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 existence of commutative singular cochains vs quasi iso between cochains on BT and its cohomology
I believe that there isn't a commutative model for the DGA of cochains on a space, because of cohomology operations. This question has some nice explanations of why this is so. For example, there is a ...
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Long exact sequence of Quillen derived functors
Let $F:\mathcal A\to \mathcal B$ be an additive right exact functor between abelian categories $\mathcal A$ and $\mathcal B$. Then for a short exact sequence
$$
0\to A\to B\to C\to 0
$$
in $\mathcal A$...
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How "commutative" are Cech cochains of a sheaf of commutative (dg) algebras?
Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative dg-algebras over $X$. Let $\mathfrak{U}$ be a fixed open covering of $X$ and $C^\bullet(\mathfrak{U},\mathcal{F})$ be the ...
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Criteria for being representation-infinite for subcategories of quiver algebras
Let $A$ be a quiver algebra over a field $K$ (maybe we need algebraically closed?).
Then the following is two statements are well known:
In case $A$ is representation-infinite, every Auslander-Reiten ...
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Strong Convergence vs Conditional Convergence for Spectral Sequences (Is there a simple explanation?)
I am curious if there is a relatively simple explanation of what is the difference between strong convergence and conditional convergence for Spectral Sequences?
(Hopefully a simpler explanation than ...
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K-flat, K-flabby resolution
Let $X$ be a topological space and $F$ a flat sheaf of abelian groups. It is well known that taking the Godement resolution of $F$ gives rise to a "flabbyflat" or "flasqueflat" resolution, in the ...
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Derived tensor products and Tor of commutative monoids
Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: $$\text{Hom}(M\otimes N,L)...
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Does the de Rham complex induce a functorial soft resolution of the category of cochain complexes of sheaves of vector spaces on a smooth manifold?
I apologize in advance if this is pretty straightforward; I'm a differential geometer and physicist by training so my homological algebra and homotopy theory are a bit weak.
Question: Let $M$ be a ...
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What is the relationship between Frobenius extensions and Separable extensions
Let $R\to S$ be an extension of possibly non-commutative rings. I am interested in the relationship between $R\to S$ being Frobenius and it being separable.
If it is a Frobenius extension, then there ...
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What is the algebro-geometric or measure-theoretic "content" of Dhillon and Mináč's motivic Artin symbols over an arbitrary ground field?
1. Short version. In this text, Dhillon and Mináč define motivic Artin symbols. Having fixed a ground field $k$ and a smooth projective curve $Y$ over $k$ equipped with the action of a finite group $G$...
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Ext$^1(D(A),A)$ for hereditary algebras
Let $A$ be a hereditary (non-semisimple) finite-dimensional algebra over a field $K$. Let $M:={\rm Ext}^{1}_A(D(A),A)$ ($ \cong D(\tau((D(A)))$ as left modules) and let $A^e$ be the enveloping algebra ...
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Recovering an A-infinity structure on an Ext-algebra from a quiver presentation
Let $A=KQ/I$ be a basic finite dimensional algebra given by a quiver with relations. Let $S$ denote the direct sum of the corresponding simple modules.
According to [Keller: A-infinity algebras in ...
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Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: references and convergence
Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $...
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Ext^1 for a local finite dimensional selfinjective algebra
Is there a nonprojective module $M$ over a finite dimensional local selfinjective algebra with $Ext^{1}(M,M)=0$? I asked this question also here:
http://arxiv.org/pdf/1609.00588.pdf.
There it is ...
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divided power structure on Hocschild cohomology?
Does Hochschild cohomology of a cocommutative Hopf algebra over a field of positive characteristic have a natural divided power structure?
If not, perhaps a certain natural extra structure on the ...
- 1,506
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Description of connecting maps of Derived functors
Let $C$ be an abelian category with enough injectives and $F$ be a left exact additive functor. Consider the short exact sequence $0 \to A' \to A \to A ''\to 0$. Therefore, we have connecting maps $\...
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Partial formality of A-infinity structure implies formality
Let $A$ be a (finite dimensional, unital, associative) $k$-algebra, where $k$ is a (algebraically closed) field. Let $M$ be a (finite dimensional) $A$-module. Then, it is known that $\operatorname{Ext}...
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Do differential objects form triangulated categories?
Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\...
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Second homology of mapping class group via universal coefficient theorem
Let $S_g$ be the closed, orientable surface of genus $g\ge 2$ and $Mod(S_{g,1})$ the mapping class group of homeomorphisms fixing a base point $x_0\in S_g$ modulo isotopies fixing $x_0$.
It is known ...
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Hochschild Cohomology of the Quantum Torus
I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...
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Can triangulated categories be "approximated by countable subcategories" (that are triangulated but not full!)?
For a given (finite) set of (objects and) morphisms $f_i$ in a triangulated category $C$ I am interested in a (non-full!) triangulated subcategory $C'\subset C$ of "small size" that would contain them....
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Can we obtain a derived category from an additive category? Like a category of Banach modules?
Let $A$ be a Banach algebra, let $A$-mod be the category of left Banach modules (as defined in Helemskii's "Banach and locally convex algebras"), $A$-mod is an additive category, but not abelian ...
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Projective dimension of ring over its center
If $A$ is a ring and $Z(A)$ is its center then what is a sufficient condition for the projective dimension of $A$ over $Z(A)$ (ie: $pd_{Z(A)}(A)$) to be finite?
(Assuming that $A\neq Z(A)$).
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Vanishing of Andre-Quillen homology and injective dimension
Let $(A,m,k)$ be a commutative local ring.
Assume that for all $n\ge 3$, the Andre-Quillen homology modules $H_n(A,k,k)$ vanish.
Does this imply that $A$ has finite injective dimension over itself?
...
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Definition of modules over $C_\infty$-algebras ("commutative $A_\infty$-algebras")
Let $\Lambda$ be a finite-dimensional associative algebra. We can think of this as an $A_\infty$-algebra with vanishing $m_i$ for $i \ne 2$ and consider $A_\infty$-modules $M$ over it. A list of ...
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interpretation of homology of "non-commutative Koszul complex"
Let $A = Sym^*(V)$ be a polynomial ring. The Koszul complex
$\cdots \to \wedge^2 V \otimes A \to V \otimes A \to A$
gives a resolution of the residue field $k$, so for any $A$-module $M$, the ...
- 10.2k
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Sign problem for the shift functor on DG modules
Let $\mathcal{A}$ be a Differential graded $k$ category, where $k$ is a commutative ring. I want to extend the shift functor $[1]$ on chain complexes to dg modules over $\mathcal{A}$. Now a right dg ...
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Do exact functors commute with spectral sequences ?
Let $F: \mathcal{A} \to \mathcal{B}$ be an exact covariant functor of abelian categories and let
$$\mathscr{C}: A \to A \to B \to A$$ be an exact couple in $\mathcal{A}$ with corresponding spectral ...
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Exactness is often an open condition. How often?
Let $\mathcal C$ denote an abelian category. Consider a formal $\mathbb Z$-graded object $V_\bullet$ in $\mathcal C$, by which I simply mean a $\mathbb Z$-indexed list $n\mapsto V_n$ of objects. A ...
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Minimal Koszul-Tate resolutions
In what generality of commutative associative algebras does there exist a minimal Koszul-Tate resolution? Or what is the most general condition known?
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Defining the cup product in Ext using a Kunneth formula
I want to make a Kunneth product of sorts on Ext. In particular, letting $C_*$ be a $R$-free resolution for $k$ over a $k$-hopf algebra $R$, elements in $Ext_R(k,k)$ are represented by maps in $Hom_R(...
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an exercise in the Rotman book
I read the following exercise in the book of Rotman: "an introduction to homological algebra"
9.21: In the situation $(_RA, _SB_R, _SC)$ with $B$ $R$-projective, use the adjoint isomorphism to obtain ...
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Coherence of the monoid algebra of a non-finitely generated monoid
Let $P$ be an integral, sharp, finitely generated commutative monoid (say even torsion-free and saturated if you like), and consider the "rational cone" $P_\mathbb{Q}\subseteq P^{gp}\otimes_\mathbb{Z}...
- 1,020
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Alternative approaches to the universal coefficient theorem
Let $A$ be a chain complex of free $R$-modules over a PID $R$, and let's assume $A$ has finite cohomological type, by which I mean $H^\ast(A)$ is finitely generated in each dimension and $0$ for large ...
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Pullbacks of Abelian Categories and their Ext-Groups
Let $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ be abelian categories and $F: \mathcal{A} \to \mathcal{C}$ and $G: \mathcal{B} \to \mathcal{C}$ be functors between them. It is then possible to ...
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Extensions of maps between graded modules
Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\...
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Question about unbounded derived categories of quasicoherent sheaves
This is a bit of a strange question since I more or less want to ask the MO crowd whether I've understood the situation correctly. If you have an unbounded complex of quasicoherent injective sheaves $...
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A smooth twisted tensor product of dg algebras?
I want to consider a Z/2Z dg algebra. As an algebra, it is generated over $\mathbb{Q}$ by two elements where x is even and e is odd with the relations $xe=ex$ and $e^2=1$(this makes it in particular ...
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Is a certain A-infinity algebra (homologically) smooth?
An A-infinity algebra $A$ is smooth a'la Kontsevich if it is perfect as an $A$-$A$-bimodule. I am wondering about the standard tricks to show smoothness of given algebras. A relatively basic example ...
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$\mathbb{Z}[T]$-Solidification in light condensed setting
In the lectures to "Analytic Stacks" Scholze and Clausen introduced a new concept of "light" condensed mathematics. In Lecture 7 Clausen introduces the derived $T$-solidification ...
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An $E_{\infty}$-algebra is a $C_{\infty}$-algebra?
Past this question in MO have raised the following questions for me.
Question
In characteristic $0$, it is well-known that a Kadeishvili‘s $C_{\infty}$-algebra is an $E_{\infty}$-algebra.
However, do ...
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Hochschild cohomology of path algebra as a cohomology of simplicial complex
M. Gerstenhaber and S. D. Schack have shown that a cohomology of simplicial complex can be expressed as a Hochschild cohomology of path algebra constructed from this complex (link).
Is the opposite ...
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Infinite dimensional homology theory for submanifolds of Hilbert and Banach spaces
Is there a version of homology theory for spaces for which explicitly infinite dimensional "cells" are allowed?
The spaces in question include e.g.
\begin{equation}
X = (x: x \in l_2: p_i(x) ...
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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)) =...
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Finitistic dimension conjecture — why artin algebras?
As I understand it, the finitistic dimension conjecture says that if a ring $A$ is finitely generated over a commutative artinian ring $K$, then the finitistic dimension of $A$ is finite.
My question ...
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base change property of Topological Hochschild homology
What is the "base change property" of topological Hochschild homology?
In Proposition 11.10 of Bhatt-Morrow-Scholze's paper "Topological Hochschild homology and integral p-adic Hodge ...
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Lifting theorem for modules over a DGA
In the survey paper "Cartan's Constructions, the homology of $K(\pi,n)$'s, and some later developments" by J. C. Moore ( http://www.numdam.org/item/AST_1976__32-33__173_0/ ) he states a ...
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Classification of 2-periodic triangulated categories
Let $T$ be an algebraic triangulated (k-linear over a field, Hom-finite, idempotent-complete) category. Call $T$ 2-periodic if $\Omega^2(X) \cong X$ for all $X \in T$.
Question 1: Is there a ...
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Regular coherence of tensor algebras
Under what conditions on a bimodule $M$ over a noetherian commutative ring $R$ is the tensor algebra $T=T_R(M)$ regular coherent? A theorem of Gersten says this is true for a free bimodule $M$. If $M$ ...
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Multi-variable cohomology operations
Intuitively, cohomology operations are ways to locally compute a cocycle $\alpha\in H^i(X, G)$ from any cocycle $\beta\in H^j(X, H)$. Formally, they are in one-to-one correspondence with homotopy ...
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