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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Every surface of sufficiently large genus separates

Let $M^3$ be a smooth closed orientable manifold. Does there exist a non negative integer $g_0$ such that every closed orientable embedded surface $\Sigma \subset M$ of genus $g \geq g_0$ represents ...
2 votes
1 answer
236 views

How to define cohomology of algebraic structures?

I learned that the Hochschild cohomology of an associative algebra $A$ with a bimodule $M$ is defined using the cochain \begin{align*} \cdots \rightarrow C^n(A,M) \stackrel{d^n}{\longrightarrow} C^{n+...
2 votes
1 answer
87 views

On image of map $\text{Ext}^1_R(X,F)\to \text{Ext}^1_R(X,G)$ induced by $R$-linear map of free modules $F\to G$ with entries in the maximal ideal

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $F,G$ be finitely generated free $R$-modules and $f:F\to G$ be an $R$-linear map such that $f(F)\subseteq \mathfrak m G$. Let $X$ be a finitely ...
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1 vote
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40 views

Induced exact sequence on symmetric bilinear forms on abelian groups

Let $F=\operatorname {Hom}(S^2(\,\_\,),\mathbb{Q}/\mathbb{Z}):\mathsf{Ab} \to \mathsf{Ab}$ be the functor that sends an abelian group to the group of symmetric bilinear forms on this group. As far as ...
5 votes
1 answer
117 views

Restriction vs. multiplication by $n$ in Tate cohomology

$\DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\Cor}{Cor}$ This question was asked in MSE. It got no answers or comments, and so I post it here. Let $H$ be a subgroup of a finite group $G$, and ...
1 vote
0 answers
90 views

Prove that $B$ is a directing module

Let $A\cong\mathbb{K}Q/I$ be a finite dimensional, associative, basic $\mathbb{K}$ algebra, where $\mathbb{K}$ is algebraically closed and $Q$ is a finite Gabriel quiver on $n$ vertices and $I\...
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4 votes
1 answer
95 views

Vanishing of higher limits

Let $I$ be a directed set and let $X_I$ be a corresponding inverse system of, say, (complex) vector spaces or abelian groups (in my case in general not finite-dimensional, resp. not finitely generated)...
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4 votes
1 answer
146 views

Injective modules

Let $A$ be a finite dimensional $k$-algebra and let $I$ be an injective module。My question is whether $I$ is a direct sum of finite-dimensional injective modules。
0 votes
0 answers
145 views

Distinguished triangles as generalizations of short exact sequences

If you'll have patience with me, I understand that this is not the first time that a variation of this question is being asked on MathOverflow, but alas, I am unable to truly make sense of those ...
3 votes
1 answer
336 views

Is this exact sequence known?

$\newcommand{\Tors}{{\rm Tors}} \newcommand{\tf}{{\rm\, t.f.}} \newcommand{\Gt}{{\Gamma\!,\,\Tors}} \newcommand{\Gtf}{{\Gamma\!,\tf}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\Z}{{\mathbb Z}} \...
6 votes
0 answers
143 views

Iterating exact triangles (particularly in Floer homology)

There are several different Floer-homological invariants of 3-manifolds (and knots). The most prominent of these are Heegaard Floer homology, monopole Floer homology, and instanton Floer homology. It ...
4 votes
1 answer
370 views

Generalizations of global Euler characteristic formula

Let $ K $ be a number field, $ S $ a finite set of primes of $K $ including the archimedean primes and $ G_{K,S} $ be the Galois group of the maximal extension of $K$ unramified outside $ S $. Assume ...
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4 votes
1 answer
257 views

Does $\mathrm{Ind}(\mathcal{C})$ have enough injectives, if $\mathcal{C}$ is an abelian category?

Question. Let $\mathcal{C}$ be a small abelian category. Does the category $\mathrm{Ind}(\mathcal{C})$ of ind-objects of $\mathcal{C}$ have enough injectives? I have seen many times that $\mathrm{Ind}(...
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7 votes
1 answer
364 views

Long exact sequences for parametrized cohomology

I'm reading Michael Shulman's articles on cohomology in HoTT here and here, as well as Floris van Doorn's thesis here. Given $E: Z \to \mathsf{Spectrum}$ a family of spectra over a homotopy type $Z$, ...
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2 votes
0 answers
89 views

Graded global dimension of a graded algebra

Let $k$ be an algebraically closed field of characteristic $0$. Let $A := k \langle x,x^{-1},y \rangle /(xy-qyx, x^{d_1}-ay^{d_2})$, where deg$(x)>0$, deg$(y)>0$, $q,a \in k^*$ and $d_1\text{deg}...
2 votes
1 answer
123 views

Mayer-Vietoris sequence in group cohomology for arbitrary pushout squares of groups?

Suppose that we have (not necessarily injective) group homomorphisms $H \to G_1$ and $H \to G_2$, and we construct the pushout (i.e. amalgamated free product) $G_1 \sqcup_H G_2$. Suppose that we have ...
  • 5,453
0 votes
0 answers
39 views

Cohomology groups of the complex of sets whose convex hull not containing 0 in $\mathbb{R}^d$

Wegner proves that let $K$ be a finite family of convex sets in $\mathbb{R}^d$, then the Nerve of $K$, which is a simplicial complex and is defined as $$D=N(K):=\{ \{S_1,\dots, S_k\}: S_i\in K \text{ ...
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4 votes
1 answer
301 views

Surjectivity of natural map of rings

$\DeclareMathOperator\Hom{Hom}$Let $A$ be an integral domain and $P$ be a prime ideal in $A$. We denote $B=A/P$ then is the following natural map $$\Hom_A(P,A)\otimes_A B\to \Hom_A(P,B)$$ surjective? ...
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2 votes
0 answers
73 views

Representation finite Hopf algebras up to stable equivalence

It is well known that every representation-finite group algebra $KG$ is stable equivalent to a symmetric Nakayama algebra. Question: Is it true that every representation-finite Hopf algebra is stable ...
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6 votes
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On the derived categories of coherent sheaves on a quadric

I am reading On the derived categories of coherent sheaves on some homogeneous spaces - Kapranov for the proof for quadrics. Consider a vector space $V=\mathbb{C}^{n+2}$ with the standard quadratic ...
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1 vote
1 answer
93 views

Even and odd part of the Hochschild and cyclic homology of a super-algebra

Let $A$ be a $\mathbb Z_2$-graded $k$-algebra, where $k$ is a field of characteristic $0$. Then we know that the tensor product of $A$ with itself is also $\mathbb Z_2$-graded by $$(A\otimes_k A)_0:=...
1 vote
1 answer
166 views

A result of Schofield in the case of quivers with relations

Let $Q$ be a quiver without oriented cycles. A result of Schofield says that, for dimension vectors $\alpha$ and $\beta$ of $Q$, $\beta\hookrightarrow\alpha$ iff $\operatorname{ext}(\beta, \alpha-\...
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1 vote
1 answer
64 views

Subrings, submodules, and flatness

Let $R$ be a ring, and $S$ a subring of $R$. Let $M$ be a right $R$-module, and $N$ a right $S$-submodule of $M$. If $N$ is flat (or faithfully flat) as a right $S$-module, does it then follow that ...
7 votes
1 answer
264 views

Homological dimensions of rings of smooth functions

What is the global dimension of the algebra $C^\infty\mathbb R$ of smooth functions $\mathbb R\to\mathbb R$? What is the global dimension of the algebra $(C^\infty\mathbb R)_0$ of germs of smooth ...
  • 295
1 vote
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108 views

Finitely presented homology group

Given a finitely presented $i$th singular homology group over $\mathbb Z$ of a topological space $X$. If one knows the family of $i$th singular homology groups of $X$ over all possible fields, can one ...
1 vote
1 answer
163 views

Difference between $K(1)$-local K theory and l-adic completion of etale $K$ theory

Let $X$ be an scheme. Fix a prime $l$ which is invertible in $X$. Consider the $K(1)$-localization at prime $l$ of algebraic K theory $L_1K(X)$ and $l$-adic completion of etale K theory $K^{et}(X)$. ...
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2 votes
0 answers
107 views

Hochschild cohomology and outer automorphisms

Given an algebra $A$, I believe that the "conjugation" action of $\mathrm{Aut}(A)$ on $\mathrm{HH}^*(A)$ factors through $\mathrm{Out}(A)$. I’m looking for a reference.
0 votes
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71 views

Determining the free part of integral homology group

I would like to ask whether there is a way to determine the rank of the free part of a k-th singular homology group over $\mathbb Z$ of a manifold from the collection of k-th singular homology groups ...
3 votes
1 answer
158 views

Is the normalized simplicial bar construction of an operad a cooperad?

Suppose that $\mathcal{P}$ is a connected, unital operad in $\mathbb{k}$-vector spaces (or complexes), i.e. $\mathcal{P}(1)=\mathbb{k}$ and the unit map for $\mathcal{P}$ is the identity. One may form ...
6 votes
1 answer
374 views

Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?

Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
7 votes
2 answers
325 views

Derived functors out of an unbounded derived $\infty$-category

Let $\mathcal A$ be an abelian category. In this lecture, Thomas Nikolaus Defines the unbounded derived category $\mathcal D(\mathcal A)$ as $\mathcal K(\mathcal A)[W^{-1}]$, where $\mathcal K(\...
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2 votes
0 answers
98 views

When does Tate spectral sequence degenerate at $E_2$?

For a spectrum $M \in \text{Sp}^{B\mathbb{S}^1}$ with a circle group action, there is Tate spectral sequence $$ E_2^{ij}=\pi_{-i}(H(\pi_{-j}M))^{t\mathbb{S}^1} \Longrightarrow \pi_{-i-j}(M^{t\mathbb{S}...
1 vote
1 answer
176 views

How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?

Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...
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5 votes
1 answer
170 views

Periodic objects in Frobenius categories

Let $A$ be a finite dimensional Gorenstein algebra and $C$ the stable module category of $A$. Question: Does there always exist an indecomposable periodic object $X$ in C, that is an object with $\...
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1 vote
0 answers
135 views

Do we have a left adjoint of $i^*$ for a closed immersion $i$?

Let $i: X\hookrightarrow Y$ be a closed immersion of varieties. We have the derived pullback functor $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$. My questions is: can we construct a left adjoint of $i^*$ in ...
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0 votes
0 answers
40 views

Examples of almost complete intersection ideals

I am stutying some articles about perfect ideals and there are interesting results when $I$ is an ideal over a noetherian local ring that satisfies the following conditions : $\operatorname{grade}(I,R)...
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5 votes
0 answers
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A Galois connection arising from discussion concerning flat module and pure exact sequence

There is some sort of symmetry in the definition of flat module and pure short exact sequence which can be made precise as follows. Let $R$ be a ring (with unit), $\mathcal{R}$ be the class of all ...
1 vote
0 answers
74 views

A question about mapping cone and resolutions

I am studying this papper https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-44/issue-3/Betti-numbers-of-almost-complete-intersections/10.1215/ijm/1256060413.full By Daniel ...
7 votes
0 answers
294 views

What is a morphism of Tannakian categories?

I feel that this question is interesting but has not received enough attention; possibly because it's in MSE. So, the present question is mainly a repost, in the hopes of getting a good answer. (If ...
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1 vote
0 answers
78 views

Spectral sequences associated to cohomologies of simplicial type and derived-functor type: Proof of convergence

Assume I have two cohomology theories $\mathrm{\tilde{H}^{*}}$ and $\mathrm{H^{*}}$, the latter being defined over a Grothendieck site $X$ as the derived functor of some left-exact covariant functor $\...
7 votes
1 answer
241 views

Under which conditions is the bar construction a conservative functor?

The bar construction is a functor $A\mapsto Bar(A)$ from the category of augmented differential graded algebras over a commutative ring $R$ to the category of chain complexes of $R$-modules. It sends ...
6 votes
1 answer
444 views

How to prove that topological Hochschild homology of a smooth proper stable k-linear infinity category is dualizable?

Let $k$ be a perfect field of characteristic $p$. I heard that the Topological Hochschild homology of a smooth proper stable infinity category (or dg-category) is dualizable as a THH(k)-module ...
6 votes
0 answers
282 views

“Cohomological equation” in dynamical systems

Let $$\dot{x}=Ax+v_r(x)+v_{r+1}(x)+ \dots$$ with $x \in \mathbb{C}^n$ and $v_r: \mathbb{C}^n \to \mathbb{C}^n$ a homogenous, polynomial function of order $r.$ Then, being able to find a suitable $h$ ...
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2 votes
0 answers
55 views

$\mathbb Z_n[x_1^\pm,\dots,x_D^\pm]$-modules extended from $\mathbb Z_n$

Let $n$ be a positive integer and let $\mathbb Z_n=\mathbb Z/n \mathbb Z$. Consider the ring of Laurent polynomials $R=\mathbb Z_n[x_1^\pm,\dots,x_D^\pm]$. $R$-modules of the form $M=M_0 \otimes_{\...
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3 votes
0 answers
87 views

Criterion for representation-finite algebras

Let $A=KQ/I$ a quiver algebra with acyclic $Q$. Question: Is $A$ representation-finite if and only if $\tau^{-n}(A)=0$ for some $n \geq 1$? Here $\tau$ is the Auslander-Reiten translate of $A$. This ...
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1 vote
0 answers
81 views

A question about minimal system of generators and regular sequences

Let $(R,\mathfrak{m},k)$ a Noetherian local ring and $I$ an ideal. Suppose that: $\mu(I)=\operatorname{grade}(I,R)+1$ and $\operatorname{pd}_R(R/I)=\operatorname{grade}(I,R)$. (Some people says $I$ is ...
5 votes
1 answer
204 views

Localization of a ring and the Hom functor

Let $R=\mathbb{Z}[x,x^{-1}]$ be the ring of Laurent polynomials in $x$, $\mathfrak{p}=(1-x)$ be an ideal in $R$ and $R_\mathfrak{p}$ be the localization. I want to know what $\text{Hom}_R(R_\mathfrak{...
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5 votes
1 answer
213 views

About a recent paper of Rickard on finitistic dimension

Apologizes if this is a basic question, but I am new to the area of finite dimensional algebras. I am reading the paper "Unbounded derived categories and the finitistic dimension conjecture" ...
4 votes
1 answer
134 views

Finite lattices that are Koszul

Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$. It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no ...
  • 22.5k
4 votes
1 answer
124 views

When does $FP_n(R)$ imply $F_n$?

It is known that if a group $G$ is of type $F_2$ (finitely presented) and of type $FP_n(\mathbb{Z})$, then $G$ is of type $F_n$. However, is this true also for other rings which are not $\mathbb{Z}$? ...
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