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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Non-triviality of a morphism

Let $X$ be a smooth Gushel–Mukai fourfold and $Y$ a smooth hyperplane section, which is a Gushel–Mukai threefold. I consider semi-orthogonal decomposition of $X$ and $Y$: $$D^b(X)=\langle\mathcal{O}_X(...
user41650's user avatar
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Connes’ homology

Is Connes’ homology $H_*^\lambda(F)$ of algebras $F=\mathbb{Z}_p[x]$ or $F=\mathbb{Z}_p[x]/(x^p)$ known for $p$ prime where $\mathbb{Z}_p= \mathbb{Z}/p$ ?
Victor's user avatar
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Homological algebra generalization of covering map

I would like to know if there exists an operation in homological algebra that generalizes the notion of covering maps for abstract chain complexes (over any field or ring, or maybe just certains where ...
Virgile Guemard's user avatar
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Does a functor preserving injectives also preserve K-injective complexes?

Let $F:A\to B$ be an exact functor of Grothendieck abelian categories. If $F$ preserves injective objects, then does the exact functor $F:K(A)\to K(B)$ preserves K-injective complexes? For example, ...
Doug Liu's user avatar
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Does Poincaré duality link topological study and representation study of a given Lie group?

The Poincaré duality for an oriented n-manifold M takes the form : $$H^\star(M) \simeq H_c^{n-\star}(M)^\vee.$$ Instead of M take now a real Lie group G. We can basically study it by looking at its ...
TopGenAx's user avatar
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Is any deformation of an acyclic complex gauge equivalent to a trivial one?

This question is related to this question. Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator ...
Zhaoting Wei's user avatar
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Does homotopy equivalence between deformations of cochain complexes imply gauge equivalence?

Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator $d+\epsilon: C^{\bullet}\otimes_k m\to C^{\...
Zhaoting Wei's user avatar
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Group cohomology of ${\rm SL}_{n}(\mathbb{F}_p)$ acting on trace zero matrices over $\mathbb{F}_p$

Let $p>5$ be a prime. For any integer $n\geq 2$, let $M_{n}^{0}(\mathbb{F}_p)$ denote the $n\times n$ matrices with trace $0$ over a finite field $\mathbb{F}_p$ of order $p$. Then we have an action ...
stupid boy's user avatar
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2 answers
336 views

Zeros of higher Ext functors

I'm trying to understand module classes that are defined as the kernels of higher Ext functors (e.g., arising here; as this paper suggests, I'm coming at this problem outside of module theory). One ...
Will Boney's user avatar
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Sections of a vector bundle taking values in a gerbe

We can make sense of sections of a vector bundle taking values in a line bundle; an oversimplified answer might be sections of the vector bundle tensored with the line bundle. However, I can't see the ...
Partha's user avatar
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Is there a strictly coassociative resolution of polynomial growth, for a finite group?

Let $G$ be a finite group and $k$ a field of characteristic $p$. It is well known, thanks to the work of Quillen, that the trivial $kG$-module $k$ has a projective resolution of polynomial growth. To ...
Dave Benson's user avatar
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Motivation for working with augmented objects in homological or higher algebra

I would like to understand if there is deeper reason/motivation behind augmentations in homological algebra. Recall classically in homology if there is a complex of free $R$-modules ($R commutative ...
user267839's user avatar
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Connected components of $Q(\mathrm{s\tau-tilt}A)$

I'm reading about support $\tau$-tilting modules and their mutations. I'm trying to understand the mutation quiver. Let $A$ be a finite dimensional algebra over an algebraically closed field, which is ...
It'sMe's user avatar
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Exact sequences with two FL-modules

Let $R$ be a ring. An $R$-module $M$ is called FL (FP) if it has a finite resolution consisiting of finitely generated free (projective) modules. Given an exact sequence of $R$-modules, $0\to M_1\to ...
Andrei Jaikin's user avatar
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A question about surjective maps between quadratic algebras

Let $V$ be a finite-dimensional vector space and $$ U \subseteq W \subseteq V \otimes V $$ be a proper inclusion of vector subspaces. Then take the tensor algebra $$ T(V) = \bigoplus_{i=1}^{\infty} V^{...
Pierre Dubois's user avatar
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A map in group cohomology from $H^n(G,G^{\vee})$ to $H^{n+1}(G,U(1))$

Let $G$ be a finite abelian group and denote by $G^{\vee}=\mathrm{Hom}(G,U(1))$ its Pontryagin dual. For any positive integer $n$ one can define a homomorphism of abelian groups $$ f:H^{n}(G,G^{\vee})\...
Andrea Antinucci's user avatar
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290 views

What is the geometric interpretation of the first Hochschild homology group of path algebra constructed from a directed graph?

Let $\mathcal{G} = (V, E, s, t)$ is a directed graph, where $V$ - the set of its vertices, $E$ - the set of its edges, $s: E \rightarrow V, s((v_1, v_2)) = v_1$ and $t: E \rightarrow V, s((v_1, v_2)) =...
Alexander's user avatar
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Example of a triangular string algebra that is rep infinite, but $\tau$-tilting finite

Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$. Hence, $A$ can be realized as the path algebra of a bound quiver $(Q,I)$, where $I\subseteq\mathbb{K}Q$ is an ...
It'sMe's user avatar
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Does maximally incompleteness cause nonvanishing of the extension of maximal ideal of a valuation ring by rank 1 free module?

In B. Bhatt's lecture notes[1], Remark 4.2.5 says ... $\operatorname{Ext}_R^2(k,R)$ is non-zero if $K$ is not spherically complete. which amounts to the following pure algebraic question. Statement ...
XYC's user avatar
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Rep infinite, but $\tau$-tilting finite

Let $A$ be a finite dimensional algebra over an algebraically closed field. I'm trying to better understand the difference between $A$ being representation infinite and $A$ being $\tau$-tilting ...
It'sMe's user avatar
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dg-Künneth formula for qcqs schemes

Let $X$ and $Y$ be qcqs schemes over a field $k$ (or I am happy to assume any nice condition up to smooth (quasi-)projective varieties if that makes the folloiwng question true). Let us define $Perf(X)...
P. Usada's user avatar
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Chain complexes indexed over measurable subsets of $\mathbb{R}$: Towards a measurable notion of Euler Characteristic

I have for a while tried to generalize the notion of a chain complex in a way to obtain a "continuous" or at least "measurable" notion of Euler Characteristic. I have come up with ...
The Thin Whistler's user avatar
2 votes
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186 views

What is a quasi-isomorphism of complexes of vector bundles?

Consider a homomorphism $f$ between two complexes of vector bundles over a fixed smooth manifold $M$. $$ \cdots \to V_{i - 1} \xrightarrow{\delta_{i-1}} V_i \xrightarrow{\delta_i} V_{i + 1} \to \cdots ...
Mattis Bakken's user avatar
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What is the flat dimension of $I(R)$ for $R$ the free algebra in $n$ variables?

Let $R$ be the free non-commutative algebra in $n$ variables and let $I(R)$ be the injective envelope of the regular module $R$. Question 1: Is $I(R)$ flat? Question 2: More generally, let $R=KQ$ be ...
Mare's user avatar
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7 votes
1 answer
257 views

Semi-projective complexes of modules over a finite group

Let $G$ be a finite group and $k$ a field of characteristic $p$ dividing $|G|$. A perfect complex of $kG$-modules is by definition a finite complex of finitely generated projective ($=$ injective $=$ ...
Dave Benson's user avatar
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Is the subcategory of strict morphisms abelian?

Let $A$ be an additive category with kernels and cokernels. A morphism $f$ is called strict if the natural morphism from the coimage to the image is an isomorphism. In Schneiders: Quasi-abelian ...
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Faithfully injective projective modules

An $R$-module I is called faithfully injective if it is injective and the functor $Hom_R(-, I)$ has the image of a complex being exact if and only if the original complex is exact. I wonder if it is ...
Projective injective's user avatar
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Multiplicativity of Euler–Poincaré characteristics of cohomology of pro-$p$ groups

While reading a paper, I found a mentioning that for an extension $1 \rightarrow H \rightarrow G \rightarrow N \rightarrow 1$ of pro-$p$ groups, the Euler–Poincaré characteristics $\chi(H)$, $\chi(G)$,...
gualterio's user avatar
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3 votes
1 answer
223 views

Can we define $\operatorname{Ext}$ groups in the category of short exact sequences?

Let $\mathcal A$ be an Abelian category. We can assume it has enough injectives. Let $\operatorname{SES}_{\mathcal A}$ denote the category, of which objects are short exact sequences in $\mathcal A$, ...
Display Name's user avatar
1 vote
0 answers
81 views

Image of the boundary maps in the homological spectral sequence of a filtration of a chain complex

I'm trying to understand the construction of the homological spectral sequence of a filtration given in C.A.Weibel ''An introduction to homological algebra''. Here, they start with a filtration of a ...
Marcos's user avatar
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2 votes
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230 views

Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=0$ for $i>0$?

Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
Gabriel's user avatar
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topological functor of tor functor

The framework of Quillen's model categories gives us a very general way of defining things as derived functors. For instance, in this way one can realise the singular homology as Andre-Quillen ...
Li Guanyu's user avatar
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5 votes
1 answer
447 views

Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\mathbb{G}_a,\mathbb{G}_m)=0$ for $i>0$?

Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
Gabriel's user avatar
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2 votes
3 answers
477 views

Canonical product in sheaf cohomology

EDIT: Let $\mathcal{F},\mathcal{G}$ be sheaves of abelian groups on a topological space $X$. Then there exists a canonical cup product $$H^i(X,\mathcal{F})\otimes_\mathbb{Z}H^j(X,\mathcal{G})\to H^{i+...
asv's user avatar
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1 vote
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Does this ring homomorphism have finite flat dimension?

Let $k$ be a field and consider the ring homomorphism $f:k[x,xy,xy^2]\rightarrow k$ defined by mapping $x,xy,xy^2$ to zero in $k$. I am trying to show that this ring homomorphism has finite flat ...
Boris's user avatar
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2 votes
0 answers
75 views

Equivalence of two descriptions of differentials of Koszul complex

My question comes from learning the paper [BGS96] Koszul duality patterns in representation theory by Beilinson, Ginzburg and Soergel, published in 1996. Let $A=T_{A_0}A_1/\langle R\rangle$ be a ...
L. Yhui's user avatar
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-1 votes
1 answer
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When morphism of complexes is homotopic to 0?

Let $f\colon A^\bullet\to I^\bullet$ be a morphism of bounded below complexes in an abelian category. Assume all $I^i$ are injective objects. Assume also that $f$ induces the zero map on cohomology. ...
asv's user avatar
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5 votes
1 answer
915 views

Basic question on the de Rham theorem

There is a modern nice proof of the de Rham theorem based on sheaf theory. The de Rham theorem says that for a smooth manifold $M$ there is a canonical isomorphism $$H^i_{dR}(M,\mathbb{R})\simeq H^i_{...
asv's user avatar
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3 votes
0 answers
131 views

Certain vanishing of Hom in the derived category of a finite dimensional algebra

Let $A$ be a finite dimensional algebra and ${\rm D}(A)$ be the derived category of $A$. Let a fixed finitely generated $A$-module $N$ have the following property: for every complex ${\bf X}$ of ...
H. Ali's user avatar
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3 votes
0 answers
62 views

Turning a Frobenius algebra into a symmetric algebra via tensor products

Let $A$ be a finite dimensional Frobenius algebra over a field $K$, which means that $A \cong D(A)$ as right $A$-modules. Being symmetric means that $A \cong D(A)$ as $A$-bimodules. Here $D(-)=Hom_K(-,...
Mare's user avatar
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3 votes
0 answers
82 views

Exterior powers of the Cartan matrix and Dyck paths

(This question can be formulated purely combinatorially in terms of Dyck paths, which is done in the second part of the question. But I am more interested whether this can be explained by some sort of ...
Mare's user avatar
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1 vote
1 answer
170 views

Additivity of group cocycles?

In Juven Wang, Zheng-Cheng Gu, and Xiao-Gang Wen - Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond, the authors calculated many ...
Ruizhi liu's user avatar
1 vote
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96 views

On Serre's condition and singular locus of determinantal rings

Let $R$ be a Commutative Noetherian ring. Let $\mathbf X:=[X_{ij}]_{1\le i \le r, 1 \le s \le t}$ be a matrix of indeterminates. Let $t>1$ be an integer, and $I_t(\mathbf X)$ denote the ideal in $...
Snake Eyes's user avatar
0 votes
1 answer
209 views

Infinite dimensional homology and spectral sequences

I am new to spectral sequences, so I'm not sure about the difficulty of this question. Suppose we have a filtration of a chain complex $\emptyset=D_{-1}\subset D_{0}\subset D_1\subset\dots D_n= C$ and ...
Marcos's user avatar
  • 577
3 votes
1 answer
163 views

$\Omega$ for noetherian semiperfect rings

Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$. Let $\Omega^n(mod A)$ be the category of $n$-...
Mare's user avatar
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4 votes
1 answer
215 views

Decompose an unbounded (cochain) complex in the homotopy category

Let $\mathcal{A}$ be an abelian category, it is known that any complex $A^{\bullet}$ admits a distinguished triangle $$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[...
user avatar
1 vote
0 answers
165 views

Grothendieck trace formula for schemes with étale fundamental groups that have no dense cyclic subgroup

This question may be more of a philosophical rather than mathematical nature. Assume I have a scheme $X$ and an endomorphism $F:X\longrightarrow X$. For instance, $X$ might be of finite type over $\...
The Thin Whistler's user avatar
4 votes
1 answer
101 views

Simplicial enrichment on unbounded algebras over an operad

In his paper "Homological Algebra of Homotopy Algebras" V.Hinich introduced a simplicial structure on algebras in unbounded chain complexes over arbitrary $\Sigma$-split operads. Not to get ...
Grisha Taroyan's user avatar
2 votes
0 answers
171 views

Is there a homotopical analogue of short exact sequence?

For $R$-modules for a commutative ring $R$, submodules and quotients are put on roughly the same footing; the kernel of a quotient is an injection into the source, and the cokernel of this injection ...
Keith Millar's user avatar
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6 votes
1 answer
427 views

Leray spectral sequence and pullbacks

I am trying to find a reference for the following well-known result on the functoriality of the Leray spectral sequence: Let $\pi:X\to Y$ and $\pi':X'\to Y'$ be morphisms of schemes and denote by $E_2^...
Victor de Vries's user avatar

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