# Questions tagged [flat-cohomology]

The flat-cohomology tag has no usage guidance.

**1**

vote

**0**answers

126 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^...

**5**

votes

**0**answers

132 views

### Does an fppf cohomological class annihilated by an etale cover come from etale cohomological group?

Let $X$ be a scheme, $F$ a sheaf on the fppf site of $X$, and $\alpha\in H^i_{\mathrm{fppf}}(X,F)$ such that it is trivialized by an etale cover of $X$. Does $\alpha$ lie in the image of the canonical ...

**3**

votes

**0**answers

136 views

### Finiteness of $H^2(X,\mu_n)$

Let $X$ be a proper curve over $k$ (algebraically closed) of characteristic $p>0$.
When is $H_{fl}^2(X,\mu_n)$ is a finite group?
It's true when $X$ is smooth but are there any more general ...

**3**

votes

**1**answer

150 views

### Known techniques to compute flat cohomology after base change

Let $f$ be some homogenous polynomial of degree $d>2$.
Let $X = \operatorname{Proj} (k[x,y,z]/(f))$ where $k$ is algebraically closed field of characteristic $p >0$.
Now let $R$ be a $k$-...

**1**

vote

**0**answers

126 views

### Group scheme representation from action of a group scheme on a variety?

Let f be some homogenous polynomial of d.
Let $X = \operatorname{Proj} (k[x,y,z]/(f))$
where $k$ is algebraically closed field of characteristic $p>0$.
Now $G$ is a group scheme acting on $X$.
...

**1**

vote

**0**answers

167 views

### Cohomology and base change

Let $f$ be some homogenous polynomial of $d$.
Let $X = \operatorname{Proj} (k[x,y,z]/(f))$ where $k$ is algebraically closed field of characteristic $p >0$.
Now let $R$ be a $k$-algebra.
What ...

**4**

votes

**0**answers

142 views

### What is the map induced by Frobenius on the cohomology group $H_{fppf}^1(X, \mathbb{G}_a)$?

Let $X$ be a projective curve of degree $p$ in $\mathbb{P}^2_k$, where $k$ is a field of charcteristic $p>0$. $X$ may not be reduced. I'm trying to compute $H_{fppf}^1(X, \alpha_p)$ using the ...

**8**

votes

**0**answers

215 views

### étale vs syntomic vs flat cohomology

Let $\mathscr{A}/X$ be an abelian scheme over $X$ of characterisitic $p$. The étale topology is not fine enough for the Kummer sequence for $\mathscr{A}$ to be (right) exact, but the syntomic and flat ...

**8**

votes

**1**answer

432 views

### flat/crystalline cohomology of abelian variety

Let $A/k$ be an abelian variety over an algebraically closed field and $\ell \neq \mathrm{char}\,k$.
In http://jmilne.org/math/articles/1986b.pdf, Theorem 15.1(b) it is proved that $$H^r_{et}(A, R) = ...

**8**

votes

**0**answers

412 views

### Weil pairing and Tate module for $p$-torsion in characteristic $p$

Let $A/\mathbf{F}_{p^n}$ be an Abelian variety with $\bar{A} = A \times_{\mathbf{F}_{p^n}} \bar{\mathbf{F}}_{p^n}$.
If $\ell \neq p$ is prime, there is a natural isomorphism $$\mathrm{H}^2_{\mathrm{...