Loading web-font TeX/Math/Italic

The Stacks project

Lemma 59.102.5. With notation as above.

  1. For X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{ph}) and an abelian torsion sheaf \mathcal{F} on X_{\acute{e}tale} we have \epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F} and R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0 for i > 0.

  2. For a proper morphism f : X \to Y in (\mathit{Sch}/S)_{ph} and abelian torsion sheaf \mathcal{F} on X we have a_ Y^{-1}(R^ if_{small, *}\mathcal{F}) = R^ if_{big, ph, *}(a_ X^{-1}\mathcal{F}) for all i.

  3. For a scheme X and K in D^+(X_{\acute{e}tale}) with torsion cohomology sheaves the map \pi _ X^{-1}K \to R\epsilon _{X, *}(a_ X^{-1}K) is an isomorphism.

  4. For a proper morphism f : X \to Y of schemes and K in D^+(X_{\acute{e}tale}) with torsion cohomology sheaves we have a_ Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_ X^{-1}K).

Proof. By Lemma 59.102.4 the lemmas in Cohomology on Sites, Section 21.30 all apply to our current setting. To translate the results observe that the category \mathcal{A}_ X of Cohomology on Sites, Lemma 21.30.2 is the full subcategory of \textit{Ab}((\mathit{Sch}/X)_{ph}) consisting of sheaves of the form a_ X^{-1}\mathcal{F} where \mathcal{F} is an abelian torsion sheaf on X_{\acute{e}tale}.

Part (1) is equivalent to (V_ n) for all n which holds by Cohomology on Sites, Lemma 21.30.8.

Part (2) follows by applying \epsilon _ Y^{-1} to the conclusion of Cohomology on Sites, Lemma 21.30.3.

Part (3) follows from Cohomology on Sites, Lemma 21.30.8 part (1) because \pi _ X^{-1}K is in D^+_{\mathcal{A}'_ X}((\mathit{Sch}/X)_{\acute{e}tale}) and a_ X^{-1} = \epsilon _ X^{-1} \circ a_ X^{-1}.

Part (4) follows from Cohomology on Sites, Lemma 21.30.8 part (2) for the same reason. \square


Comments (0)

There are also:

  • 3 comment(s) on Section 59.102: Comparing ph and étale topologies

Your email address will not be published.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.