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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colimits of spectral sequences

I'm looking for some references about colimits of spectral sequences. More precisely: let $X : I \longrightarrow \cal{C}$ be a functor from a filtered category $I$ to the category of double cochain ...
Agustí Roig's user avatar
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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 $\...
Mare's user avatar
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Uses for (Framed) E2 algebras twisted by braided monoidal structure

$\newcommand{\C}{\mathcal{C}}$ $\newcommand{\g}{\mathfrak{g}}$ If $\C$ is a monoidal category (not necessarily a symmetric monoidal category), it's possible to define the notion of an algebra object $...
Dmitry Vaintrob's user avatar
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Global dimension 2 Nakayama algebras , 321-avoiding permutations and Fibonacci combinatorics

A $n$-LNakayama algebra (for $n \geq 2$) is a list $[c_0,c_1,...,c_{n-1}]$ with $c_{n-1}=1$, $c_i \geq 2$ for $i \neq n-1$ and $c_i -1 \leq c_{i+1}$ for all $i$. They are in bijection with Dyck paths, ...
Mare's user avatar
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Functoriality of filtered spectral sequences

What is the appropriate functoriality statement of a filtered chain map between filtered spectral sequences? Suppose that we have two filtered chain complexes $C,C'$ and a filtered chain map $f\colon ...
Onkar Singh Gujral's user avatar
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Simplicial Complex Induced by a Morphism

Let $(C, \otimes, I)$ be a symmetric monoidal abelian category. For an object $M$ in $C$ with a map $\rho : M \rightarrow I$, we can form the chain complex $M_*$ where $M_n = \otimes_{i = 1}^n M$ and $...
Ronald J. Zallman's user avatar
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Homology of bar complex vs homology of indecomposables

$\require{AMScd}$ Background: This question is about the bar and cobar constructions, and their relationship with the indecomposables of a dg-algebra. A brief summary of the bar and cobar ...
Julian Chaidez's user avatar
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Properties of right rejective subcategories

I am reading this paper finiteness of representation dimension, on page 1012 there is a place I don't understand: Why $\mathcal{C}(, \mathbb{F}(X)) \rightarrow [\mathcal{C'}](,X)$ is an isomorphism? ...
Xiaosong Peng's user avatar
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689 views

Direct limits of a matrix and its transpose

Let $A \in M_n(\mathbb Z)$ and $A^T$ denote the transpose of $A$. Define the direct limits $$H_1 = \mathrm{colim} (\mathbb Z^n \xrightarrow{A} \mathbb Z^n \xrightarrow{A} \mathbb Z^n \xrightarrow{A} \...
Toke Nørgård-Sørensen's user avatar
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When is the category of Gorenstein projective $R$-modules Frobenius?

Let $R$ be a ring (associative with unit, but not necessarily commutative, and definitely not necessarily Noetherian.) Then the category $\operatorname{GP}(R)$ consists of those $R$-modules having a ...
Matthew Pressland's user avatar
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How to compute this $\mathrm{Ext}^1$?

Let $A$ be a regular local $\mathbb{C}$-algebra of dimension $2$, such as the localization of $\mathbb{C}[x,y]$ at $(x,y)$, and let $\nu=(\nu_1\geq\nu_2\geq\cdots\geq\nu_{\ell}\geq0)$, $\mu=(\mu_1\geq\...
Qfwfq's user avatar
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Extension-closed subcategories of triangulated categories as "almost exact" categories

Did anybody study those subcategories of triangulated categories that are closed with respect to "extensions" (in the sense of distinguished triangles; in particular, any such $B$ is additive)? If we ...
Mikhail Bondarko's user avatar
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Homogeneous polynomial vector fields tangent to the unit sphere

This question has something to do with that one. Let $n\ge1$ and $d\ge1$ be two given integers. Consider the polynomial vector fields $v=(v_1,\ldots,v_n)$ whose components $v_j$ are homogeneous of ...
Denis Serre's user avatar
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"as close to being semisimple as it can possibly be."

I had originally asked this question on math stack exchange but I think maybe it's more appropriate to ask it here. In the paper of Beilinson, Ginzburg and Soergel entitled "Koszul Duality Patterns......
Anette's user avatar
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Hochschild homology and change of non-ground ring

Let $k$ be a field, $R$ is a commutative algebra over $k$ and $A$ is an associative algebra over $R$. There is a morphism of commutative algebras $R \to T$. Is it possible to reduce calculation of ...
Sasha Pavlov's user avatar
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Why is the transfer map Tate-dual to restriction ?

In one of their papers (before Theorem 7.2), Benson and Carlson state that the transfer map is Tate-dual to the restriction homomorphisms (also see Remark 1.3 of this recent paper). More precisely:...
Demin Hu's user avatar
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quasi-isomorphism

Is the distinction between quasi-isomorphism and `weak homotopy equivalence' ONLY that the first means inducing an isomorphism in homology and the second to an isomorphism of homotopy groups?
Jim Stasheff's user avatar
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Comparing stabilization of stable category modulo injectives and a Verdier localization

Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as ...
Snake Eyes's user avatar
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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_{...
asv's user avatar
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Can we construct a filtered chain complex from a spectral sequence?

Suppose $\{(E_r,d_r)\}$ for $r>0$ forms a spectral sequence over a field $\mathbb{F}$, i.e. for any $r$, $(E_r,d_r)$ is a chain complex over $\mathbb{F}$ and $E_{r+1}=H(E_r,d_r)$. For simplicity, ...
Faniel's user avatar
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What is the meaning of this coboundary homomorphism for group hypercohomology?

$\require{AMScd}$ Let $\Gamma=\{1,\gamma\}$ be a group of order 2. In my problem from Galois cohomology of real reductive groups I came to a commutative diagram of $\Gamma$-modules (abelian groups ...
Mikhail Borovoi's user avatar
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Rigid monoidal and closed monoidal categories

I am trying to understand the relationship between rigid monoidal categories and closed monoidal categories. First every rigid monoidal category is closed, with an adjoint to the functor $X \otimes -$ ...
Jake Wetlock's user avatar
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Hochschild cohomology of an Azumaya algebra

Let $k$ be a field. Given a commutative $k$-algebra $Z$ and an associative algebra $A$ that is Azumaya over $Z$, do we have an isomorphism of Hochschild cohomologies: $HH^*(A) \cong HH^*(Z)$? This is ...
MathManiac's user avatar
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Sullivan minimal model in the case of $H^1(V)\neq 0$

Is there a simple construction of a Sullivan minimal model $\Lambda U \rightarrow V$ in the case that $H^1(V)\neq 0$? Do you have a reference? I envisage a degree-wise construction as in the case of $...
Pavel's user avatar
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On tilting and cotilting modules

Let A be an Artin algebra and assume all modules are basic, then a classical result says that tilting modules $T$ are in bijection with complete cotorsion pairs $(T^{\perp}, \check{ add(T)})$ (with ...
Mare's user avatar
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Projective module which splits off sequence of submodules, but not the sum

Does there exist an example of a module $X$ over some ring $R$ together with submodules $T_i$ such that: $X$ is projective, $X$ splits as an internal direct sum $X\cong T_1\oplus T_2\oplus \ldots \...
nikola karabatic's user avatar
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Strøm model structure on nonnegatively graded chain complexes

Let $\newcommand{\Ch}{\mathsf{Ch}}\Ch_{\ge 0}(R)$ be the category of $\mathbb{N}$-graded chain complexes over some ring $R$, and $\Ch(R)$ the category of $\mathbb{Z}$-graded chain complexes. The ...
Najib Idrissi's user avatar
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Derived equivalences of Artin algebras with finitistic dimension zero

Let $A$ be an Artin algebra of finitistic dimension zero and $B$ an algebra derived equivalent to $A$. Does $B$ also have finitistic dimension zero? In case this is true, this might generalise the ...
Mare's user avatar
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Subcategories of the Verdier quotient?

Let $\mathcal T$ be a triangulated category and $\mathcal C$ a thick triangulated subcategory. We consider the Verdier quotient $\mathcal T/\mathcal C$. Is there a bijective correspondence between ...
Triangulated's user avatar
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Functoriality of the formality quasi-isomorphism of E-polydifferential operators

Given a smooth manifold $M$ and a Lie algebroid $E\rightarrow M$ we can consider the $E$-polydifferential operators $D_E$ and the $E$-polyvectorfields $T_E$ as $$D_E:=\bigoplus_{k=-1}^\infty\mathcal{...
Niek de Kleijn's user avatar
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Rank four quadratic Form with non trivial discriminant in I(k)

Im sure this is a beginners question. Let $k$ be a field and $I(k)$ the fundamental ideal in the Witt-ring W(k). The Arason-Pfister-Hauptsatz states: "If $\varphi$ is any anisotropic class in $I^n(...
nxir's user avatar
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2 answers
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bialgebra cohomology

It seems that the Gerstenhaber-Schack (bi)complex of an associative bialgebra carries a homotopy e_3-algebra structure and a degree 2 Lie algebra bracket, up to homotopy. Does anyone know about a ...
anita naolekar's user avatar
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550 views

Perverse vs real formality?

Let $\cal A$ be an abelian category, say linear over a field, with enough injectives and $\cal P$ be the heart of a t-structure on the bounded derived category $D^b(\cal A )$. Assume that $\cal P$ ...
Jan Weidner's user avatar
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Chain Homotopy classes as n-homology of a double complex

Hey all, I try to understand an argument in Lücks "A basic introduction to surgery theory" on page 51 which goes as follows: Let $\mathbb{Z} \pi$ be the group ring where $\pi$ denotes the ...
Alex_K's user avatar
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Extending the definition of "pure of dimension n" from simplicial complexes to simplicial sets?

Recall that a (combinatorial) simplicial complex $X$ is said to be $n$-dimensional if it contains at least one face of dimension $n$ and no faces of dimension $n+1$. Further, an $n$-dimensional ...
Harry Gindi's user avatar
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Line bundles in abelian $\otimes$-categories

By an abelian $\otimes$-category I mean a symmetric monoidal category $(\mathcal{A},\otimes,\mathcal{O})$, such that $\mathcal{A}$ also is an abelian category and for every $M \in \mathcal{A}$ the ...
Martin Brandenburg's user avatar
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269 views

Classifying Algebra Extensions over a fixed extension?

There are lots of "Ext groups" in homological algebra which measure extensions of various things. I'm sure there must be a homological algebra machine for computing the following, and I'm hoping that ...
Chris Schommer-Pries's user avatar
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1 answer
473 views

Why two definitions of localization of categories coincide?

Let $\mathsf{C}$ be a category and $S$ be a collection of morphisms in $\mathsf{C}$. In this generality, we can construct the localization $S^{-1}\mathsf{C}$ by posing its objects to be the same as ...
Gabriel's user avatar
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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{...
cellular's user avatar
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On universally closed morphisms of reduced schemes

In this question I'd like to examine some properties of universally closed morphisms. The question is self-contained. It can also be seen as a follow-up to this question. Let $R$ be a discrete ...
user avatar
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117 views

The tensor product of two topological complexes with closed range

A Künneth formula by Grothendieck/Schwartz states the following: Let $A, B$ be chain complexes of nuclear Fréchet spaces. If the differentials $d_A, d_B$ are topological homomorphisms (meaning in ...
Lukas Miaskiwskyi's user avatar
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1 answer
128 views

Covariant splittings of Hopf algebra projections

What is an example of a pair of Hopf algebras $(A,B)$ with a surjective Hopf algebra map $\phi:A \to B$ such that $\phi$ does not admit a $B$-bi-comodule splitting $s:B \to A$? To be clear, the right $...
Jake Wetlock's user avatar
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(Stable) Auslander algebras in a specific example

Let $k$ be a field and $Q$ be the quiver with two vertices 1 and 2 and three arrows: $a$ from 2 to 1, $b$ from 2 to 1 and $c$ from 2 to 2. Let $I_1=\langle ab-c^2,ba\rangle$ and $I_2=\langle ab-c^2,c^...
Mare's user avatar
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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 ...
Mare's user avatar
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Existence of non-trivial reflexive modules

Recall that a module $M$ over a ring $R$ is reflexive in case the natural evaluation map $f_M:M \rightarrow M^{**}$ (where $M^{*}=Hom_R(M,R)$) is an isomorphism, where $f_M(m)=g$ with $g(h)=h(m)$, ...
Mare's user avatar
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Is the sheaf associated to a differential structure of a specific type?

On a set $X$, let us define a set $\mathcal{D}$ of functions from $X$ to $\mathbb{R}$. Consider first the initial topology $\tau_\mathcal{D}$ on $X$ with respect to $\mathcal{D}$, i.e. the coarsest ...
Christophe Wacheux's user avatar
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Definition of Non-commutative de-Rham-Cohomology

Let $A$ be a (not necessarily commutative, associative) $k$-algebra. The bimodule of non-commutative one-forms $\Omega^1_A$ is the free $A$-bimodule generated by symbols $da$, $a \in A$, subject to ...
Matthias Ludewig's user avatar
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325 views

Two approaches to periodic cyclic cohomology

Cyclic cohomology may be defined in several ways: the easiest way to define it is via a subcomplex $C^*_{\lambda}$of Hochschild complex consisting from cyclic cochains. There are also other ...
truebaran's user avatar
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2 answers
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Algebras with all simples reflexive

Let $A$ be a ring. Recall that a module $M$ is called reflexive in case the canonical evaluation map $f_M : M \rightarrow M^{**}$ is an isomorphism. Here $(-)^{*}$ denotes the functor $Hom_A(-,A)$. In ...
Mare's user avatar
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General existence theorem for cup products

I'm curious if it is possible to formulate cup products and prove that they exist in a general way which would subsume a lot of examples: e.g. group cohomology, sheaf cohomology for sheaves on ...
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