# Questions tagged [etale-cohomology]

for questions about etale cohomology of schemes, including foundational material and applications.

**6**

votes

**0**answers

94 views

### Extension of sheaves of $\infty$-algebras

Let
$(\mathcal{C},\tau)$ be a site, whose Grothendieck topology is $\tau$
$F : \mathcal{C}\to D(k)$ a sheaf of $\infty$-$k$-algebras, with $k$ a ring and $D(k)$ the derived category of $k$-modules.
...

**3**

votes

**0**answers

121 views

### Functoriality of base change morphisms

Consider a commutative diagram of morphisms of schemes:
$$\begin{array}{ccccccccc}
X_2 & \xrightarrow{j_2} & Y_2 \\
f'\downarrow & & \downarrow f \\
X_1 & \xrightarrow{j_1} &...

**3**

votes

**0**answers

97 views

### On the existence of nice hypercovers

Let $\mathcal{C}$ be a site and $X$ a sheaf of sets on $\mathcal{C}$.
Then there exists a hypercover $K_{\bullet}$ of $X$ such that $K_n$ is a coproduct of representable presheaves on $\mathcal{C}$ ...

**9**

votes

**2**answers

1k views

### Étale cohomology of morphism whose fibers are vector spaces

Let $X\rightarrow Y$ be a morphism (may not be smooth) of varieties such that the fibres are vector spaces. Are the $l$-adic cohomologies of $X$ and $Y$ equal?
If not, under what condition (other ...

**3**

votes

**0**answers

156 views

### Does there exist trace maps between $\ell$-adic cohomology groups for finite flat morphisms?

Let $\operatorname{ProjSmooth}_k$ be the category of smooth projective varieties over a field $k$, and consider the $\ell$ cohomology theory $H^*(-)$ ($l \not = \operatorname{char} k$), how to define ...

**3**

votes

**1**answer

196 views

### Nearby cycles and extension by zero

Let $f: X\to \text{Spec}(R)$ be a proper and smooth morphism, with $R$ a strictly henselian dvr. Call $s = \overline{s}$ the closed point and $\eta$ the geometric point of $\text{Spec}(R)$.
Call $i_s ...

**2**

votes

**0**answers

124 views

### Berthelot’s comparison theorem and functoriality

Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$.
Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...

**4**

votes

**1**answer

143 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 ...

**14**

votes

**1**answer

548 views

### GAGA for henselian schemes

In this paper, F. Kato recollects basic facts on henselian schemes and proves some partial results towards GAGA in the context of henselian schemes.
Let $I$ be a finitely generated ideal in a ...

**6**

votes

**1**answer

171 views

### Vanishing of higher direct image of finite morphisms relative to the fppf topology

Let $f:X \to Y$ be a finite morphism of schemes.
Let $\mathcal{F}$ be a sheaf of abelian groups on the the etale site of $X$ then we know that $R^{i}f_{*} \mathcal{F} = 0$. Is this statement also true ...

**1**

vote

**0**answers

131 views

### Cohomology of constant sheaves

Let $X= spec(k)$ where $k$ is an algebraically closed field. Consider the constant sheaf $\mathbb{Z}$ on the fppf site of $X$. I'm interested in computing $H^1_{fppf}(X, \mathbb{Z})$. I know that $H^...

**1**

vote

**1**answer

177 views

### Swan-conductor and base change

Let $C$ be a proper smooth curve over a perfect field $K$ of positive characteristic $p$, $u: U \hookrightarrow C$ strictly open and $\mathfrak{F}$ a lisse (lcc) $\mathbb{F}_l $-sheaf $(l \neq p)$ on $...

**1**

vote

**0**answers

63 views

### Definition for equivariant $l$-adic sheaves

What is the definition of equivariant $l$-adic or ($\mathbb{Z}_l$-) sheaves?
Suppose $G$ acts on $X$, I could pick a $G$-equivariant etale sheaf of $\mathbb{Z}/l^n$ module on $X$ for each $n$, and ...

**2**

votes

**1**answer

157 views

### Vector bundles on henselian schemes

Let $X$ be a smooth and projective scheme over $\mathbf{Z}_p$.
We call $\mathfrak{X}$ the ringed space whose topological space is the topological space of the special fiber of $X$, and whose ...

**12**

votes

**1**answer

341 views

### Some basic questions on crystalline cohomology

Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ and ${X}$ its base change to an algebraic closure $k$ of $\mathbf{F}_q$.
Crystalline cohomology $H^*_{\rm cris}(X) := H^*((X/W(k))_{\rm ...