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.
2,611
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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 $Y$:
$$D^b(X)=\langle\mathcal{O}_X(...
0
votes
0
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96
views
Connes’ homology
Is Connes’ homology $H_*^\lambda(F)$ of algebras $F=\mathbb{Z}_p[x]$ or $F=\mathbb{Z}_p[x]/(x^p)$ known for $p$ prime where
$\mathbb{Z}_p= \mathbb{Z}/p$ ?
2
votes
0
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254
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Homological algebra generalization of covering map
I would like to know if there exists an operation in homological algebra that generalizes the notion of covering maps for abstract chain complexes (over any field or ring, or maybe just certains where ...
3
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0
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113
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Does a functor preserving injectives also preserve K-injective complexes?
Let $F:A\to B$ be an exact functor of Grothendieck abelian categories. If $F$ preserves injective objects, then does the exact functor $F:K(A)\to K(B)$ preserves K-injective complexes?
For example, ...
1
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0
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71
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Does Poincaré duality link topological study and representation study of a given Lie group?
The Poincaré duality for an oriented n-manifold M takes the form : $$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 ...
0
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0
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40
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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 ...
6
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134
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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^{\...
9
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185
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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 ...
3
votes
2
answers
336
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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 ...
2
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0
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75
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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 ...
9
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114
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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 ...
2
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0
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144
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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 ...
0
votes
0
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86
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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 ...
4
votes
1
answer
169
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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 ...
1
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1
answer
173
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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^{...
1
vote
1
answer
177
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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})\...
4
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0
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290
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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)) =...
2
votes
0
answers
83
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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 ...
1
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0
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98
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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 ...
2
votes
1
answer
140
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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 ...
2
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0
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122
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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)...
1
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0
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93
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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 ...
2
votes
0
answers
186
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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
...
0
votes
0
answers
54
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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 ...
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 $=$ ...
3
votes
1
answer
304
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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 ...
4
votes
1
answer
335
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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 ...
1
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0
answers
115
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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 $\chi(H)$, $\chi(G)$,...
3
votes
1
answer
223
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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 $\mathcal A$, ...
1
vote
0
answers
81
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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 ...
2
votes
0
answers
230
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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,\...
1
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0
answers
187
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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 ...
5
votes
1
answer
447
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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,\...
2
votes
3
answers
477
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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
$$H^i(X,\mathcal{F})\otimes_\mathbb{Z}H^j(X,\mathcal{G})\to H^{i+...
1
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0
answers
68
views
Does this ring homomorphism have finite flat dimension?
Let $k$ be a field and consider the ring homomorphism $f:k[x,xy,xy^2]\rightarrow k$ defined by mapping $x,xy,xy^2$ to zero in $k$.
I am trying to show that this ring homomorphism has finite flat ...
2
votes
0
answers
75
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Equivalence of two descriptions of differentials of Koszul complex
My question comes from learning the paper [BGS96] Koszul duality patterns in representation theory by Beilinson, Ginzburg and Soergel, published in 1996.
Let $A=T_{A_0}A_1/\langle R\rangle$ be a ...
-1
votes
1
answer
169
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When morphism of complexes is homotopic to 0?
Let $f\colon A^\bullet\to I^\bullet$ be a morphism of bounded below complexes in an abelian category. Assume all $I^i$ are injective objects. Assume also that $f$ induces the zero map on cohomology.
...
5
votes
1
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915
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Basic question on the de Rham theorem
There is a modern nice proof of the de Rham theorem based on sheaf theory.
The de Rham theorem says that for a smooth manifold $M$ there is a canonical isomorphism
$$H^i_{dR}(M,\mathbb{R})\simeq H^i_{...
3
votes
0
answers
131
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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 ...
3
votes
0
answers
62
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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(-,...
3
votes
0
answers
82
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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 ...
1
vote
1
answer
170
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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 ...
1
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0
answers
96
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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 $...
0
votes
1
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209
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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 ...
3
votes
1
answer
163
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$\Omega$ 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 $n$-...
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}[...
1
vote
0
answers
165
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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 $\...
4
votes
1
answer
101
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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 ...
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 ...
6
votes
1
answer
427
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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^...