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Let $X$ be a scheme (say of finite type over a field), $i: Y \hookrightarrow X$ a closed immersion and $j : U \hookrightarrow X$ be the open complement.

Let $F$ be an étale sheaf over $X$ ($\overline{\mathbf{Q}}_l$-sheaf if it helps). In the book Weil conjectures, perverse sheaves and the l-adic fourier transform of Kiehl and Weissauer they claim (at least I think that's what they claim) that it is obvious that

$$\text{dim}(\text{Supp}(F)) \leq i$$ is equivalent to $$\text{dim}(\text{Supp}(j^*F)) \leq i \text{ and } \text{dim}(\text{Supp}(i^*F))\leq i$$ That the first line implies the second seems reasonable to me but i'm not sure how to show the other implication ?

I'm guessing the reasoning is that first of all it's just a statement about topological space : i.e. we are saying that a closed subspace $Z$ of $X$ has dimension less and $i$ if and only if $Z \cap U$ and $Z \cap Y$ have dimension less than $i$. But i'm not sure how to prove this nor that the claim above reduces to this.

karénine
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  • I don't think the topological statement at the end is true: Take $Z = X$ to be the spectrum of a DVR, $Y$ the closed point, and $i = 0$. – Elle Najt Apr 12 '16 at 05:24
  • For varieties over a field, some topological statement like that should be true. (See Hartshorne II.3.20e.) You can (probably) reduce to the integral case and then use the characterization of dimension in terms of the transcendence degree of the function field over the base field. (If the variety intersects this open $U$, then we see its function field using just the open set. Otherwise, we can compute its dimension purely in $U^c$.) (I'm not at all comfortable with Etale sheaves so I am hesitant to say whether or not this actually solves your problem.) – Elle Najt Apr 12 '16 at 15:48
  • thanks a lot for your comments, i'll look at Hartshorne :) – karénine Apr 12 '16 at 17:39

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