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7 votes
0 answers
156 views

Invariants of etale topological type that are not homotopy invariants

Artin--Mazur theory attaches etale homotopy type to reasonable schemes. Associated to this homotopy type are certain invariants of the scheme, such as etale fundamental group and higher homotopy ...
geometer's user avatar
  • 723
7 votes
0 answers
155 views

Do residues commute with transverse base change?

Fix a number $n > 0$. Given a smooth $\mathbb{Z}[1/n]$-scheme $X$ (i.e., a smooth scheme such that $n$ is invertible in its ring of functions), we may consider the étale sheaf $\mu_n$ on $X$ which ...
Yonatan Harpaz's user avatar
6 votes
0 answers
305 views

Does one need l to be invertible in S in order to consider the l-adic cohomology of S-schemes and motives?

When Ivorra defines the $l$-adic realization of $S$-motives (i.e. of Voevodsky's motives over a scheme $S$) he demands $l$ to be invertible in $S$. Is this condition really necessary? What happens ...
Mikhail Bondarko's user avatar
5 votes
0 answers
220 views

Is there a simple counterexample to étale proper base change on the unbounded derived category?

The best non-derived version of proper base change on the étale site of a scheme I know is that for $f : X \to Y$ proper and $g : Y' \to Y$ arbitrary, the base change morphism $g^{-1} R f_\star \...
C.D.'s user avatar
  • 605
5 votes
0 answers
238 views

Cohomology groups on small fppf site and small etale site are not the same

Let $F$ be a quasi-coherent sheaf on a scheme $X$. Is there an example where cohomology groups of $F$ on small fppf site of $X$ and small etale site of $X$ are not isomorphic?
geometer's user avatar
  • 723
5 votes
0 answers
735 views

Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?

This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases? Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to ...
Mikhail Bondarko's user avatar
4 votes
0 answers
416 views

Henselization of normal rings (Milne's EC)

The usual way to define the Henselization $A^h$ of a local ring $(A, \mathfrak{m})$ is by taking the direct limit $\varinjlim (B, \mathfrak q)$ over all étale neighborhoods of $A$ (i.e. pairs $(B,\...
user267839's user avatar
  • 6,028
3 votes
0 answers
274 views

Ind-etale vs weakly etale

In this article Bhatt and Scholze consider ind-etale and weakly etale maps of affine schemes. We have two (easy) statements, proven in Prop.2.3.3(1) and (5): -- any ind-etale map is weakly etale, -- ...
AlexIvanov's user avatar
2 votes
0 answers
99 views

Geometric generic point of a complete linear system

In the following context: Let $S$ be a connected smooth projective surface over $\mathbb{C}$, and let $\Sigma$ be the complete linear system of a very ample divisor $D$ on $S$. Let $d=\dim(\Sigma)$ ...
Roxana's user avatar
  • 519
2 votes
0 answers
416 views

Henselization and completions of local rings & schemes

That's the second part of my coarse becoming acquainted with Henselizations of fields and local rings. (in this question we focus on local rings as it is more algebro geometric motivated). So let $(R,...
user267839's user avatar
  • 6,028
2 votes
0 answers
220 views

When inverse image is conservative; a reference or a generalization?

I am interested in the following question: for $f$ being a morphism of schemes, which conditions ensure that $Rf^*_{et}$ is conservative? This is true if $f$ has a section or if $f$ is an \'etale ...
Mikhail Bondarko's user avatar
1 vote
0 answers
165 views

Discriminant ideal in a member of Barsotti-Tate Group

Let $S = \operatorname{Spec} R$ an affine scheme (in our case latter a complete dvr) and $p$ a prime. Then Barsotti-Tate group or $p$-divisible group $G$ of height $h$ over $S$ is an inductive system ...
user267839's user avatar
  • 6,028
1 vote
0 answers
56 views

local acyclicity when restricting to an hypersurface

Let $X$ be a smooth scheme over $\mathbb{C}$ and a constructible sheaf $K$ of complex vector spaces on $X\times\mathbb{A}^1$ and a function $g:X\rightarrow \mathbb{A}^1$. Suppose that $K$ is locally ...
prochet's user avatar
  • 3,472
1 vote
0 answers
180 views

Weaker version of smooth base change for étale sheaves

Consider the cartesian square of schemes $$ \require{AMScd} \begin{CD} X' @>{g'}>> X \\ @V{f'}VV @VV{f}V \\ S' @>>{g}> S \end{CD} $$ and the base change map $$ \eta : ...
Michele's user avatar
  • 11
1 vote
0 answers
189 views

Verdier duality on excellent schemes

Let $f:X\rightarrow Y$ be a regular morphism between $k$-schemes which are noetherian and excellent with a funcion of dimension. In the book by Illusie-Laszlo-Orgogozo, there is a theorem (4.4.1 in ...
prochet's user avatar
  • 3,472
0 votes
0 answers
188 views

constructibility for pushforward

Let consider a quasicompact open $j:U\rightarrow\mathbb{A}^{\mathbb{N}}$ over a field $k$, Is there an example where $Rj_{*}\mathbb{Z}/n\mathbb{Z}$ is not constructible, where $n$ is prime to the ...
prochet's user avatar
  • 3,472