All Questions
Tagged with homological-algebra reference-request
271 questions
3
votes
0
answers
67
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Fracturing $t$-structures
$\def\tee{\mathfrak{t}}$ Let $\tee_1,\tee_2$ be two $t$-structures on a triangulated category $\cal T$; call them fracturing if the two fiber sequences $\tau^\le_1X\to X\to \tau^\ge_1X$ and $\tau^\...
- 13.6k
15
votes
2
answers
2k
views
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 ...
- 3,726
10
votes
1
answer
657
views
Cap product on Leray-Serre spectral sequences
Let $\ p:X\to B\ $ be a fibration of spaces with fiber $F$. Then there are homological and cohomological Leray-Serre spectral sequences $E^{r}_{pq}$ and $E_r^{pq}$ that converge to $H_*(X)$ and $H^*(...
CommunityBot
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2
votes
0
answers
222
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Reference request for a "truncated version" of the de Rham algebra
Let's start on the $n$-torus for sake of simplicity.$\newcommand{\T}{\mathbb T}$
If I understand the relevant definitions correctly, the usual de Rham algebra of smooth differential forms on $\T^n$ is ...
- 25.8k
11
votes
2
answers
1k
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Relation between Gerstenhaber bracket and Connes differential
Let $C$ be an arbitrary algebra (more generally, a linear 1-category). The following structures are well-known:
A degree-0 product on the Hochschild cohomology $HH^*(C)$
$$
HH^*(C) \otimes HH^*(C) \...
- 3,686
4
votes
1
answer
273
views
Homological characterisation of standardly stratified algebras using Ext
Let A be a finite dimensional algebra and $S_1,S_2,...,S_n$ the simple $A$-modules and $P_1,..,P_n$ the indecomposable projective $A$-modules. For $i=1,...,n$, define the standard module $\Delta_i$ as ...
- 3,686
7
votes
0
answers
228
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Terminology for vanishing of Hochschild homology with symmetric coefficients?
In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then
it should hopefully be understood by most readers as saying $H^n(A,M)=0$...
- 25.8k
3
votes
1
answer
353
views
Reference for constructing tensor products of finitely presented functors
I need references related to the construction of tensor product between functors
Let $k$ be a commutative ring, $C$ a small $k$-linear category and $A$ cocomplete abelian category. Let $A^C$ denote $...
- 15.9k
3
votes
0
answers
188
views
Can one complete a morphism of commutative triangles to a "commutative cube" in a triangulated category?
This question is a continuation of Can one extend a morphism of commutative triangles to a morphism of octahedral diagrams?.
I am deeply grateful for the contributions there; they roughly say that ...
CommunityBot
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5
votes
0
answers
225
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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....
- 16.9k
1
vote
0
answers
70
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On (universal) additive functors making a given complex contractible: examples?
Let $M=(M^i)$ be a (cohomological) complex of objects of some additive category $A$ (I am mostly interested in "short" complexes; yet one may also consider an unbounded $M$). I am interested in those ...
- 16.9k
6
votes
1
answer
616
views
reference for "Topological algebra of Grothendieck"
I would like to have some references for Grothendieck's theory of "Topological algebra": a synthesis of homotopical and homological algebra,
with special emphasis on topoi.
- 9,627
9
votes
0
answers
383
views
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 $...
- 525
5
votes
1
answer
740
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soft: Reference/ Suggested Read: Homological Algebraic techniques in PDEs
I was reading this article on wikipiedia and was interested by the apparent link between Homological Algebra and PDEs. What is an accessible reference which showcases the link between these topics?
CommunityBot
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49
votes
2
answers
12k
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Kunneth formula for sheaf cohomology of varieties
What is a good reference for the following fact (the hypotheses may not be quite right):
Let $X$ and $Y$ be projective varieties over a field $k$. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent ...
- 18.7k
4
votes
0
answers
162
views
When there exists some "cone" of a morphism of (ind-representable) cohomological functors?
I am interested in cohomological functors from a certain small triangulated category $C$ to abelian groups.
The question is: given a tranformation $F\to G$ of two functors of this sort, is it ...
- 16.9k
7
votes
0
answers
116
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A "lower-central" filtration of Steenrod algebra?
$\renewcommand{\Atwo}{\mathcal{A}_2}$ So, a lot of good work has been accomplished by filtering the Steenrod algebras $\mathcal{A}_p$ in powers of the Augmentation ideal; For reasons partly ...
- 731
8
votes
1
answer
2k
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Is the derived category of perfect complexes idempotent complete?
Let $\mathcal{C}$ be a category. We call a morphism $\alpha: X\rightarrow X$ an idempotent if $\alpha^2=\alpha$ in $\mathcal{C}$. We call $\mathcal{C}$ is $\textit{idempotent complete}$ if any ...
- 16.5k
4
votes
0
answers
445
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When do we have $D_{\text{perf}}(\text{Qcoh}(X))\simeq D_{\text{perf}}(X)$?
Let $(X,\mathcal{O}_X)$ be a scheme (or more generally a ringed space). We know that in general the derived category of complexes of quasi-coherent modules $D(\text{Qcoh}(X))$ is not equivalent to the ...
CommunityBot
- 1
3
votes
0
answers
422
views
What kind of ringed space $X$ has the property that a locally free sheaf is projective in Qcoh$(X)$?
It is well known that for an affine scheme $X$, every finitely generated locally free sheaf $\mathcal{E}$ is projective in the category Qcoh$(X)$. i.e. the functor $\text{Hom}_{\text{Qcoh}(X)}(\...
- 9,019
15
votes
1
answer
3k
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Is a locally free sheaf projective in the category of $\mathcal{O}_X$-modules when $X$ is an affine scheme?
Let $X$ be an affine scheme and $\mathcal{E}$ a finitely generated locally free sheaf on $X$. It is obvious that $\mathcal{E}$ is a projective object in the category Qcoh$(X)$ since we can pass to ...
CommunityBot
- 1
11
votes
0
answers
668
views
Pairing of cohomology and homology Künneth formulas
Let $k$ be a field, and let $X$ and $Y$ be CW-complexes of finite type (although the question makes sense for $k$ a ring and for more general chain complexes of finitely generated free abelian groups)....
- 35.9k
6
votes
1
answer
362
views
What does an endomorphism in a triangulated category give rise to?
Let $D\xrightarrow[]\varphi D\xrightarrow[]kE\xrightarrow[]j\Sigma D$ be an exact triangle in a triangulated category. I am trying to figure out what structure emerges from this on the base of the ...
- 8,972
5
votes
1
answer
675
views
Resolution of a module as an $A_\infty$ module over resolution of an algebra
The following question looks like a basic question on resolutions over commutative rings, unfortunately I was not able to find a reference.
Let $A$ be a regular commutative noetherian ring (and ...
CommunityBot
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8
votes
0
answers
256
views
(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 ...
- 1,276
3
votes
0
answers
210
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]^{\...
- 595
13
votes
0
answers
680
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Singular chains generated by manifolds with corners --- does it really work?
Usually we define singular homology via the complex of singular chains:
$$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$
where the right hand side denotes the free abelian ...
- 18.7k
3
votes
1
answer
422
views
Tor dimension in polynomial rings over Artin rings
I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
- 706
35
votes
4
answers
3k
views
References for sign conventions in homological algebra
There is no shortage of sign conventions in homological algebra. And once these conventions are set out, there is no shortage of diagrams where an obvious commutative diagram on the underlying ...
CommunityBot
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1
vote
1
answer
359
views
Cohomology after completion
I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if it'...
- 3,051
3
votes
1
answer
575
views
What is the "higher version" of chain homotopy in singular homology?
In basic algebraic topology, we know the following well-known chain homotopy theorem:
Let $X$ be a topological space and $I=[0,1]$ be the unit interval. Let $S_*(X)$ and $S_*(X\times I)$ be the ...
- 9,627
1
vote
2
answers
534
views
Are there some websites for self learining of advanced mathematics? [closed]
Are there some websites for self learining of advanced mathematics?
For example, some great lecture vedio of differential geometry, Lie group , Lie algebra, algebraic topology and so on. Thanks
- 2,437
2
votes
1
answer
203
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Deformations of a complex trivial up to quasi-isomorphism
Let $(C, d_0)$ be some cochain complex over commutative ring $R$ and $R$ is an algebra over a field $k$, then I could consider complex $(C[t], d_0+td_1)$ over ring $R \otimes k[t]$ where $d_1$ is some ...
CommunityBot
- 1
6
votes
0
answers
465
views
Lagrangian (classical) BRST cohomology groups
I'm trying to understand BRST complex in its Lagrangian incarnation i.e. in the form mostly closed to original Faddeev-Popov formulation. It looks like the most important part of that construction (...
- 1,545
3
votes
0
answers
333
views
Global dimension of endomorphism algebra of a coherent sheaf
Let $X$ be a smooth projective variety over algebraically closed field of characteristic zero. In one of the versions of definition of tilting object $\mathcal{F}$ on $X$ there is a requirement that ...
- 1,545
6
votes
0
answers
313
views
Extension to a right exact functor
Setup. Let $k$ be a field, $\mathcal{C}$ be a $k$-linear abelian category, $\mathcal{D}$ an arbitrary $k$-linear category. Let $\mathcal{C}'$ be a full subcategory of $\mathcal{C}$ with the following ...
- 63.1k
12
votes
2
answers
523
views
A question on some computation of group cohomologies
Let $G=H\times J$, where $H\cong J\cong C_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group ...
CommunityBot
- 1
6
votes
3
answers
2k
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Künneth formula for Ext groups
Setup: Let $X,Y$ be quasi-compact quasi-separated schemes defined over a field $k$. If necessary, you can also assume that $X,Y$ are noetherian, but I don't want to assume that $X,Y$ have the ...
CommunityBot
- 1
6
votes
2
answers
647
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The 2-group of extensions
Let $A,B$ objects of an abelian category. Then we can define the abelian group $\mathrm{Ext}^1(A,B)$ as the set of isomorphism classes of extensions $0 \to B \to E \to A \to 0$, endowed with the Baer ...
- 4,519
3
votes
1
answer
432
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Sequences of maps between modules such that $\ker(d_n) \subseteq \text{im}(d_{n+1})$
Consider a sequence of maps between $R$ modules (where $R$ is a ring with unity) $$\cdots \rightarrow M_{n+1} \xrightarrow{d_{n+1}} M_{n} \xrightarrow{d_{n}} M_{n-1} \rightarrow \cdots$$ such that $\...
CommunityBot
- 1
3
votes
2
answers
744
views
Why every complex of injectives is homotopically injective (provided that, the injective dimension is finite)?
Let $\scr A$ be an abelian category with exact products and a cogenerator (e.g. $\scr A$ is a category of modules). Let ${\mathbf K}(\scr A)$ be the homotopy category of cochain complexes over $\scr ...
- 15.2k
4
votes
4
answers
284
views
Stratifications and Cohomology Computations
I am interested in references and suggestions concerning the use of stratifications in topology to inductively compute topological invariants. I would appreciate a fairly introductory reference on the ...
- 34.7k
4
votes
1
answer
598
views
Additive functors and Derived Categories
I have been learning about derived categories from Hartshorne's "Residues and Duality". One of the main theorems, as it is presented there seems to be the canonical isomorphism $RF \circ RG \cong R(F \...
CommunityBot
- 1
5
votes
0
answers
675
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Do exact functors commute with spectral sequences ?
Let $F: \mathcal{A} \to \mathcal{B}$ be an exact covariant functor of abelian categories and let
$$\mathscr{C}: A \to A \to B \to A$$ be an exact couple in $\mathcal{A}$ with corresponding spectral ...
- 16.2k
5
votes
2
answers
715
views
Examples of tilting objects that don't come from exceptional sequences
This is a question on geometric tilting theory. On smooth projective variety it is possible to define in general tilting object as perfect complex that satisfy some properties, but are there examples ...
- 5,062
3
votes
2
answers
353
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Morphisms between $K_0$
I suppose this is a question with a well known answer. Suppose $A$ and $B$ are two algebras over some field and there is a map
$$
f: \operatorname{K_0}(A) \to \operatorname{K_0}(B)
$$
is it ...
- 15.2k
9
votes
2
answers
796
views
Recovering an abelian category out of its derived category
I'm trying to learn more about derived category stuff and my curiosity has made me to ask these questions. Sorry if I'm being sloppy, I'm a new learner.
In Wikipedia it has been stated that since ...
- 1,273
6
votes
3
answers
2k
views
DG-projective vs. K-projective complexes
Hello!
I'm a student learning the basics of working with the unbounded derived category $D(\mathcal{A})$. I arrived at the natural question, "is every K-projective complex formed out of projective ...
- 109
7
votes
0
answers
275
views
Not isomorphic varieties with isomorphic tilting algebras
Let $X$ be a smooth projective variety over a field, than tilting object $T$ on $X$ is a perfect complex that is a compact generator of the derived category $\operatorname{D}(QCoh(X))$ and satisfies ...
- 1,545
5
votes
1
answer
448
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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:...
- 16.2k