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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
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 ...
34 votes
2 answers
5k views

Example Wanted: When Does Čech Cohomology Fail to be the same as Derived Functor Cohomology?

I want to know exactly how derived functor cohomology and Cech cohomology can fail to be the same. I started worrying about this from Dinakar Muthiah's answer to an MO question, and Brian Conrad's ...
Chris Schommer-Pries's user avatar
31 votes
1 answer
3k views

What was the error in the proof of Roos' theorem?

Background: In 1961, Roos (who, sadly, apparently passed away just last month) purported to prove [1] that in an abelian category with exact countable products (AB4${}^\ast_\omega$), limits of inverse ...
Tim Campion's user avatar
  • 63.9k
29 votes
4 answers
3k views

origin of spectral sequences in algebraic topology

I have the following somewhat vague question. I am not sure if it is appropriate for this forum, please feel free to close (or migrate to stackexchange). I have been "brought up" as an algebraic ...
Tom Bachmann's user avatar
  • 1,961
21 votes
6 answers
3k views

A ring such that all projectives are stably free but not all projectives are free?

This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
Hailong Dao's user avatar
  • 30.5k
21 votes
1 answer
2k views

A spectral sequence for computing cohomology of a space from that of its strata

Let $X$ be a smooth complex variety (not necessarily compact) and let $D$ be a normal crossings divisors with components $D_1$, $D_2$, ..., $D_N$. For a set of indices $I$, let $D_I = \bigcap_{i \in I}...
David E Speyer's user avatar
17 votes
1 answer
1k views

Who wrote `if only I could understand the equation $d^2=0$'?

I remember reading something like if only I could understand the equation $d^2=0$ as an epigraph to a memoir on homological algebra. I think the author was Henri Cartan, and the epigraph may have ...
Matthieu Romagny's user avatar
16 votes
1 answer
360 views

Moduli space of boundary maps with prescribed chain and homology groups?

Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C_n$ and $H_n$ of $R$-modules, which are tame in every ...
Vidit Nanda's user avatar
  • 15.5k
11 votes
1 answer
793 views

Is the pairing between contours and functions perfect (modulo the kernel given by Stokes' theorem)?

Let $s: \mathbb C^n \to \mathbb C$ be a homogeneous degree-$d$ polynomial which is nonsingular (in the sense that the hypersurface it defines in $\mathbb{CP}^{n-1}$ is smooth; equivalently the ...
Theo Johnson-Freyd's user avatar
10 votes
1 answer
679 views

Homology of the free loop space of a Grassmanian

Is there any reference for calculation of the rational homology of the free loop space $H_*(\mathcal{L}Gr(k,n),\mathbb{Q})$ of a complex Grassmanian? More precisely, I am interested in computing ranks ...
Filip's user avatar
  • 1,677
10 votes
0 answers
813 views

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 ...
Dan Petersen's user avatar
  • 40.2k
9 votes
1 answer
457 views

Deformations of Ext rings

Let $k$ be a base ring and $k[x]$ the ring of polynomials in an indeterminate $x$ over $k$. Consider a (not necessarily commutative) algebra $A$ over $k[x]$ and two $A$-modules $M$ and $N$. Then for ...
Craig Westerland's user avatar
8 votes
0 answers
394 views

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 ...
Patrick Elliott's user avatar
7 votes
1 answer
812 views

How to prove that topological Hochschild homology of a smooth proper stable k-linear infinity category is dualizable?

Let $k$ be a perfect field of characteristic $p$. I heard that the Topological Hochschild homology of a smooth proper stable infinity category (or dg-category) is dualizable as a THH(k)-module ...
Keiho Matsumoto's user avatar
6 votes
1 answer
425 views

Why does $p_*p^! A$ deserve to be called homology with coefficients in $A$?

Let $p:X\to S$ be the unique map from a (locally compact) topological space $X$ to a point. Since $\underline{\hom}(\underline{\mathbb{Z}},-)$ is the identity functor, we have that $\Gamma(X,-)=\hom(\...
Gabriel's user avatar
  • 711
5 votes
1 answer
782 views

Resolutions by free Differential Graded Algebras

I am struggling to find references for explicit computations of things like symmetric algebras and resolutions in the dg context. (any pointers in this direction would be highly appreciated!) I have ...
Yosemite Sam's user avatar
  • 1,889
5 votes
1 answer
163 views

Relative flasqueness?

It is known that a flasque sheaf on a topological space has trivial cohomology. Suppose that we are in a relative situation of a smooth fibration $\pi: X \to S$ and $F$ is a sheaf on $X$. Is there ...
Vladimir Baranovsky's user avatar
5 votes
0 answers
290 views

About the left adjoint of $f^*$

In lots of different cases (Verdier duality, Grothendieck duality, étale cohomology, ...) the very existence of a (right) adjoint to the sheaf functor $f_!$ gives useful information. (I'm going to ...
Gabriel's user avatar
  • 711
4 votes
2 answers
228 views

Is there a Čech-like way of computing $H^\bullet(X,M^\bullet)$ or even $\mathsf{R}f_* M^\bullet$?

Let $X$ be a topological space (or a site) and let $M$ be a sheaf on $X$. If $X$ is paracompact, or if $X$ is a noetherian separated scheme and $M$ is quasi-coherent, or if $X$ is quasi-projective ...
Gabriel's user avatar
  • 711
4 votes
1 answer
592 views

Six functor formalism for quasi-coherent $D$-modules

Let $X$ be a smooth scheme over a field $k$ and let $\mathsf{D}_{\text{qc}}(\mathcal{D}_X)$ be the full subcategory of $\mathsf{D}(\mathcal{D}_X\mathsf{-Mod})$ composed of the complexes of $\mathcal{D}...
Gabriel's user avatar
  • 711
4 votes
1 answer
417 views

Second differential in long exact sequence for Cech cohomology for nonabelian groups

I do not really think it fits MO, but I posted it in MathStackExchange with little success, so... Assume that we have a short exact sequence of, say, Lie groups $1\rightarrow A\rightarrow B\...
berndt's user avatar
  • 49
4 votes
0 answers
234 views

Is there an analogue to the koszul complex for constructible sheaves?

Given a variety $X$ and a complete-intersection morphism $$ Y \to X $$ is there an analogue of the Koszul complex for $\mathcal{O}_Y \in \textbf{Coh}(X)$ in the setting of constructible sheaves? ...
54321user's user avatar
  • 1,716
4 votes
0 answers
299 views

homology theory for affine and projective algebraic sets?

Given $f_1,\ldots,f_r\in K[x_1,\ldots,x_m]$, resp. homogeneous $f_1,\ldots,f_r\in K[x_0,\ldots,x_m]$, is there a chain complex built from these polynomials, such that any polynomial map $\varphi\!: Z(...
Leo's user avatar
  • 1,589
3 votes
0 answers
189 views

Pontryagin square on spin and non-spin manifold

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
annie marie cœur's user avatar
3 votes
0 answers
418 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 ...
Joaquín Moraga's user avatar
2 votes
0 answers
137 views

details of a dévissage argument for constructible sheaves

I am working on the following Künneth-type isomorphism from [SGA5, exposé III, 2,3]: $\mathrm{Settings}.$ Let $X_1, X_2$ be separated finite type schemes over the spectrum of a field $S=\mathrm{Spec}...
Wilhelm's user avatar
  • 375
2 votes
0 answers
133 views

Formulation of cap product in group-equivariant sheaf cohomology + applications?

Originally asked on Math SE but it was suggested I move it here. Suppose one has a distinguished cocycle in the group-equivariant sheaf cohomology $\Phi \in H^n(X, G, \mathcal{F})$ for a "nice&...
xion3582's user avatar
  • 101
2 votes
0 answers
371 views

How to deduce Künneth from its relative version (in cohomology of sheaves)

Let $p:X\to S$ and $q:Y\to S$ be morphisms of "spaces" over $S$. We have an isomorphism $$f_!(M\boxtimes N)=p_! M\otimes q_!N$$ in the derived category of "sheaves" over $S$, where ...
Gabriel's user avatar
  • 711
2 votes
0 answers
486 views

An alternative proof of Künneth spectral sequence, independent of Künneth formula for homology

I am currently reading Künneth spectral sequence, which is given below. Let $R$ be a ring and A$=\big\{A_n,d_n:A_n\longrightarrow A_{n-1}\big|d_{n-1}\circ d_n=0\big\}_{n\in \Bbb Z}$ be a chain ...
Sumanta's user avatar
  • 632
1 vote
1 answer
609 views

The Krull dimension of the tensor product of rings

The Krull dimension of a ring $R$ is defined as the length of the longest chain of prime ideals in it. Let $R_i$, for $i\in\mathbb{N}$ denote a sequence of commutative Noetherian rings of Krull ...
rr314's user avatar
  • 35
1 vote
1 answer
310 views

Realize as homology a given polynomial ring

I am wondering if one can realize a polynomial ring as the homology of some chain complex in the same sense that the homology groups of a space with a cell complex structure is the homology of its ...
MathBug's user avatar
  • 333
1 vote
0 answers
96 views

Chain complexes indexed over measurable subsets of $\mathbb{R}$: Towards a measurable notion of Euler Characteristic

I have for a while tried to generalize the notion of a chain complex in a way to obtain a "continuous" or at least "measurable" notion of Euler Characteristic. I have come up with ...
The Thin Whistler's user avatar
1 vote
0 answers
93 views

Spectral sequences associated to cohomologies of simplicial type and derived-functor type: Proof of convergence

Assume I have two cohomology theories $\mathrm{\tilde{H}^{*}}$ and $\mathrm{H^{*}}$, the latter being defined over a Grothendieck site $X$ as the derived functor of some left-exact covariant functor $\...
The Thin Whistler's user avatar