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A map of locales $f : X \rightarrow Y$ is closed if it satisfies the reciprocity relation $f_*(f^*(X) \vee Y) \cong X \vee f_* (Y)$.

How can we express a that a map of schemes $f : X \rightarrow Y$ is closed in terms of the direct image $f^*$ and inverse image $f_*$? Is this equivalent to the same reciprocity relation?

Is there some tweaking of the setting for locales which gives rise to a similar reciprocity condition for schemes?

Note: here $f_*$ and $f^*$ are the direct image and inverse image functors between categories of quasicoherent sheaves on $X$ and $Y$.

Any insights are appreciated.

The map $f^*((f_* M) \otimes N) \rightarrow M \otimes f_* (N)$ arises as the adjunct of the map $$ f^* ((f_* M ) \otimes N) \rightarrow (f^* f_* M) \otimes (f^* N) \rightarrow M \otimes (f^* N)$$

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  • $\begingroup$ What exactly do you mean by $f^*$ and $f_*$? If you mean on all (abelian) sheaves, it's just a question about topological spaces. Or are you thinking specifically of quasi-coherent sheaves or sheaves of $\mathcal O_X$-modules? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Sep 10, 2020 at 14:15
  • $\begingroup$ quasi-coherent sheaves of $\mathcal{O}_X$-modules $\endgroup$
    – user30211
    Commented Sep 10, 2020 at 14:21
  • $\begingroup$ I don't know what locales are but there is a projection formula for intersection theory of cycles which looks superficially similar. $\endgroup$
    – Asvin
    Commented Sep 10, 2020 at 22:00
  • $\begingroup$ @Asvin thanks, I'll look into that. $\endgroup$
    – user30211
    Commented Oct 7, 2020 at 17:16

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