4
$\begingroup$

I have a question about following argument on page 3 in paper arxiv.org/abs/1702.04404 by Will Sawin (Proposition 3):

The claim is that for a finite, dominant map $f:X \to Y$ between varieties $X,Y$ over $k =\Bbb F_p$ composition of the adjunction counit and trace maps

$$ \Bbb{Q}_l \to Rf_*f^* \Bbb{Q}_l=Rf_{!}f^*\Bbb{Q}_l \to \Bbb{Q}_l $$

on level of derived cat of constructible $\ell$-sheaves ($\ell$ coprime to $p$) admits a splitting, ie $\Bbb{Q}_l$ as constant $\ell$-sheaf is a summand of the middle term (middle equality is due to finite is proper).

Question: Why here can it be deduced that it splits?

A result in SGA 4 shows that this composition is given by multiplication by degree of $f$. But the question is why this implies splitting? Note that derived category – as additive triangulated category – is not abelian so we cannot just use splitting argument.

Motivation: This appears to be an example for a sheaf for which admitting the splitting behavior I was looking for here some time ago.

Improve this question
$\endgroup$
5
  • $\begingroup$ Since $f$ is finite, the pushforwards are concentrated in degree zero, so you can just argue in the usual abelian category of constructible sheaves. BTW you also need $f$ dominant (which is assumed in Sawin's paper). $\endgroup$
    – Satan's Minion
    Commented Aug 26 at 10:26
  • $\begingroup$ @Satan'sMinion: Nice.Maybe towards the more general linked question this philosophy is linkd - to identify certain pieces in degree $0$-pieces in this or $t$- structure sense - gives riseto having a core/heart living as abelian subcategory in derived cat (which is only additive), such that once we know our objects are living there, we suddenly have the full ansenal of abelian cat tools like this splitting lemma. A potential problem is - do you mabe know it such inclusion is full & faithfull? (...in order not to "loose" the morphisms estalished $\endgroup$
    – user267839
    Commented Aug 26 at 10:48
  • $\begingroup$ once in derived category when restricting ourself to abelian subcategory in order to ponder there about splittings "down there"?) $\endgroup$
    – user267839
    Commented Aug 26 at 10:50
  • 1
    $\begingroup$ The heart of any t-structure sits fully faithfully inside the ambient category, by definition. I suggest you study the basics more carefully. :) $\endgroup$
    – Satan's Minion
    Commented Aug 26 at 17:41
  • $\begingroup$ @Satan'sMinion: Ok, so the trick was really just to realize this counit-trace composition $c \circ t$ actually never leaves the abelian cat of constructible sheaves and to observe that any multiplication of const sheaf $\Bbb Q_l$ by a nonzero number is actually an iso, and so we can find an inverse map $r$, ie $r \circ t \circ c= id_{\Bbb Q_l}$, and then we can apply just the splitting lemma linked above? $\endgroup$
    – user267839
    Commented Aug 27 at 14:04

0

Reset to default

You must log in to answer this question.