All Questions
Tagged with etale-cohomology algebraic-groups
22
questions
2
votes
1
answer
262
views
Commutative group scheme cohomology on generic point
Setup:
Let $k$ be an algebraically closed field.
Let $C$ be a smooth connected projective curve over $k$.
Let $J$ be a smooth commutative group scheme over $C$ with connected fibers.
Let $j:\eta\to C$ ...
- 123
4
votes
1
answer
405
views
Étale group schemes and specialization
If $A$ is an abelian variety over a finite field $\mathbf{F}_q$, then $A(\mathbf{F}_q)$ (resp. $A(\overline{\mathbf{F}}_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}_q(t))$ is a ...
user501361
2
votes
0
answers
164
views
Explicit construction of a presentation of a constructible sheaf of $\mathbb{Z}$-modules
This question was prompted by the two following:
Constructible étale sheaves on X are étale algebraic spaces over X
Naive question about constructing constructible sheaves
If I have a ...
- 617
1
vote
0
answers
319
views
Any finite flat commutative group scheme of $p$-power order is etale if $p$ is invertible on the base
This question is immediately related to Discriminant ideal in a member of Barsotti-Tate Group
dealing with Barsotti–Tate groups and here I
would like to clarify a proof presented by
Anonymous in the ...
- 5,780
1
vote
0
answers
162
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
...
- 5,780
0
votes
0
answers
208
views
Endomorphism rings of flat group schemes
Let $R$ be a commutative ring and $X$ be a flat $R$-group scheme.
We call $\text{End}_R(X)$ the ring of endomorphisms of the $R$-group scheme $X$, defined over $R$.
Let $R\to S$ be a ring map ...
user147445
9
votes
1
answer
446
views
Structure of the variety of $n$-tuples of $m \times m$ matrices with zero product
Consider the functor sending a commutative ring $R$ to $\{(A_1,\dots,A_n) \in ( M_m(R) )^n | A_1 \dots A_n =0 \}$ which defines a scheme over $\mathbb Z$, let $X$ be its base change to $\mathbb C$.
...
- 6,188
6
votes
1
answer
511
views
How to compute Galois representations from etale cohomology groups of a generalized flag variety?
Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $...
- 6,188
4
votes
0
answers
151
views
Traces of Frobenius Endomorphism on Etale Cohomology and $G$-torsors
I have a smooth, projective, and rigid Calabi-Yau threefold $X$ defined over $\mathbb{Q}$. Such spaces always have integral models. Let's assume we have an action on $X$ by a finite abelian group $G$...
- 1,701
6
votes
1
answer
279
views
Cokernel of map of étale sheaves
Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ ...
- 115
7
votes
1
answer
524
views
Vector space objects in schemes - confusion
Let $R$ be the ring $\mathbf{C}\times\mathbf{C}$, and consider the affine line $\mathbf{A}^1_R$.
$\mathbf{A}^1_R$ can be given the structure of additive group scheme over $R$, denoted $(\mathbf{G}_a)...
user128448
2
votes
0
answers
256
views
etale cohomology of tori
Let $k$ be an algebraically closed field.
Let $A$ be a strictly henselian local ring which is a $k$-algebra.
Let $T$ a torus over $k((t))$.
Can we compute $H^{1}(A((t)),T)$?
- 3,452
8
votes
1
answer
403
views
Is $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$ true?
Let $X$ be a smooth geometrically connected scheme over a field $k$ of characteristic 0 (but not necessarily algebraically closed, I can take it to be a number field). Let $F$ be a finite algebraic ...
- 83
5
votes
1
answer
684
views
Cohomological interpretation of G-equivariant line bundles
In the theory of reductive algebraic groups, there is the following map (notation: $G$ reductive over an algebraically closed field, $T$ a maximal torus, $B$ a Borel, $X(T)$ the characters of $T$, $...
- 1,986
2
votes
1
answer
371
views
When is a $\overline{\mathbb{Q}}_{\ell}$-local system the inverse image of a $\overline{\mathbb{Q}}_{\ell}$-local system?
I am trying to learn character sheaf theory, and encounter the following question:
(*) Let $f\colon X\rightarrow Y$ be a morphism of quasi-projective smooth varieties over $\overline{\mathbb{F}}_q$, ...
- 1,606