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.
1,097 questions with no upvoted or accepted answers
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Isomorphisms between minimal $A_\infty$-algebras having identical $k$-truncations
Let $A_m =(A,0,m_2,m_3,\dots)$ and $A_n=(A,0,n_2,n_3,\dots)$ be two $A_\infty$-structures on a vector space $A$. Assume that
i) $A_m$ and $A_n$ are isomorphic, and
ii) $A_m$ and $A_n$ have the same ...
- 523
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Finite dimensionality of Ext(M,N)
Let $K$ be a field of characteristic $0$ and let $R$ be a (noncommutative) Noetherian $K$-algebra. Let $M$ and $N$ be simple left $R$-modules and assume further the following conditions on $R$:
(...
user91931
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Dimensions of dual vector spaces
Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...
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On functoriality of the Leray spectral sequence
The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
- 40.2k
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Complexes of representations with complementary central charges
This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary ...
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How acyclic models led to idea of model categories
The Wikipedia article about Acyclic models notices that the way that they were used in the proof of the Eilenberg–Zilber theorem laid the foundation stone to the idea of the model category.
Could ...
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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 ...
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3x3 lemma in triangulated categories
I am currently reading Le Stum's Rigid Cohomology and have encountered the following passage (proof of Proposition 5.2.16):
The deduction made here seems to be purely "triangulated category-...
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A characterisation of symmetric algebras using Hochschild (co)homology
A finite dimensional (connected if needed) $K$-algebra $A$ over a field $K$ is called symmetric when $A \cong Hom_K(A,K)$ as $A$-bimodules. Symmetric algebras are Frobenius algebras and include for ...
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How did Jouanolou define the cup product with no finiteness hypotheses in SGA 5?
In SGA 5 Exposé VII, at the beginning of §2, Jouanolou lets $X$ and $Y$ denote two schemes, $f:X\rightarrow Y$ a morphism, and $A$ the ring $\mathbf{Z}/\nu\mathbf{Z}$ where $\nu$ is an integer prime ...
- 1,217
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Cartan determinant of stable categories
Let $A$ be a finite dimensional algebra with finitely many indecomposable non-projective modules $M_1, M_2,...,M_n$.
Let $a_{i,j}:=\dim(\underline{Hom_A}(M_j,M_i))$ (the dimension of the stable Homs ...
- 26.5k
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Does a morphism which induces an isomorphism between Hochschild homology also induce an isomorphism between cyclic homology?
In a 1998 paper by B. Keller, the author consider the following problem in Section 1.4: Let $k$ be a commutative ring and $X$ a scheme over $k$. We can consider the cyclic homology as well as the ...
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Are dualizable objects in the derived category of a ringed topos perfect?
Recall that an object $a$ in a symmetric monoidal category $(\mathcal{C}, \otimes, e)$
is dualizable if there exists an object $b$ and morphisms $\varepsilon\colon b \otimes a \to e$
and $\eta\colon e ...
- 1,538
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Dualizable objects in homotopy category of chain complexes
The proposition 1.9 from "Duality, Trace and Transfer" by Dold and Puppe states that:
Given a commutative ring $R$, a chain complex of $R$-modules is strongly dualizable in $Ho(Ch(R))$, the homotopy ...
user95283
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(Torsion in) homology of free nilpotent groups
It is known that for free $k$-step nilpotent group on $r$ generators $N(r, k)$ its integral homology is torsion-free in degrees $\leq 3$ (obvious for 1 and 2, Igusa&Orr computations for 3). ...
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Categorification of definitions in the context of the derived category of quasi-coherent sheaves
Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned ...
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On the definition of triangulated categories
Triangulated categories were introduced in the 1960s by Grothendieck and Verdier in order to develop homological algebra in the framework of derived categories. An example of a triangulated category ...
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Derived functor of Malcev completion ( = "rational K-theory")
By a theorem of Keune (which is more of definition rather then theorem per se) K-groups of ring $R$ are Puppe's derived functors of pronilpotent completion of $GL(R)$. They are derived in the most ...
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Existence of universal extension between two modules?
I need a reference for the following fact:
Let $R$ be (for simplicity) an algebra over the field $k$, let $A, B$ be $R$-modules. Let $E = Ext^1_R(B, A)$. There is a natural isomorphism between $...
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The global dimension of fields
In the absence of the Axiom of Choice, it is not necessarily true that all vector spaces over a field have bases.
What are the possible global dimensions of fields in a model of ZF in which AoC ...
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Evens norm as a derived functor
In the theory of cohomology of groups there are several important maps between cohomologies: restriction, inflation, transfer and Evens norm. The first three of them can be clearly seen on the level ...
- 1,484
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On the invariance of the Kaledin class
In Formality of DG algebras (after Kaledin), Lunts introduces an $A_\infty$-Hochschild cohomology class, called the Kaledin class, controlling formality of an $A_\infty$-algebra up to a certain order. ...
- 6,777
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Projective $BP_\ast$-dimension of the $BP$-homology of classifying spaces of finite groups
Fix a prime $p$ and let $G$ be a finite group. Do we know the projective dimension of $BP_\ast (BG)$ as a graded $BP_\ast$-module? Or at least that it is finite?
My guess is the following:
The ...
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268
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What exactly goes wrong with $f_!$ outside of locally compact spaces?
Let $f:X\to Y$ be a morphism of ringed spaces. We define a functor $f_!:\mathcal{O}_X\mathsf{-Mod}\to\mathcal{O}_Y\mathsf{-Mod}$ as
$$\Gamma(U,f_!\mathscr{F}):=\{s\in \Gamma(f^{-1}(U),\mathscr{F})\:|\:...
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A model structure on CDGAs
Assume our base ring $k=\mathbb{Z}_{(p)}$ or $\mathbb{F}_p$ and
for simplicity
let us consider non-negative-graded connected algebras $A$ with an augmentation, i.e. $A^0=k,A^{<0}=0$.
The question: ...
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Rlim versus tensor product
Let $R$ be a coherent ring, and let $(M_n)_{n\geq 1}$ and $(N_n)_{n\geq 1}$ be two inverse systems of finitely generated flat $R$-modules. If $R^1 \lim M_n=R^1 \lim N_n = 0$, is it true also that $R^1 ...
- 13.1k
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Dyck paths of Dynkin type
(The conjecture is a homological algebra question, but question 2 is a pure combinatorics question given that the conjecture is true)
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...
- 26.5k
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Chevalley-Eilenberg cohomology of polynomial vector fields on $\mathbb{A}^2$
I have a question similar to one given here.
What is the cohomology of the Lie algebra of polynomial vector fields on an affine space $\mathbb{A}^2$ over a field of characteristic $0?$ (I'm interested ...
- 1,374
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Cohomology of constructible sheaves via exit paths
Let $X$ be a stratified space, with stratification $S$ (we will ignore technicalities).
The category of exit paths $Ex(X,S)$ is a directed refinement of the path groupoid of $X$ accounting for the ...
- 1,635
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Bringing cohomology recipes from algebra to topology?
In algebra, cohomology theories are often defined very roughly like this. Start with a scheme $X$ you want to study. Form a category-with-extra-structure (a site) $\mathcal{C}$ whose objects are, ...
- 1,150
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"Differential graded homological algebra" by Avramov, Foxby and Halperin
I am looking for "Differential graded homological algebra" by L. Avramov, H. Foxby, and S. Halperin, which is widely cited as a preprint o as a manuscript, e.g. https://scholar.google.com/scholar?q=L....
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Stability property for differential graded algebras
For a short exact sequence $1 \to A \stackrel{\iota}{\to} B \stackrel{\pi}{\to} C \to 1$ we have the situation that the square
is both a pushout in groups and also a pullback.
This would be wrong if ...
- 1,382
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homotopy MC element
A homotopy MC element is the Linfty analog of a Maurer-Cartan
element for a Lie algebra. Where is anything written about
homotopy MC elements as perturbations of strict MC elements?
- 3,880
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$n$-fold tensor products of $D(A)$ for finite dimensional algebras
Let $A$ be a finite dimensional quiver algebra over a field $K$ and let $D(-):=Hom_K(-,K)$ denote the natural duality (assume algebras are connected).
Define $\psi_A:=sup \{ n \geq 1 | D(A)^{\otimes ...
- 26.5k
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On constructible Hall algebra and instantons
I heard in a talk by Yan Soibelman that by starting with a quiver $Q$ with a set of vertices $I$ we can either symmetrize or anti-symmetrize its Euler-Ringel form $\chi_Q$. He claims that anti-...
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Fractal homological algebra
The usual definition of a chain complex requires for the indices to be integer numbers. However, taking inspiration from the theory of Hausdorff dimension, one can think of 'fractal' chain complexes (...
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Is every $R^iF(M)$ isomorphic to some $F(N)$?
Let $A$ and $B$ be abelian categories. Assume that $A$ has enough injectives. Is there a "useful" (take that to mean what you will) condition on $A$ and $B$ such that the following is true?
For all ...
- 3,079
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Functorial classes in Brauer group
For a smooth variety $X$ over a perfect field of characteristics $p$ the sheaf of differential operators is an Azumaya algebra(etale locally is isomorphic to endomorphisms of its center, which is ...
- 7,377
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Homology of inverse limits over inverse systems more complicated than towers
Most textbooks discussions of homology of inverse limits of chain complexes consider only “towers,” i.e. inverse systems indexed by the natural numbers. I’d like to find a reference that explains what ...
- 5,489
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Non-Standard Derived Equivalences of Non-Flat Algebras
I read that for algebras $R$ and $S$ (over a commutative ring), assuming that $R$ or $S$ is flat, the existence of a derived equivalence $\mathcal{D}(R) \to \mathcal{D}(S)$ implies the existence of an ...
- 156
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Kunneth spectral sequence
In Rotman's Homological Algebra, 1st edition, there is written:
Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also hold ...
- 1,589
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Does abstract nonsense of model categories determine the "nonlinear" morphisms of $L_\infty$ algebras?
Recall that a Lie algebra is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a ...
- 54.6k
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Do the Adeles Split?
I asked this question about a week ago here https://math.stackexchange.com/questions/288955/splitting-the-exact-sequence-of-the-idele-class-group, but got no answer so I thought I'd aske here and see ...
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Depth for non-commutative rings
The depth of a ring or module is one of the most basic invariants in commutative ring theory.
Q1: Is there also a powerful notion of depth for non-commutative rings ?
By a search in mathscinet, I ...
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Reference/ elementary proof of a result about projective dimension in group rings
Hello- I've had to use a result that sounds like it should be well-known, but I couldn't find any references, and my proof is rather unsatisfactory. I was hoping someone here could help! The problem ...
- 298
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(Reduced) cyclic homology of a free product of unital algebras
Shameless upfloat of 1-year old question - the motivation is that in general the corresponding Banach version is false, so I am trying to see where the proof breaks down, and what (if anything) can be ...
- 25.8k
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Twisting cochain intuition
I'm currently reading through Ed Brown's paper "Twisted tensor products, I", (MR105687, Zbl 0199.58201) and I couldn't find any simple examples of twisting cochains. I understand all ...
- 171
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Applications of Banach space homology
There is a well-developed theory of Banach space homology. What are some of its useful applications to Banach space theory and which important questions can one answer using it? In other words, how ...
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Homotopy theory of differential objects
In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...
- 6,025
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Maps in the Künneth theorem for K-theory of C*-algebras
The following is named the Künneth theorem for tensor products in the book by Blackadar on K-theory for operator algebras: If $A$ and $B$ are C*-algebras and $A$ is in the bootstrap class, then there ...
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