All Questions
Tagged with homological-algebra etale-cohomology
17 questions
2
votes
0
answers
137
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Leray spectral sequence for étale homology
Let $X$, $Y$, $Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. ...
- 12.9k
2
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0
answers
168
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When do we have $\bigoplus_{i + j = n} R^i f_* \mathbb{Q}_\ell \otimes_{\mathbb{Q}_\ell} R^j g_* \mathbb{Q}_\ell \cong R^{i + j} h_* \mathbb{Q}_\ell$?
Milne, Étale Cohomology, theorem 8.5 states the following version of the Künneth formula (in slightly greater generality). Let $\Lambda$ be a finite commutative ring. Let $X, Y, S$ be schemes with $S$ ...
- 531
3
votes
1
answer
241
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Could I get an interpretation for application of Euler characteristics in number theory？
As a beginner who just get in touch with Euler characteristics in this field, could I get some intuition for the arithmetic meaning of Euler characteristics of bounded complexes, for example Selmer ...
- 10.9k
2
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0
answers
137
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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}...
- 375
5
votes
1
answer
408
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On universally closed morphisms of reduced schemes
In this question I'd like to examine some properties of universally closed morphisms.
The question is self-contained. It can also be seen as a follow-up to this question.
Let $R$ be a discrete ...
CommunityBot
- 1
1
vote
1
answer
303
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A question about a truncated object
I was hoping someone could help me with the understanding of a particular truncated object. Here are some background:
For any object $A$ in an abelian category $\mathcal{A}$, we can view $A$ as an ...
- 40.2k
2
votes
0
answers
174
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Interpretation of some maps involving cohomology groups
I've asked some questions on Math Stackexchange regarding areas around my research but I received very little success with responses, so I thought I might try to share some of my other problems here ...
- 895
1
vote
0
answers
166
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To see that the fundamental class of a local complete intersection is independent of choice of regular sequence
In SGA 4½ ‘Cycle,’ Grothendieck defines (among other things) the fundamental class of a local complete intersection $Y\subset X$ ($X$ simply a noetherian scheme) of codimension $c$ locally as the cup-...
- 1,217
9
votes
0
answers
300
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How did Jouanolou define the cup product with no finiteness hypotheses in SGA 5?
In SGA 5 Exposé VII, at the beginning of §2, Jouanolou lets $X$ and $Y$ denote two schemes, $f:X\rightarrow Y$ a morphism, and $A$ the ring $\mathbf{Z}/\nu\mathbf{Z}$ where $\nu$ is an integer prime ...
- 1,217
3
votes
1
answer
267
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Morphism of sites and abelian sheaf cohomology
Let $f : \mathcal{C}\to\mathcal{D}$ be a morphism of sites (see the Stacks Project) with induced morphism of topoi
$$(f^{-1}, f_*) : Sh(\mathcal{D})\to Sh(\mathcal{C}).$$
By assumption, $f^{-1}$ is an ...
CommunityBot
- 1
2
votes
0
answers
325
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A question on direct limits of rings, and descent of ideals
Motivated by an étale cohomology calculation I am going to do, here is a question that should have positive and not too hard answer.
Let $A$ be a ring, $A_0$ a ring such that $A_0$ is equipped with ...
CommunityBot
- 1
2
votes
0
answers
20
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Cohomology of $\mathbf{G}_a\otimes_{\mathbf{Z}}\mathbf{G}_a$
Let $X$ be a smooth projective variety over a field.
Is $$H^p(X_{Zar},\mathbf{G}_a\otimes_{\mathbf{Z}}\mathbf{G}_a)$$
at all related to $H^p(X_{Zar},\mathbf{G}_a) = H^p(X,\mathcal{O}_X)$ via tensor ...
CommunityBot
- 1
4
votes
0
answers
432
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Reference request: sheaf-theoretic operations in the classical topology?
Like many graduate students before trying to learn something about étale cohomology and Deligne's proof(s) of the Riemann hypothesis part of the Weil conjectures, I am hunting for references detailing ...
- 273
8
votes
0
answers
286
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Functorial classes in Brauer group
For a smooth variety $X$ over a perfect field of characteristics $p$ the sheaf of differential operators is an Azumaya algebra(etale locally is isomorphic to endomorphisms of its center, which is ...
- 7,377
3
votes
0
answers
510
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Is there a Hochschild-Serre spectral sequence for unramified cohomology?
Similar to the Hochschild-Serre spectral sequence for etale cohomology ($H^p(G, H^q_{et}(X_L, \mathcal F|_{X_L})) \Rightarrow H^{p+q}_{et}(X, \mathcal F)$ for a Galois field extension $L/k$ with ...
- 31
8
votes
1
answer
714
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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 ...
CommunityBot
- 1
5
votes
1
answer
163
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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 ...
- 16.1k