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.
2,619
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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,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
- 39.3k
2
votes
1
answer
556
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Interpretation of Hochschild Homology groups
In all the literature I've come across there are many concrete interpretations of the first few Hochschild Cohomology groups. For example $HH^1(A,M)\cong Derivation/Inner Derivations$ etc....
In ...
- 4,969
2
votes
1
answer
523
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Castelnuovo Mumford Regularity
Consider the polynomial ring $S = C[x_1,\ldots,x_n]$. Let
$$
0 \rightarrow E_{n-1} \rightarrow \cdots \rightarrow E_1 \rightarrow E_0 \rightarrow I \rightarrow 0 ,
$$
be a ...
- 73
11
votes
1
answer
1k
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Motivation behind the definition of hochschild cohomology
For an associative algebra $A$ one can define the Hochschild cohomology of $A$ as $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$ (this definition also works for the graded and dg cases as ...
- 595
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0
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292
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generalisation of the universal coefficient spectral sequence
Suppose I have a bounded chain complex $C_{\ast}$ over the group ring $\mathbb{Z}G$ for a finite group $G$. In topology we are usually in a situation when $C_{\ast}$ is a complex of projective modules....
- 71
1
vote
3
answers
436
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Smooth affine algebras are Calabi-Yau
Are all smooth affine algebras over a field Calabi-Yau?
I'm thinking yes since they satisfy Van den Bergh duality with dualizing module themselves (have I made a mistake in this reasoning)/
- 27
1
vote
1
answer
597
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Formally smooth map from a regular ring
Let $A,B$ be two commutative noetherian rings. Let $f:A\to B$ be a formally smooth homomorphism. If $A$ is a regular ring (in the sense that all its localizations are regular local rings), does this ...
- 13
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331
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Pursuing an abelian categorical proof of the Zassenhaus Lemma
Fix an abelian category and suppose $M',M,N',N$ are sub-objects of a given object such that $M'\subset M$ and $N'\subset N$. Then there exists a canonical isomorphism
$\frac{M'+(M\bigcap N)}{M'+(M\...
- 41
3
votes
0
answers
540
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Tensor product of pullbacks of abelian categories
Does the tensor product of abelian categories commute with pullbacks? In more detail:
Let $k$ be a field. We consider $k$-linear small abelian categories $\mathcal{B},\mathcal{A}_0,\mathcal{A}_1,\...
- 61.5k
4
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0
answers
243
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When does a commutative DGA have a finitely generated quasi-free resolution?
Suppose that $A$ is a commutative dg-algebra (say over base $k$) which is bounded in non-positive degree (with cochain complex conventions). There exists a quasi-free resolution of $A$. My question is,...
- 15.3k
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214
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Coherent sheaves and Mitchell's embedding theorem
Let $S$ be a scheme. By Freyd-Mitchell's theorem the category of (quasi)coherent sheaves of $\mathcal{O}_S$-modules is equivalent to a full subcategory of the category of left $R$-modules for some ...
- 1,001
5
votes
0
answers
169
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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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2
votes
2
answers
225
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Does semi-free behave well under totalization
Suppose I have a dg algebra $(A,d)$ and a chain complex $M^\bullet$ of semi-free $(A,d)$ modules. I am hoping it is true that $ Tot^\coprod (M^\bullet)$ is again a semi-free $(A,d)$ module. Is this so?...
- 595
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1
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Known norm varieties and the Bloch-Kato conjecture
The Bloch-Kato conjecture states that
$K_M^n(k)/l \simeq H^n(k,\mu^{\otimes n}_l)$ for every $n,l$,while $l$ is invertible in $k$.
A important part in the proof of the Bloch-Kato conjecture is to ...
- 1,419
5
votes
1
answer
226
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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(...
- 1,419
6
votes
2
answers
470
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Interpretations of differentials in hypercohomology spectral sequences as Yoneda products
I would like to know whether the differentials in a particular hypercohomology spectral sequence can each be interpreted, in some natural way, as Yoneda products between extension groups.
More ...
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1
vote
1
answer
227
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Bounded algebras of finite global dimension
Let $k$ be a field, $Q$ an acyclic finite quiver and $I$ an admissible ideal of $kQ$.
I am looking for a reference for the fact that the bounded algebra $kQ/I$ has finite global dimension.
- 11
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votes
0
answers
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A factorization system on ${\rm Ch}(R)$
This is a sort of follow up to this MO question.
Let $R$ be a ring (eventually with good properties) and $\mathrm{Ch}(R)$ be the category of chain complexes of $R$-modules (eventually bounded).
...
- 13.2k
1
vote
0
answers
245
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Global dimension of a subalgebra with all units
(All rings here are always assumed to be unital and associative).
Setup
Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ satisfying:
If $u$ is a unit in $...
- 4,969
5
votes
2
answers
491
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Homotopy factorization of morphisms of chain complexes
This is a sort of follow up to this MO question.
Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). ...
- 6,719
8
votes
1
answer
695
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DG enhancements of $\ell$-adic derived categories
This question is similar in flavor to Existence of dg realization for 6 functors
Let $X$ be a complex variety and $D(X)$ the bounded derived category of constructible sheaves (the Euclidean topology ...
- 1,585
8
votes
1
answer
535
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Is a Gorenstein ring a quotient of a local complete intersection
The title says it all - Suppose you are given a noetherian Gorenstein local ring $(A,m,k)$ of finite Krull dimension.
Does there exist a local complete intersection ring $B$ such that $A$ is a ...
- 103
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0
answers
161
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How to understand $\mathcal{L}BG \simeq G/^{\text{ad}}G$ in term of simplicial sets?
First let $G$ be a topological group and $BG$ its classifying space. Let $\mathcal{L}BG=\text{Map}(S^1, BG)$ be the free loop space of $BG$.
We can see that $\mathcal{L}BG$ has the homotopy type of $...
- 8,707
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votes
1
answer
166
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Can André–Quillen homology detect the property of being Gorenstein?
Let $(A,m,k)$ be commutative noetherian local ring.
Can one detect if $A$ is a Gorenstein ring from the André–Quillen homologies $H_n(A,k,-)$?
- 103
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votes
0
answers
212
views
Trivial extensions by torsion-free groups
Let $A$ be an abelian group. Recall that $A$ is
($\bullet$) a Whitehead group if $\text{Ext}(A,\mathbb Z)=0$,
($\bullet$) a free abelian group if $\text{Ext}(A,D)=0$ for every abelian group $D$.
...
3
votes
0
answers
207
views
What is a morphism of $B_\infty$ algebra
Let $k$ be a commutative ring and $C$ a $ \mathbb{Z}$ graded $k$ module. A $B_\infty$ structure on $C$ is the datum of a differential and a multiplication on $ BC:= \bigoplus^\infty_{i =0} C[1]^{\...
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Cohomological dimension of the category of sheaves
Let $X$ be an $n$-dimensional manifold. Then for any sheaf $\mathcal{F}$ on $X$, the cohomology $H^i(X; \mathcal{F})$ vanishes for $i > n$.
Let $k$ be a field, and let $\mathrm{Shv}_k(X)$ be the ...
- 25.3k
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votes
0
answers
164
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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 ...
- 10.2k
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votes
0
answers
825
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Hochschild cohomology and bar resolutions
I asked the following question on mathstack but didn't receive any comments, so I thought I'd try my luck here.
Let $A$ be an associative algebra over a field $k$. One can define $HH^n (A,A)$ as $ ...
- 595
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votes
1
answer
131
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Injective dimension over enveloping algebra
Let $k$ be a field, and let $A$ be a commutative noetherian $k$-algebra.
If a finitely generated $A$-module $M$ has finite injective dimension over $A$, does this imply that $M\otimes_k M$ has finite ...
- 61
3
votes
1
answer
384
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Are there analogs of String Homology structure in cyclic homology?
I was reading John D.S. Jones' paper "Cyclic homology and equivariant homology" where he introduces a variant of cyclic homology that is isomorphic (as modules over the ring $K[u]$) to equivariant ...
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1
answer
955
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Exact triple yields a distinguished triangle in derived category
In Methods of Homological Algebra before Proposition III.3.5 there is a short comment:
"The next proposition shows that any exact triple [of complexes] can be competed to a distinguished triangle".
...
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0
answers
384
views
Reference needed: Homology of the blow-up
Given an algebraic variety $X$ over the complex numbers.
Let $V$ be a subvariety of $X$ and $\pi_V \colon X' \rightarrow X$
be the blow-up of $X$ at $V$.
It is posible in general to compute the ...
- 1,585
3
votes
1
answer
840
views
Is the space of smooth functions with compact support a DF space?
Is there a good criterion, when a (nuclear) LF space is DF (DFN)? What are references to find out about that? If not, is there a known way, to construct a projective resolution of LF spaces? Prosmans ...
- 31
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votes
0
answers
286
views
Flat and injective quasi-coherent sheaves
Let $X$ be a quasi-compact semi-separated scheme and
$$\varepsilon: 0\to A \to B \to C \to 0$$
be a short exact sequence of quasi-coherent sheaves. $\varepsilon$ is called a (categorical) pure exact ...
- 141
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votes
0
answers
4k
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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 ...
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votes
0
answers
215
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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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0
answers
442
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Are dualizable modules finitely generated?
Let $A$ be a commutative noetherian ring, and assume that $A$ has a dualizing complex $R$. Let $D(-) := \operatorname{RHom}_A(-,R)$ be the associated dualizing functor, and let $M$ be an $A$-module.
...
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votes
2
answers
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Spectral Sequences reference
What is the best reference for spectral sequences for mathematicians who are not experts at the subject, but would just like to open a book and find the SS they need, without going in to deep.
I'm ...
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votes
1
answer
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Cohomology ring of classifying space of spin group $\mathrm{BSpin}(n)$
$\DeclareMathOperator\BSpin{BSpin}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\BSO{BSO}\DeclareMathOperator\BO{BO}\DeclareMathOperator\BPin{BPin}$In the answer for question: Homology of ...
- 4,716
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vote
0
answers
200
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Example H-unital algebra which is not unital
What is an example of an algebra which is H-unital (that is its Bar complex is acyclic) and yet it is not unital?
- 11
2
votes
1
answer
170
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dg-flat complexes and their characters
Let $\otimes$ denotes the usual tensor products of complexes and symbols live in the category of chain complexes of $R$-modules. Let $X$ be a dg-flat complex (i.e. $X_n$ is flat for each n and $X\...
- 141
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vote
1
answer
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Finite universal delta-functors
Let $F^\bullet : \mathcal{A} \to \mathcal{B}$ be a cohomological delta-functor which vanishes in degree strictly greater than $d$.
Thus, $F^{d-\bullet}$ is a homological delta-functor.
Now assume ...
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0
answers
469
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Soft Question: What does periodic cyclic theory measure?
Ex1) The cyclic homology of $\mathbb{C}[X,Y]$ and that of the algebra of functions on the sphere $S^2$ have the same periodic cyclic homology. Clearly, however, these objects are topologically very ...
- 4,969
4
votes
1
answer
452
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Original sources for two theorems by Bass, Matlis and Papp
It is an interesting fact that a commutative ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective, and if and only if every injective $R$-module is a direct sum of ...
- 6,670
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votes
1
answer
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Why can't one modify Kaplansky's proof to conclude that every projective module is a direct sum of its finitely generated projetive submodules?
Due to the examples given in the answer to this question, I know that the conclusion is of course incorrect. But by reading Kaplansky's proof of theorem 1 in this paper and replacing every occurrence ...
- 950
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0
answers
181
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Chain homotopy of non-abelian category
How can one define the chain homotopy in non-abelian category? (The category I have in mind is the category of chain complexes of monoids.)
user42154
7
votes
3
answers
6k
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is the tensor product of projective modules again projective?
Let $R$ be a commutative ring and let $A_1$ and $A_2$ be (not necessarily commutative) $R$-algebras. Under which conditions on $A_1$ and $A_2$ is the following true:
For every projective $A_1$-module $...
- 93
3
votes
0
answers
137
views
Homology of the fixed points of the singular complex of a G-space
I posted the following to stackexchange a while ago [1], without any answers. Maybe the question is too unmotivated, but it seems very natural to me.
Suppose $X$ is a topological space and $G$ a ...
- 1,951
3
votes
1
answer
222
views
Exposition of the Calabi complex
I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature".
The complex is referenced as "Calabi complex" in various citing ...
- 4,776