All Questions
Tagged with etale-cohomology homotopy-theory
16
questions
2
votes
0
answers
207
views
Using the Dold-Thom Theorem to define \'etale cohomology
For reasonable spaces $X$, the Dold-Thom Theorem states that $\pi_i(SP(X)) \cong \tilde{H}_i(X)$ where $SP(X) = \bigsqcup_i \mathrm{Sym}^i(X)$. There is a purely algebro-geometric realization of this ...
- 7,716
1
vote
0
answers
178
views
Does the Gross-Hopkins period map have a natural interpretation coming from derived algebraic geometry?
The Gross-Hopkins period map is a map on the $W(k)$-points of $LT_n \to P^{n-1}$, where $k = F_{p^n}$. It sends a lift $G$ of the Honda formal group to the 1-$d$ subspace $\omega_G$ of the Dieudonn'e ...
- 448
2
votes
1
answer
490
views
Computation of cohomology of Eilenberg-Maclane spaces
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Spf{Spf}$Background:
If $E$ is a complex-oriented spectrum, then $E^*(K(\mathbb{Z}/p^k,1))$ sits inside a long exact ...
- 448
2
votes
0
answers
148
views
When is map of $E_{\infty}$-ring spectra etale iff certain condition is fullfilled
When is it true that a map of $E_{\infty}$-ring spectra $R \to S$ is etale (in Lurie's sense) if and only if, $\operatorname{TAQ}^R(S) = 0$ and $ \pi_*(R)\otimes_{\pi_0(R)} \pi_0(S) = \pi_*(S)$?
- 448
3
votes
0
answers
206
views
Étale homotopy equivalent varieties are deformation equivalent
Let $k$ be an algebraically closed field of characteristic $p>0$.
Let $V_1$ and $V_2$ be étale simply-connected smooth proper varieties over $k$. Assume there is an isomorphism between the prime-to-...
user145520
1
vote
0
answers
117
views
Essential Image of the Étale Homotopy type
For any scheme $X$ we can associate the étale homotopy type $Et(X)$, which is a pro-object in the homotopy category of CW-complexes. My question is, do we have a good understanding of the essential ...
- 4,286
5
votes
0
answers
410
views
Modern context for hypercohomology spectra
In Thomason's paper Algebraic K-theory and étale cohomology, (Ann. ENS 1985, Numdam link) Thomason develops an elaborate theory of hypercohomology spectra, $\mathbb{H}(X,\mathcal{F})$ for presheafs of ...
- 255
10
votes
1
answer
1k
views
Derived base change in étale cohomology
Given a commutative square of ringed topoi
$$\begin{array}{ccc}X'\!\! & \overset{f'}\to & Y'\!\! \\ \!\!\!\!\!{\small g'}\downarrow & & \downarrow{\small g}\!\!\!\! \\ X & \...
- 27.9k
3
votes
1
answer
259
views
Morphism of sites and abelian sheaf cohomology
Let $f : \mathcal{C}\to\mathcal{D}$ be a morphism of sites (see the Stacks Project) with induced morphism of topoi
$$(f^{-1}, f_*) : Sh(\mathcal{D})\to Sh(\mathcal{C}).$$
By assumption, $f^{-1}$ is an ...
- 181
3
votes
1
answer
445
views
Spectral sequence for tensor product of complexes
Let $X$ be a scheme, $K^{\bullet}$ and $P^{\bullet}$ bounded complexes of abelian sheaves on $X_{\rm ét}$.
I want to compute the hypercohomology:
$$\mathbb{H}^*(X_{\rm ét}, K^{\bullet}\otimes^L_{\...
user113452
5
votes
1
answer
567
views
Derived completion of complexes
Suppose $K$ is a bounded above complex of free abelian groups, and take its derived $\ell$-adic completion $K^{\wedge,\ell} = R\lim (K/\ell^n)$ in the derived category, for $\ell$ a prime.
If $K\to L$...
user120812
6
votes
1
answer
301
views
Homotopy equivalence between two basepoints of the etale homotopy type of the one-torus
Let $T = \mathbb{G}_m$ be the torus, and let $\tilde{T}$ be its étale universal cover (a pro-object in schemes of finite type). Then both $T$ and $\tilde{T}$ have a well-defined étale homotopy type. ...
- 7,912
9
votes
0
answers
446
views
Etale homotopy types of schemes over positive characteristic fields
The étale homotopy type is a construction due to Artin and Mazur that generalizes the étale fundamental group. If $X$ is a scheme over a separably closed field $k$, then the étale homotopy type of $X$ ...
- 2,640
7
votes
1
answer
788
views
Etale and Algebraic K-theory with rational coefficients
We know that the Quillen-Lichtenbaum conjecture, now proved by Rost, Voevodsky, and Weibel, says that for smooth finite type $k$-schemes $X$, etale and algebraic $K$-theory with finite coefficient $\...
- 71
1
vote
0
answers
116
views
Etale homotopy type of simplicial preheaves
Etale topological type of a simplicial presheaf $X$ on the etale site of a scheme
is a pro-space. A simplicial presheaf can be written as a coequalizer of presheaves $P_n$, and each such presheaf is ...
- 143