All Questions
Tagged with homological-algebra reference-request
271 questions
7
votes
0
answers
116
views
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
views
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 ...
- 9,019
4
votes
0
answers
445
views
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 ...
- 9,019
3
votes
2
answers
1k
views
An alternative definition of pseudo-coherent complex
Let $(X,\mathcal{O}_X)$ be a scheme or a general ringed space. First recall that a complex of $\mathcal{O}_X)$-modules $\mathcal{E}^{\bullet}$ is called strictly perfect if $\mathcal{E}^{\bullet}$ is ...
- 9,019
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
views
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 ...
- 9,019
11
votes
0
answers
667
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 ...
- 18.4k
9
votes
3
answers
1k
views
Poincaré duality for (co)homology of Lie algebras?
Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements.
In Algebra, Geometry, and Software Systems by Joswig & Takayama on p.200, it ...
- 1,589
3
votes
2
answers
490
views
Standard homology result on bicomplexes
Suppose you have got a bicomplex in an abelian category with objects $(A_{rs},d_{rs})$ such that $A_{rs} = 0$ for $r < 0$ or $s < 0$. Suppose furthermore that the rows $(A_{r\bullet},d_{r\bullet}...
- 5,327
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
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 ...
- 1,589
12
votes
1
answer
2k
views
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 ...
- 4,826
4
votes
1
answer
472
views
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 ...
- 6,700
3
votes
1
answer
228
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 ...
- 5,327
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 ...
- 165
13
votes
0
answers
680
views
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
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'...
- 1,049
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 ...
- 9,019
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
- 977
5
votes
2
answers
377
views
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 ...
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
5
votes
1
answer
674
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 ...
- 1,545
2
votes
1
answer
203
views
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 ...
- 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 ...
- 123
6
votes
3
answers
2k
views
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 ...
- 63.1k
6
votes
2
answers
646
views
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 ...
- 63.1k
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 ...
- 666
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 ...
- 4,920
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 \...
- 41
3
votes
1
answer
432
views
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 $\...
user32666
5
votes
0
answers
675
views
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
3
votes
2
answers
353
views
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 ...
- 1,545
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 ...
- 1,661
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 ...
- 329
5
votes
2
answers
714
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 ...
- 1,545
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
views
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:...
- 569
5
votes
2
answers
590
views
An explicit homotopy equivalence between the de Rham complex and the Cech-de Rham total complex
I'm currently in need an explicit formula in classical cohomology which I'm pretty sure is well known, but which I've been unable to find in the references I am aware of.
Let $X$ be a smooth ...
- 6,777
3
votes
1
answer
149
views
Interesting examples of a 4-torsion X in a triangulated category such that $2 End(X/2X)\neq 0$?
It is well-known that for the sphere spectrum $S$ in the ('topological') stable homotopy category the object $S/2S$ i.e. the cone of $S\stackrel{\times 2}{\to}S$, is not $2$-torsion.
So I wonder ...
- 16.9k
3
votes
0
answers
244
views
Reference Request: The Categorification of $\mathbb{Z}$ as cochain complexes of vector spaces
Just as the fact that a categorification of $\mathbb{N}$ is the category of finite dimensional vector spaces, a categorification of $\mathbb{Z}$ (in my mind) is the category of bounded cochain ...
- 9,019
13
votes
1
answer
694
views
Classification of long exact sequences
Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.
The category $\mathcal C$ is naturally additive as a subcategory of ...
- 3,174
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 ...
- 25.8k
10
votes
3
answers
2k
views
Where can I find a proof of the de Rham-Weil theorem?
Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
- 360
14
votes
1
answer
625
views
$A_\infty$ structure on Ext-algebras well defined?
Let $M$ be an object in an $k$-linear abelian category with enough projectives. Then one can construct an $A_\infty$-structure on the Ext algebra
$$Ext^\bullet(M,M)$$
as follows: One chooses ...
- 13.2k
3
votes
2
answers
543
views
Earliest/most standard reference for derived categories of hereditary algebras
Let $A$ be a hereditary algebra and let $\mathcal{D}$ be the derived category of bounded complex of finitely generated $A$-modules. Then, for any complex $C_{\bullet}$ in $\mathcal{D}$, we have $C_{\...
- 156k
6
votes
2
answers
684
views
Somewhat general question that includes: "Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?"
Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree?
Example: Given a dg ...
- 54.6k
2
votes
2
answers
2k
views
Torsion-free modules over a general ring
I want to know how to prove that a torsion-free module over a general ring is flat. In Lectures on Rings and Modules, T.Y. Lam proves this in the case where your ring is an integral domain. Can you ...
- 17