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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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 ...
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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 $\...
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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 $...
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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, ...
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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 ...
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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 $...
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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 ...
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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?
...
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689
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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} \...
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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 ...
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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\...
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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 ...
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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 ...
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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......
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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 ...
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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:...
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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?
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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 ...
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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_{...
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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, ...
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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 ...
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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 $X \otimes -$ ...
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229
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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 ...
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288
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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 $...
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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 ...
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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 \...
5
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194
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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 ...
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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 ...
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327
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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 ...
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1
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189
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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{...
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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(...
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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 ...
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550
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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 $\cal P$ ...
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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 ...
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1
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509
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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 ...
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612
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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 ...
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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 ...
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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 ...
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1
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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{...
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389
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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 ...
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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 ...
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128
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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 $...
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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^...
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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 ...
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340
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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 $g(h)=h(m)$, ...
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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 ...
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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 ...
5
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325
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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 ...
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2
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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 ...
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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 ...