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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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45 views

Upper bound for embedding of submodules of projective modules

Assume we have a finite dimensional algebra $A$ with the following property: Every indecomposable submodule of a projective module embedds into $A^n$ for a fixed $n$. Is there a good method to ...
4
votes
1answer
114 views

Global dimension of the tensor algebra

Let $R$ be a semisimple ring with a non-zero $R$-bimodule V. Let $T_R(V):= \bigoplus\limits_{k=0}^{\infty}{V^{\otimes_k}}$ be the tensor algebra of $V$. Question 1: Is there a simple proof that $...
3
votes
1answer
118 views

Global dimension of a graded algebra

Let $A= \bigoplus\limits_{n=0}^{\infty}{A_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A_0$. Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext_A^i(...
3
votes
1answer
192 views

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 ...
4
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0answers
62 views

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 ...
4
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0answers
108 views

Sections of sheaves on limit spaces

Let $\{U_{\nu}\}_{\nu\in I}$ be an inverse system of topological spaces over a filtered index set $I$ with continuous transition maps. Let $A_0$ be a sheaf of abelian groups on $U_{\nu_0}$, for some $...
3
votes
0answers
49 views

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 ...
3
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0answers
81 views

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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0answers
94 views

is the homology of a free complex again free?

Let $k$ be a field and $R$ be a finitely generated graded $k$-algebra (e.g. a polynomial ring in some variables). Let $M$ be a finitely generated graded vector space so the graded module $\widetilde{M}...
2
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0answers
71 views

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 ...
3
votes
1answer
74 views

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 $R$ ...
4
votes
1answer
255 views

Kernels and cokernels of multicomplex homomorphisms

Let $\mathcal A$ be a (complete and cocomplete) Abelian category. A multicomplex in $\mathcal A$ is a bigraded object $X^{(\bullet,\bullet)}$ with differentials $$ d^{(i,j)}_r\colon X^{(i,j)}\to X^{(...
2
votes
1answer
123 views

An example of a local ring which is not CM and a MCM module over it

I am looking for an example of a commutative noetherian local ring $(A,m)$, and a maximal Cohen-Macaulay module $M$ over $A$ (in particular $M$ is finitely generated over $A$), such that for some $p \...
3
votes
1answer
121 views

Question on $\operatorname{Ext}$ in a local Frobenius algebra

Let $A$ be a finite dimensional local Frobenius algebra with simple module $k$ and an indecomposable non-projective module $M$ (that is also finite dimensional). Question: Is there an example of ...
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1answer
168 views

Flat module and Projective Module

In Rotman's book "Intro to homological algebra" Theorem 3.62 Let $0\rightarrow K\rightarrow\ F\rightarrow A\rightarrow 0$ be an exact sequence of right R-modules, where $F$ is free. The following are ...
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0answers
49 views

Tensor product of mapping cones

If $A^*_i \to B^*_i \to C^*_i \to A^*_i[1]$ is a distinguished triangle of complexes for $i=1,2$, is there a nice way to express the derived tensor product $C^*_1 \otimes^L C^*_2$ in terms of the ...
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163 views

On the coalgebraic Homotopy Transfer Theorem

Let $A$ be a dg algebra, say over a field. The Homotopy Transfer Theorem says that $H(A)$ can noncanonically be given the structure of $A_\infty$-algebra, extending the induced multiplication on $H(A)$...
5
votes
1answer
411 views

Is there an adjoint to the inclusion of I-adically complete modules to all modules?

A module $M$ over a ring $R$ is $I$-adically complete with respect to the ideal $I$, if the canonical map $M \to \lim M/I^nM$ is an isomorphism. There exists a completion functor: $M \mapsto \lim M/I^...
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0answers
73 views

Strict units in A-infinity algebras

In Kontsevich-Soibelman's paper "Notes on $A_\infty$-algebras, $A_\infty$-categories and non-commutative geometry", $A_\infty$-algebras with strict units are defined so units act trivially on higher ...
7
votes
1answer
312 views

Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$

All varieties appearing below are assumed smooth projective over $\mathbb C$ and all vector bundles, sections etc are assumed to be algebraic/holomorphic. We use the word resolution to mean quasi-...
6
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102 views

Pin cobordism v.s. “KO” theory in low or in any dimensions

Fact: The spin cobordism is equivalent to "KO" theory in low dimension if we only consider the 2-torsion. This is related to a question and an answer supports the claim. Here we denote the $p$-...
6
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0answers
142 views

Arf-Brown-Kervaire invariant and a surjective map $G \to Pin^-$

We know that the Arf-Brown-Kervaire (abk) invariant is a bordism invariant of $$ \Omega_2^{Pin^-}(pt)=\mathbb{Z}/(8\mathbb{Z}), $$ where the $\mathbb{Z}/(8\mathbb{Z})$ is generated by a 2-manifold $M^...
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57 views

Reference Request: A “Chevalley-Eilenberg”-style formulation of the $L_\infty$ algebra minimal model theorem?

The nicest definition of $L_\infty$-algebras ---which I will call a "Chevalley-Eilenberg" style definition after the obvious analogy with the Chevalley-Eilenberg differential of Lie algebras--- is the ...
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vote
1answer
96 views

modules whose every submodule is a homomorphic image

Let $R$ be a commutative ring with unity. Let us say that an $R$-module $M$ satisfies property $\mathcal P$ if every submodule of $M$ is a homomorphic image of $M$. Can we characterize all ...
3
votes
1answer
100 views

commutative ring satisfying descending chain condition on radical ideals

Let $R$ be a commutative ring with unity which satisfies d.c.c. on radical ideals. then does $R$ satisfy a.c.c. on radical ideals ? If this is not true in general, then what happens if we also assume $...
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votes
0answers
54 views

Monoidal equivalence of categories of modules in different models of higher algebra

A result of Shipley states the following (in the language of model categories): For a differential graded algebra $A$, the $\infty$-category of (dg)-modules over $A$ agrees with the $\infty$-category ...
2
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0answers
90 views

On a conjecture about tilting modules

There is the following conjecture on tilting modules (see also History of an open problem on partial tilting modules): Given a partial tilting module $T$ over a finite dimensional algebra $A$ (that ...
4
votes
0answers
100 views

Tensor product of bimodules

Im not sure whether this question is appropriate for MO, but I do not have much experience with bimodules (or I forgot many things). Let $A$ be a finite dimensional (connected) algebra over a field $...
8
votes
0answers
109 views

$n$-fold tensor products of $D(A)$ for finite dimensional algebras

Let $A$ be a finite dimensional quiver algebra over a field $K$ and let $D(-):=Hom_K(-,K)$ denote the natural duality (assume algebras are connected). Define $\psi_A:=sup \{ n \geq 1 | D(A)^{\otimes ...
7
votes
0answers
58 views

Is the double orthogonal of a localizing Serre subcategory still a localizing Serre subcategory?

Let $R$ be an unital ring and consider the category of left $R$-modules $R$-Mod. We call a full subcategory $\mathcal{W}\subset R$-Mod a localizing Serre subcategory if $\mathcal{W}$ is closed under ...
3
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0answers
63 views

Deformations of nilpotent parts of superalgebras

I have two questions concerning some results in the article "Deformations of nilpotent parts of superalgebras" of N. van den Hijligenberg, J.Math.Phys. 35, 1427 (1994); doi:10.1063/1.530598 After ...
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0answers
83 views

Question on vanishing Hochschild cohomology

Recall that for an $K$-algebra $A$ with $A^e:=A^{op} \otimes_K A$ the Hochschild cohomology is defined as $HH^n(A,M):=Ext_{A^e}^n(A,M)$. Question: Is there a finite dimensional selfinjective ...
7
votes
1answer
110 views

Tensor product of a DGA and an $A_\infty$ algebra

In general there seems no way to naturally define the tensor product of two $A_\infty$ algebras $A$ and $B$. But, if $(A, m^A_1,m^A_2)$ is only a DGA(differential graded algebra) and $(B, m^B_k, k\ge ...
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1answer
69 views

Gorenstein projective modules of a certain triangular matrix algebra

Let $B$ be a finite dimensional selfinjective algebra over a field $k$ with a finite dimensional non-projective $B$-module $M$ and $$A=\pmatrix{k&M\\0&B}.$$ A module $N$ over an algebra $C$ ...
3
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0answers
42 views

Quotient of quasi-isomorphic nonpositively graded cdga's

I'm looking for a theorem about quotient of quasi-isomorphic cdga's: Let $A, B$ be two cdga's (commutative differential $\mathbb Z$-graded algebra) concentrated in nonpositive degree, and $\mathfrak ...
4
votes
1answer
120 views

Quotient of quasi-isomorphic cdga's

I'm looking for a theorem about quotient of quasi-isomorphic cdga's: Let $A, B$ be two cdga's (commutative differential $\mathbb Z$-graded algebra) of nonpositive degrees, and $\mathfrak m \subset A, ...
5
votes
1answer
207 views

Calculating the Ext-algebra with a computer

Given a finite dimensional quiver algebra $A$ over an arbitrary field and a module $M$ of finite injective dimension or finite projective dimension. Let $B$ be the Ext algebra of $M$, that is $B:=\...
5
votes
0answers
82 views

Derived invariant acyclic algebras

Call a connected quiver algebra $A=KQ/I$ (finite quiver Q and admissible ideal I) derived-invariant in case every quiver algebra derived equivalent to $A$ is even isomorphic to $A$. For example local ...
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votes
0answers
137 views

How to visualize the Microsupport of a Sheaf?

I am looking through Persistent homology and microlocal sheaf theory to learn a bit on barcodes. They are require the notion of a microsupport of a sheaf, looks like it could be a rather concrete ...
3
votes
0answers
110 views

Induced Homomorphism on Cohomology of Symmetric Group 3

For the symmetric group $S_3$, there is an inclusion $i:\mathbb{Z}/3\mathbb{Z}\hookrightarrow S_3$. How can I assert that the induced homomorphism $$i^{\ast}:H^{n}(S_3,\mathbb{Z})\rightarrow H^{n}(\...
3
votes
0answers
60 views

$\Omega^2(S) \cong \tau(S)$ 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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107 views

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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votes
1answer
453 views

“Exactness” of operadic cohomology

There are two somewhat widely known theorems which say if $A$ is a nonnegatively graded commutative algebra in char $0$, then forgetful map on operadic cohomology $H^*_{Harr}(A, A) \to H^*_{Hoch}(A, ...
6
votes
1answer
253 views

Understanding the functoriality of group homology

EDIT: I've decided to rephrase my question in order for it to be more concise and to the point. Let $G$ be a group, and let $F_\bullet\rightarrow\mathbb{Z}$ be a free $\mathbb{Z}[G]$-resolution of ...
5
votes
1answer
189 views

Manifold generators of O-bordism invariants

If I understand correctly, I can obtain the $O$-cobordism group of $$ \Omega^{O}_3(BO(3))=(\mathbb{Z}/2\mathbb{Z})^4, $$ The 3d cobordism invariants have 4 generators of mod 2 classes, are generated ...
2
votes
0answers
44 views

$Ext_{A^e}^i(D(A),A)$ for finite dimensional algebras

Let $A$ be a finite dimensional non-semisimple algebra over a field $K$ with enveloping algebra $A^e=A^{op} \otimes_K A$. Let $D(A)=Hom_K(A,K)$. Question: Is there always a positive integer $i>...
4
votes
1answer
110 views

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 ...
9
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1answer
343 views

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 ...
8
votes
0answers
92 views

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 ...
5
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0answers
85 views

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 ...