# Closed map of schemes and Frobenius reciprocity

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 $$f^* ((f_* M ) \otimes N) \rightarrow (f^* f_* M) \otimes (f^* N) \rightarrow M \otimes (f^* N)$$

• 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? – R. van Dobben de Bruyn Sep 10 at 14:15
• quasi-coherent sheaves of $\mathcal{O}_X$-modules – Dean Young Sep 10 at 14:21
• I don't know what locales are but there is a projection formula for intersection theory of cycles which looks superficially similar. – Asvin Sep 10 at 22:00
• @Asvin thanks, I'll look into that. – Dean Young Oct 7 at 17:16