Let $\operatorname{ProjSmooth}_k$ be the category of smooth projective varieties over a field $k$, and consider the $\ell$ cohomology theory $H^*(-)$ ($l \not = \operatorname{char} k$), how to define reasonable trace maps $\operatorname{Tr}:H^*(X) \rightarrow H^*(Y)$ for any finite flat morphism $f: X \rightarrow Y$ in $\operatorname{ProjSmooth}_k$?

In particulal, trace maps shall be compatible with base change and $ \operatorname{Tr} \circ f^* = \deg f$. In the case $f$ is etale, one can define it using adjunction between $f^!$ and $f_!$ . But what if we only assume the flatness condition?

Motivation: We can define trace maps of finite flat morphisms for coherent cohomology or algebraic de Rham cohomology.

  • $\begingroup$ The functors $f^!$ and $f^*$ agree on lisse sheaves, hence such a trace (we do not need $X$ and $Y$ smooth and projective over a field, but only that both are regular schemes). Another way to see this: $\ell$-adic cohomology is representable in Voevodsky’s triangulated category of mixed motives; in particular, it is a presheaf with transfers, whence the trace map together with the degree formula. $\endgroup$ – Denis-Charles Cisinski Feb 21 at 23:18
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    $\begingroup$ @Denis-CharlesCisinski How to see they agree if we only assume $f$ is finite flat? For your another approach, can you give me some references? Thank you :) $\endgroup$ – zzy Feb 21 at 23:27
  • $\begingroup$ They agree because of absolute purity. For the degree formula, I forgot to write that we need $f$ to be surjective as well. Over a field, the representability theorem is due to Suslin and Voevodsky in “Singular homology theory of abstract algebraic varieties” (Inventiones 123). In general, this is proved in arXiv:1305.5361 (published in Compositio) in which a trace map together with a degree formula is also discussed explicitly; see Def. 6.1.5 and Rem. 6.1.6. $\endgroup$ – Denis-Charles Cisinski Feb 22 at 6:49
  • $\begingroup$ @Denis-CharlesCisinski Thank you! I will have a look at it. $\endgroup$ – zzy Feb 22 at 16:34

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