Skip to main content
MathOverflow

Return to Question

added 17 characters in body; edited title; edited body
Source Link
Harry Gindi
Harry Gindi
  • 19.6k
  • 16
  • 123
  • 215

cristalline Crystalline cohomology via the syntomic site

Hello,

Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the sites $Y_{syn}$ and $(Y/W_n)_{cris}$ (where $W_n$ are the Witt vectors of $k$ of length $n$), of $Y$ with its syntomic topology and its cristallinecrystalline topology. Then the assignment $\mathcal O_{cris}:Z\mapsto H^0_{cris}(Y/W_n)$ is a sheaf on $Y_{syn}$. It is a fact that $H^*_{syn}(Y,\mathcal O_cris)$$H^*_{syn}(Y,\mathcal O_{cris})$ is canonically isomorphic to $H^{*}_{cris}(Y/W_n)$, but I don't see how to prove it.

So my question is: how toHow does one prove itthis fact?

Thanks!

cristalline cohomology via syntomic site

Hello,

Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the sites $Y_{syn}$ and $(Y/W_n)_{cris}$ (where $W_n$ are the Witt vectors of $k$ of length $n$), of $Y$ with its syntomic topology and its cristalline topology. Then the assignment $\mathcal O_{cris}:Z\mapsto H^0_{cris}(Y/W_n)$ is a sheaf on $Y_{syn}$. It is a fact that $H^*_{syn}(Y,\mathcal O_cris)$ is canonically isomorphic to $H^{*}_{cris}(Y/W_n)$, but I don't see how to prove it.

So my question is: how to prove it?

Thanks!

Crystalline cohomology via the syntomic site

Hello,

Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the sites $Y_{syn}$ and $(Y/W_n)_{cris}$ (where $W_n$ are the Witt vectors of $k$ of length $n$), of $Y$ with its syntomic topology and its crystalline topology. Then the assignment $\mathcal O_{cris}:Z\mapsto H^0_{cris}(Y/W_n)$ is a sheaf on $Y_{syn}$. It is a fact that $H^*_{syn}(Y,\mathcal O_{cris})$ is canonically isomorphic to $H^{*}_{cris}(Y/W_n)$, but I don't see how to prove it.

So my question is: How does one prove this fact?

Thanks!

added 7 characters in body
Source Link
Nicolás
Nicolás
  • 2.8k
  • 23
  • 28

Hello,

Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the sites $Y_{syn}$ and $Y_{cris}$$(Y/W_n)_{cris}$ (where $W_n$ are the Witt vectors of $k$ of length $n$), of $Y$ with its syntomic topology and its cristalline topology. Then the assignment $\mathcal O_{cris}:Z\mapsto H^0_{cris}(Y/W_n)$, where $W_n$ are the Witt vectors of $k$ of length $n$ is a sheaf on $Y_{syn}$. It is a fact that $H^*_{syn}(Y,\mathcal O_cris)$ is canonically isomorphic to $H^{*}_{cris}(Y/W_n)$, but I don't see how to prove it.

So my question is: how to prove it?

Thanks!

Hello,

Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the sites $Y_{syn}$ and $Y_{cris}$, of $Y$ with its syntomic topology and its cristalline topology. Then the assignment $\mathcal O_{cris}:Z\mapsto H^0_{cris}(Y/W_n)$, where $W_n$ are the Witt vectors of $k$ of length $n$ is a sheaf on $Y_{syn}$. It is a fact that $H^*_{syn}(Y,\mathcal O_cris)$ is canonically isomorphic to $H^{*}_{cris}(Y/W_n)$, but I don't see how to prove it.

So my question is: how to prove it?

Thanks!

Hello,

Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the sites $Y_{syn}$ and $(Y/W_n)_{cris}$ (where $W_n$ are the Witt vectors of $k$ of length $n$), of $Y$ with its syntomic topology and its cristalline topology. Then the assignment $\mathcal O_{cris}:Z\mapsto H^0_{cris}(Y/W_n)$ is a sheaf on $Y_{syn}$. It is a fact that $H^*_{syn}(Y,\mathcal O_cris)$ is canonically isomorphic to $H^{*}_{cris}(Y/W_n)$, but I don't see how to prove it.

So my question is: how to prove it?

Thanks!

Source Link
Nicolás
Nicolás
  • 2.8k
  • 23
  • 28

cristalline cohomology via syntomic site

Hello,

Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the sites $Y_{syn}$ and $Y_{cris}$, of $Y$ with its syntomic topology and its cristalline topology. Then the assignment $\mathcal O_{cris}:Z\mapsto H^0_{cris}(Y/W_n)$, where $W_n$ are the Witt vectors of $k$ of length $n$ is a sheaf on $Y_{syn}$. It is a fact that $H^*_{syn}(Y,\mathcal O_cris)$ is canonically isomorphic to $H^{*}_{cris}(Y/W_n)$, but I don't see how to prove it.

So my question is: how to prove it?

Thanks!