The Stacks project

Theorem 64.3.3 (The Baffling Theorem). Let $X$ be a scheme in characteristic $p > 0$. Then the absolute frobenius induces (by pullback) the trivial map on cohomology, i.e., for all integers $j\geq 0$,

\[ F_ X^* : H^ j (X, \underline{\mathbf{Z}/n\mathbf{Z}}) \longrightarrow H^ j (X, \underline{\mathbf{Z}/n\mathbf{Z}}) \]

is the identity.

Proof of Theorem 64.3.3. We need to verify the existence of a functorial isomorphism as above. For an étale morphism $\varphi : U \to X$, consider the diagram

\[ \xymatrix{ U \ar@{-->}[rd] \ar@/^1pc/[rrd]^{F_ U} \ar@/_1pc/[rdd]_\varphi \\ & {U \times _{\varphi , X, F_ X} X} \ar[r]_-{\text{pr}_1} \ar[d]^{\text{pr}_2} & U \ar[d]^\varphi \\ & X \ar[r]^{F_ X} & X. } \]

The dotted arrow is an étale morphism and a universal homeomorphism, so it is an isomorphism. See Étale Morphisms, Lemma 41.14.3. $\square$

Comments (0)

There are also:

  • 12 comment(s) on Section 64.3: Frobenii

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03SN. Beware of the difference between the letter 'O' and the digit '0'.