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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46 views

Concept of an exact ideal of a module category

Let $R$ be a ring and $\text{Mod}\,R$ the category of (left) $R$-modules. Consider an ideal $\mathcal{I}$ of $\text{Mod}\,R$. For $R$-modules $X$ and $Y$ let $\mathcal{I}(X,Y)$ be the collection of ...
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2-shifted Poisson bracket on Lie algebra cohomology

Let $\frak{g}$ be a semisimple Lie algebra, and let $({-},{-})$ be an invariant inner product on $\frak{g}$. The Chevalley–Eilenberg complex $C^*(\frak{g})$ has a natural Poisson bracket of degree $-2$...
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RIng that is flat over a subring as a right module but not as a left module

What is an example of a ring $R$ and a subring $S \subseteq R$ such that $R$ is flat as a right module but not flat as a left module. The following question is my motivation: Faithful flatness for ...
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Reference request: Étale base change of differential-graded algebras

I've asked this question on Math.StackExchange, but didn't receive an answer, so I'd like to try my luck here. I'm looking for a reference for the following fact, which I've recently stumbled upon: ...
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199 views

Relation of the first Hochschild cohomology and the outer automorphism group

Let $R$ be a ring. Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite? (It is not true, by the two answers. Is it ...
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142 views

Quillen–Suslin theorem in a more general context

Let $A$ be a finite dimensional local Frobenius algebra that is Koszul. Question: Is it true for the Koszul dual of $A$ that every finitely generated projective module is free? If not, is there a ...
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Is there a "Kunneth isomorphism" for internal hom of chain complexes?

If $X^\bullet$ and $Y^\bullet$ are chain complexes over a field, we know from Kunneth theorem that $$H^*((X\otimes Y)^\bullet)\cong H^*(X^\bullet)\otimes H^*(Y^\bullet) $$ I want to know if there is a ...
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Projective $BP_\ast$-dimension of the $BP$-homology of classifying spaces of finite groups

Fix a prime $p$ and let $G$ be a finite group. Do we know the projective dimension of $BP_\ast (BG)$ as a graded $BP_\ast$-module? Or at least that it is finite? My guess is the following: The ...
4
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1answer
201 views

Six functor formalism for quasi-coherent $D$-modules

Let $X$ be a smooth scheme over a field $k$ and let $\mathsf{D}_{\text{qc}}(\mathcal{D}_X)$ be the full subcategory of $\mathsf{D}(\mathcal{D}_X\mathsf{-Mod})$ composed of the complexes of $\mathcal{D}...
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An inequality for Ext

$\DeclareMathOperator\Ext{Ext}$Consider the following statement for a $K$-algebra $A$: $\dim(\Ext^1 (M, N )) \ge \min( \dim(\Ext^1(M, M )), \dim(\Ext^1 (N, N )) ),$ for all finite dimensional ...
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Which cocycles (function classes) compute $H^*(BG, A)$ for $G$ a compact Lie group?

Let $G$ be a compact Lie group. Given a class $F$ of functions (continuous, measurable, piecewise smooth, $L^2$, bounded ($L^\infty$), polynomial, …) one can define group cochains as in the finite ...
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3answers
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On the map $\Phi_M: M\otimes_RM^*\xrightarrow{x\otimes y\mapsto \left\{f\mapsto f(x)y\right\}}\text{Hom}_R(M^*,M^*) $

$\DeclareMathOperator\Hom{Hom}$Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Denote $(\_)^*:=\Hom_R(\_,R)$. There is a natural map \begin{align} \Phi_M: M \...
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Why does $p_*p^! A$ deserve to be called homology with coefficients in $A$?

Let $p:X\to S$ be the unique map from a (locally compact) topological space $X$ to a point. Since $\underline{\hom}(\underline{\mathbb{Z}},-)$ is the identity functor, we have that $\Gamma(X,-)=\hom(\...
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Cohomology spectral sequence of a CW complex filtered by its skeletons

Let $X$ be a CW complex. Suppose, $$\emptyset\subset X^0\subset X^1\subset \cdots \subset X^p\subset \cdots \subset X= \bigcup_{i=0}^{\infty} X^i,$$ is a filtration of $X$ by its skeletons $X^i$. Now ...
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1answer
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A non-projective rigid object in an abelian monoidal category

What is an example of a rigid object $A$ in an abelian monoidal category $\mathcal{M}$ that is not projective as an object in $\mathcal{M}$? (Since $\mathcal{M}$ is abelian projective just means that ...
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86 views

How to deduce Künneth from its relative version (in cohomology of sheaves)

Let $p:X\to S$ and $q:Y\to S$ be morphisms of "spaces" over $S$. We have an isomorphism $$f_!(M\boxtimes N)=p_! M\otimes q_!N$$ in the derived category of "sheaves" over $S$, where ...
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1answer
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K-projectivity for rings of finite homological dimension

Let $R$ be a Noetherian commutative ring. A complex of $R$-modules $P^{\bullet}$ is K-projective if for any acyclic complex $A^{\bullet}$, the complex of abelian groups $ Hom(P^{\bullet}, A^{\bullet})$...
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A projective module over a domain that is not faithfully flat?

Let $R$ be a (noncommutative) unital ring which is a domain and let $\mathcal{N}$ be a non-zero projective (right) module. Projectivity of course implies that $\mathcal{N}$ is flat, but does the fact ...
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284 views

Which finite posets are Koszul self-dual?

Let $P$ be a finite connected poset with incidence algebra $A_P$. For the definition and results on Koszul algebras for incidence algebras, see for example here Question: Which posets have the ...
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Derived category supported in a Serre subcategory of a locally noetherian category

This is a cross-post from math.stackexchange at https://math.stackexchange.com/questions/4251692/derived-category-supported-in-a-serre-subcategory-of-a-locally-noetherian-catego, since I didn't get ...
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If a module is second syzygy and has no free summand , then it can be defined via a minimal resolution?

Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Assume $M$ has no free summand, and that there exists an exact sequence $0\to M \to R^{\oplus a}\xrightarrow{f} R^...
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Classification of certain modules over the polynomial ring

Fix a char $0$ field $k$, one may assume it's $\Bbb C$ because it's case I'm interested in. Let $\mathcal U$ be the set of $k[x]$-modules $A$ such that $k[x]^2 \subset A \subset k(x)^2$. I'm pretty ...
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1answer
67 views

Differential of the Twisted complex for algebraic operads

I have a question about the proof of lemma 6.4.12 in the book Algebraic Operads (Loday-Vallette) which I do not seem to be able to fully complete on my own. Hopefully, somebody here can point out what ...
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56 views

Composition of faithfully flat ring extensions

Let $R$ be a not necessarily commutative, unital, ring, and for simplicity let module always mean right module. We say that a unital ring extension $R \hookrightarrow S$ is flat, or faithfully flat, ...
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Perfect dg-modules under faithfully flat extension

Let $k \subseteq \bar{k}$ be an extension of fields (Orlov in the reference below seems to indicate the same thing will hold for faithfully flat maps but the case of fields is enough for me). On page ...
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1answer
379 views

Does the language of fibred categories gives the commutativity of the diagrams in Residues and Duality?

In Residues and Duality, R. Hartshorne and A. Grothendieck say that there are a plethora of compatibilities that need to be shown in order to have a six functor formalism. For example, if $f:X\to Y$, $...
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1answer
155 views

Faithful flatness for rings

Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $M \otimes_R -$ from left $R$-modules to abelian groups that preserves ...
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How are symmetric functions related to Koszul duality?

Staying within the world of linear algebra, we have the following two "dualities" between exterior powers and symmetric powers. The first is that of Kozsul duality, so these two graded ...
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de Rham cohomology of a specific ring

I'm running into a certain algebraic de Rham cohomology computation I could use some help with. Specifically, what is the algebraic de Rham cohomology of: $$ \mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_n,(r^...
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100 views

Koszul differential of the complex $\bigwedge \mathfrak{g}^*$

Let $\mathfrak{g}$ be a finite dimensional Lie algebra. The definition of the Koszul differential is given in the article by Kumar and Vergne on equivariant cohomology page 133 as follow: Let us now ...
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73 views

Taking the homology of a chain complex, seen as a symmetric monoidal functor

I've found, in quite some places online (e.g. it's the last example in the wikipedia page about monoidal functors), a statement similar to this: Homology can be seen as a symmetric, monoidal functor $...
3
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2answers
237 views

Quasi-isomorphism preserves group hypercohomology

I am looking for a reference for the assertion in the title. In more detail, let $\Gamma=\{1,\gamma\}$ be a group of order 2. Let $A$ be a $\Gamma$-module (an abelian group on which $\Gamma$ acts). ...
5
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1answer
219 views

Unsplitting sequence of vector bundles

Let $V$ be a $n$-dimensional complex vector space. Using Grothendieck's notation, we define the Grassmannian $G(k,V)$ as the space of $k$-quotients of $V$ or, equivalently, as $$ G(k,V)=\{ \mathbb P W ...
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Question on the classification of Cuntz algebras via their extension groups and via their K-theory

I've recently been reading Kenneth Davidson's book on C*-algebras by example. One thing that particularly interested me was the classification of the Cuntz algebras by looking at the extensions of the ...
5
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153 views

L-theory periodicity

Let $\mathcal{A}$ be an additive category. I have two questions: Is there a conceptual explanation why $L(\mathcal{A})$ is 4-periodic, in the sense that $L_{i}(\mathcal{A})=L_{i+4}(\mathcal{A})$ for ...
5
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1answer
364 views

L-theory of additive category

Reading some articles in the field, I found the following statement: Proposition: Let $\mathcal{B}$ be an additive category and $\mathcal{A}$ a full additive subcategory of $\mathcal{B}$. If $\mathcal{...
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227 views

What exactly goes wrong with $f_!$ outside of locally compact spaces?

Let $f:X\to Y$ be a morphism of ringed spaces. We define a functor $f_!:\mathcal{O}_X\mathsf{-Mod}\to\mathcal{O}_Y\mathsf{-Mod}$ as $$\Gamma(U,f_!\mathscr{F}):=\{s\in \Gamma(f^{-1}(U),\mathscr{F})\:|\:...
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66 views

Coxeter polynomials of graphs

Let $Q$ be a finite connected and directed graph with $n$ points. Assume $Q$ is acyclic as a directed graph. Let $C=C_Q$ be the Cartan matrix of $Q$, that is the matrix with entries $c_{i,j}$ being ...
4
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1answer
232 views

Cup product of hypercohomologies

Let $X$ be a projective variety and $A$ and $B$ are two vector bundles on $X$. Let $C_{\bullet}$ denote the complex of sheaves $$ 0\rightarrow A\rightarrow B\rightarrow 0 $$ Then we have a cup product ...
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1answer
163 views

Finitely generated modules over Noetherian local ring that become isomorphic after faithfully flat base change

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S$ be a Noetherian ring which is a faithfully flat $R$-algebra. If $M,N$ are finitely generated $R$-modules such that $M\otimes_R S \cong N \...
4
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1answer
169 views

Proof of derived tensor-hom adjunction

This is a cross-post from math.stackexchange, since I didn't get any answers there. As far as I know, for $R,S,V,W$ rings and $M$ an $(R,W)$-bimodule, $N$ an $(R,S)$-bimodule and $L$ an $(S,V)$-...
5
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1answer
170 views

What is a Serre-smooth algebra?

Let $A$ be an $R$-algebra. In the book "Noncommutative Geometry and Cayley-smooth Orders" by Le Bruyn one can find the notion of "Serre-smooth" in the introduction. But no formal ...
8
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1answer
188 views

Chain complexes split in the derived category over rings of global dimension 1

Let $R$ be a ring of global dimension $1$. Then I have seen the claim (in a paper, and in this MO post When do chain complexes decompose as a direct sum?) that any chain complex over $R$ is equivalent ...
11
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1answer
413 views

Unbounded resolutions for Grothendieck abelian categories

Consider the following result: Theorem 1: Let $\mathsf{A}$ be a Grothendieck abelian category. Then every complex in $\mathsf{C}(\mathsf{A})$ has a $K$-injective resolution. As far as I know, the ...
6
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1answer
178 views

Lifting of flat lci maps

Suppose $R$ is a Noetherian ring and $I$ a nontrivial ideal of $R$, and $A_0\to B_0$ a finite faithfully flat lci map of smooth $R_0 := R/I$-algebras. We fix a smooth $R$-algebra $A$ lifting $A_0$ and ...
2
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0answers
36 views

Edge morphism of a particular spectral sequence

I am not sure if this question is too elementary for MathOverflow and should strictly belong to Math StackExchange but I shall try my luck. Please feel free to close it in this case and I will migrate ...
4
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1answer
216 views

Why do we need the axiom MS3 for localizing categories?

Background: Let $\mathsf{C}$ be a category and $S$ be a collection of morphisms (let's suppose that $S$ has all the identities and is closed under multiplication just to simplify a bit). We can ...
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121 views

Künneth formula for local cohomology with support

In "Differential operators on the flag varieties" (http://www.numdam.org/article/AST_1981__87-88__43_0.pdf) by Brylinski, he uses on page 53 a Künneth formula for local cohomology with ...
8
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2answers
585 views

Verdier duality under more general conditions

In the section 3.2 of Sheaves in Topology by A. Dimca, the author explains that if $f:X\to Y$ is a continuous map (between locally compact, $\sigma$-compact topological spaces with finite homological ...
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57 views

Projective objects in the category of simplicial objects in an abelian category

Let $S\mathcal{A}$ be the category of simplicial objects in an abelian category $\mathcal{A}$. In exercise 8.4.5 in Weibel's An Introduction to Homological algebra, it is said that $P \in S\mathcal{A}$...

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