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Questions tagged [flat-cohomology]

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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^...
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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 ...
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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 ...
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1answer
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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 $k$-...
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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 $X$. ...
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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 ...
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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 ...
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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
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1answer
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) = ...
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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{...