longrightarrow\mathcal{B}$ be
a functor between $\mathcal{V}$-enriched categories. Endowing $\mathcal{A}$ and $\mathcal{B}$ with promonoidal
structures, we obtain induced ...

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55 views

### Question on a subcategory being extension-closed

In the article "Homological theory of noetherian rings" by Idun Reiten from 1996, it was stated that it seems to be not known whether the subcategory $\operatorname{Tr}(\Omega^i(\mathrm{mod}\text{-}A))...

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65 views

### Simple, explicit, functorial cylinder object in CDGA

In the model category of graded commutative dg-algebras CDGA over $\mathbb{Q}$ (with the projective model structure) there is a simple, functorial construction of a path object given by tensoring with ...

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250 views

### B-model and Hochschild cohomology

In "On the Classification of Topological Field Theories" in Example 1.4.1, Lurie introduces the B-model with target an (even dimensional) Calabi-Yau variety $X$: The Hochschild cohomology $\...

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### Mysterious inequality for the homological dimension of modules

Let $A$ be an Artin algebra and $M$ an indecomposable $A$-module.
Let $pd(M)$ denote the projective dimension of $M$, $id(M)$ the injective dimension of $M$, $domdim(M)$ the dominant dimension of $M$ ...

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235 views

### Detecting weak equivalence on free loop space homology

Given $f:X \to Y$ a continuous map between two spaces (unpointed CW-complexes) such that $f$ induces an isomorphism in homology with integer coefficient, and $f$ induces an isomorphism on homology of ...

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237 views

### Isomorphism for Ext spaces for finite dimensional algebras

Let $A$ be an Artin algebra with enveloping algebra $A^e$.
Then we have $Hom_{A^e}(X,A^e) \cong Hom_A(D(A) \otimes_A X,A)$ for a bimodule $X$. (see for example in the article "A theorem of Green on ...

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156 views

### Universal property for derived category of coherent sheaves

Let $X$ be a scheme, and let $D^{*}(X)$ be the unbounded (resp. unbounded, resp. bounded below/above, etc) derived category of coherent sheaves on $X$.
The work of Robalo establishes a universal ...

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165 views

### On properties of an algebra as a bimodule

Let $A$ be a two-sided artinian ring.
Recall that a module $M$ is said to have dominant dimension at least $n$ in case the terms $I_i$ in the minimal injective coresolution of $M$ are projective for $...

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99 views

### Characterisation of minimal projective resolutions via the Euler characteristic

Let $A$ be a finite dimensional $K$-algebra (where $K$ is a field) and $M$ a finitely generated $A$-module.
Let $\psi: 0 \rightarrow P_r \rightarrow ... \rightarrow P_0 \rightarrow M \rightarrow 0$
...

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194 views

### Freyd-Mitchell for $k$-linear categories

I don't know much about the proof of the Freyd–Mitchell embedding theorem and I could not find an answer to my question looking naïvely online, but at the same time I feel like this is the kind of ...

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61 views

### Applying a Hochschild cocycle to a Maurer-Cartan element: how one should think of this?

Let $C^{\bullet}(A,M)$ be the Hochschild cochain complex of a DG-algebra $A$ with coefficients in a DG-bimodule $M$. Let $\zeta \in C^0(A,M)$ be a cocycle. Let $a \in A$ be a Maurer-Cartan element, $d(...

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142 views

### Hochschild homology of acyclic complex

Let $A$ be a differential graded algebra over a commutative ring $R$. Suppose that $H_*(A)=0$, i.e. $A$ is acyclic.
Question: Does this imply that the Hochschild homology $HH_*(A)$ also vanishes ...

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225 views

### Skew differential graded algebra

A sigma, or skew, derivation is a natural generalisation of the
notion of derivation depending on an algebra automorphism $\sigma$ which
when equal to $id = \sigma$ reduces to the usual notion of a
...

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151 views

### Sullivan minimal model in the case of $H^1(V)\neq 0$

Is there a simple construction of a Sullivan minimal model $\Lambda U \rightarrow V$ in the case that $H^1(V)\neq 0$? Do you have a reference? I envisage a degree-wise construction as in the case of $...

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### Yoneda Extensions in derived categories

If given an abelian category $\mathcal{A}$, we can consider the bounded derived category $D^b(\mathcal{A})$. For two objects $A,B \in \mathcal{A}$, we know that there is a natural identification ...

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315 views

### idea and intuition behind triangulated category [closed]

I have some trouble in understanding the significance of some axiom of triangulated category.
if someone could explain me each axiom with some intuition,and explain me the intuition behind the ...

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139 views

### On tilting and cotilting modules

Let A be an Artin algebra and assume all modules are basic, then a classical result says that tilting modules $T$ are in bijection with complete cotorsion pairs $(T^{\perp}, \check{ add(T)})$ (with ...

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### Functoriality of Hochschild cohomology for Drinfeld quotients

Let $C$ be a dg category and $C \to D$ a Drinfeld localization. Is there an induced pushforward map on $\operatorname{HH}^*(C) \to \operatorname{HH}^*(D)$, where $\operatorname{HH}^*$ denotes the ...

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165 views

### Cohomology of constructible sheaves via exit paths

Let $X$ be a stratified space, with stratification $S$ (we will ignore technicalities).
The category of exit paths $Ex(X,S)$ is a directed refinement of the path groupoid of $X$ accounting for the ...

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103 views

### A infinity structure on Yoneda Ext group

I am currently trying to control an $A_\infty$-algebra of the form $\mathrm{Ext}_X(F\oplus F'[2n-2],F\oplus F'[2n-2])$ where $X$ is a nice enough scheme and $F,F'$ are sheaves that are NOT locally ...

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189 views

### Is $Tor_A(k,k)$ a bicommutative Hopf algebra?

Let $A$ be a commutative (or graded commutative) algebra over a field $k.$ In some sources, such as Mcleary's book on spectral sequences, Corollary 7.12, pg. 248, it is claimed that $\text{Tor}_A(k,k)$...

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94 views

### Adjoining data about singularities to “correct” the category of pure motives?

There are a few well known constructions of potential categories of pure motives for smooth projective varieties over a field. My understanding is that modulo the standard conjectures these should be ...

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**1**answer

73 views

### Lie algebra cohomology: $H^i(R,V)=H^i(R,V^R)$ with $R$ reductive and $V$ an $R$-module

Let $R$ be a reductive, finite-dimensional Lie algebra over a field of characteristic 0, and let $V$ be a semisimple $R$-module (also finite dimensional). I have seen a reference to the fact that $H^i(...