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33 votes
1 answer
740 views

Equivalence of topological Hochschild homology and Mac Lane homology via an equivalence $QA\simeq HA \wedge_{\mathbb{S}} H\mathbb{Z}$

Mac Lane homology is a homology theory for (not necessarily commutative) rings. Given a ring $A$, Eilenberg and Mac Lane define its cubical construction $QA$ to be a certain connective chain complex, ...
Matt Booth's user avatar
6 votes
2 answers
419 views

If $A$ is a cofibrant commutative dg-algebra over a commutative ring of characteristic $0$, then its underlying chain complex is cofibrant

Let $R$ be a commutative ring with characteristic $0$, namely it contains the field of rational numbers. Higher Algebra Proposition 7.1.4.10 tells that the category of commutative $R$-dg-algebras $\...
Francesco Genovese's user avatar
7 votes
0 answers
541 views

Convergence of a spectral sequence of a double complex

In Weibel's book, a spectral sequence $E^r_{p,q}$ is said to weakly converge to a graded object $H_{\ast}$ if for every $n$ there exists a filtration $\dots \subset F_{r}H_{n} \subset F_{r-1}H_{\ast} \...
Federico Barbacovi's user avatar
9 votes
1 answer
505 views

Is the tensor product of pretriangulated dg-categories a pretriangulated dg-category?

In "Grothendieck ring of pretriangulated categories", Bondal, Larsen and Lunts define a product of perfect (pretriangulated with Karoubian homotopy category) dg-categories as $A\bullet B:=Perf(A\...
AT0's user avatar
  • 1,482
7 votes
1 answer
470 views

Twisted spin bordism invariants in 5 dimensions

[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance! The spin $G$-bordism invariant can be twisted in the way that ...
wonderich's user avatar
  • 10.5k
2 votes
2 answers
542 views

Tensor product of mapping cones

Fix a ring $R$. If $A^*_i \to B^*_i \to C^*_i \to A^*_i[1]$ is a distinguished triangle of complexes of $R$-modules, for $i=1$ and $2$ (so $C_i^* = cone(f_i^*)$ where $f_i^*: A_i^*\to B_i^*$), is ...
Yellow Pig's user avatar
  • 2,964
6 votes
0 answers
142 views

Pin cobordism v.s. "KO" theory in low or in any dimensions

Fact: The spin cobordism is equivalent to "KO" theory in low dimension if we only consider the 2-torsion. This is related to a question and an answer supports the claim. Here we denote the $p$-...
wonderich's user avatar
  • 10.5k
1 vote
2 answers
201 views

Why is the flat cotorsion pair actually a cotorsion pair?

I asked this question some while ago on Stack Exchange but didn't get an answer (link), so I am trying it here as well. Fix a ringed space $(X,\mathcal{O})$ and denote by $\mathcal{F}$ the class of ...
Rene Recktenwald's user avatar
6 votes
1 answer
980 views

Group (Co)Homology of Symmetric Group

The question concerns the group homology or group cohomology of symmetric groups. The entries in groupprops.subwiki.org and in this MO post show the results for the symmetric group S$_4$. groupprops....
wonderich's user avatar
  • 10.5k
7 votes
0 answers
434 views

spectral sequence for a complex with two filtrations

Suppose $(C,d)$ is a chain complex: an abelian group with a map $d:C \to C$ such that $d^2 = 0$ (people like to assume $C$ is graded; if that helps - feel free to do so). A filtration is an ascending ...
Just Me's user avatar
  • 353
10 votes
0 answers
414 views

Segal-Freed-Hopkins-Teleman = Atiyah-Hirzebruch/Leray-Serre?

Freed-Hopkins-Teleman (section 3.7) generalise Segal's (Proposition 5.3) spectral sequence for equivariant K-theory to more general local quotient groupoids (that is, topological groupoids locally ...
David Roberts's user avatar
  • 35.5k
6 votes
0 answers
136 views

torsion part of homology of simplicial complexes [duplicate]

Let $n$ be a fixed positive integer and let $K$ be a simplicial complex with $N$ vertices. Suppose the $n$-th integral homology group of $K$ is $$ H_n(K)=\mathbb{Z}^{\oplus i}\oplus (\oplus _{p \...
Shiquan Ren's user avatar
  • 1,990
6 votes
1 answer
563 views

a question about Bockstein spectral sequence

I find the following theorem for Bockstein spectral sequence at http://pages.vassar.edu/mccleary/files/2011/04/MC10.fin_.pdf, page 459: Question. for a fixed $k$, if $\beta$ does not hit $H_k(X;\...
Shiquan Ren's user avatar
  • 1,990
5 votes
1 answer
915 views

Doing some homological algebra in triangulated categories

It's well known that chain complexes are an abelian category, and in particular we can consider chain complexes of chain complexes, i.e. double complexes. Given a double complex $A^{\bullet\bullet} \...
Dan Petersen's user avatar
  • 40.2k
9 votes
1 answer
370 views

Analogue of cyclic homology for e_n-algebras?

Cyclic homology may be defined as the primitive part (with respect to a natural product) of the homology of the Lie algebra associated with the "stabilization" of an associative algebra $A$. Here the "...
nikitamarkarian's user avatar
72 votes
3 answers
8k views

Where do all these projection formulas come from?

I have been intrigued for a long time by the formal similarity of results from different areas of mathematics. Here are some examples. Set theory Given a map $f:X\to Y$ and subsets $X' \subset X, Y'\...
Georges Elencwajg's user avatar
90 votes
5 answers
7k views

Algorithm or theory of diagram chasing

One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ...
Greg Kuperberg's user avatar