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Mikhail Bondarko
Mikhail Bondarko
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Let $X,Y$ be smooth varieties over a field $k$ (which in my case is perfect of finite characteristic $p$; we may also assume that $X,Y$ are connected); $s:Y\to X$ is a finite morphism of degree $d$. Then the graph of $s$ could be considered both as a finite correspondence from $Y$ to $X$ and as a finite correspondence from $X$ to $Y$ (in the sense of Voevodsky, or in the sense of the 'classical' intersection theory). Now compose these correspondences to get a correspondence from $X$ to $X$ (using intersection theory); is this true that the result is $d\cdot id_{X}$? I would say yes, since the composite correspondce seems to have only the diagonal component as a cycle in $X\times X$, and it would be very strange if its multiplicity would be anything else but d. Yet I would be glad both for any explanations/references here as well as for pointing out any subtle points (since I don't know much about intersection theory). Does one require any extra conditions here? It would be ok for me to replace $X$ and $Y$ by theirsome open subsetssubvarieties, but I don't want to demand $s$ to be generically etale.

Let $X,Y$ be smooth varieties over a field $k$ (which in my case is perfect of finite characteristic $p$; we may also assume that $X,Y$ are connected); $s:Y\to X$ is a finite morphism of degree $d$. Then the graph of $s$ could be considered both as a finite correspondence from $Y$ to $X$ and as a finite correspondence from $X$ to $Y$ (in the sense of Voevodsky, or in the sense of the 'classical' intersection theory). Now compose these correspondences to get a correspondence from $X$ to $X$ (using intersection theory); is this true that the result is $d\cdot id_{X}$? I would say yes, since the composite correspondce seems to have only the diagonal component as a cycle in $X\times X$, and it would be very strange if its multiplicity would be anything else but d. Yet I would be glad both for any explanations/references here as well as for pointing out any subtle points. Does one require any extra conditions here? It would be ok for me to replace $X$ and $Y$ by their open subsets, but I don't want to demand $s$ to be generically etale.

Let $X,Y$ be smooth varieties over a field $k$ (which in my case is perfect of finite characteristic $p$; we may also assume that $X,Y$ are connected); $s:Y\to X$ is a finite morphism of degree $d$. Then the graph of $s$ could be considered both as a finite correspondence from $Y$ to $X$ and as a finite correspondence from $X$ to $Y$ (in the sense of Voevodsky, or in the sense of the 'classical' intersection theory). Now compose these correspondences to get a correspondence from $X$ to $X$ (using intersection theory); is this true that the result is $d\cdot id_{X}$? I would say yes, since the composite correspondce seems to have only the diagonal component as a cycle in $X\times X$, and it would be very strange if its multiplicity would be anything else but d. Yet I would be glad both for any explanations/references here as well as for pointing out any subtle points (since I don't know much about intersection theory). Does one require any extra conditions here? It would be ok for me to replace $X$ and $Y$ by some open subvarieties, but I don't want to demand $s$ to be generically etale.

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Mikhail Bondarko
Mikhail Bondarko
  • 16.9k
  • 4
  • 34
  • 97

A composition of a finite morphisms with the transpose correspondence: is it the multiplication by the degree?

Let $X,Y$ be smooth varieties over a field $k$ (which in my case is perfect of finite characteristic $p$; we may also assume that $X,Y$ are connected); $s:Y\to X$ is a finite morphism of degree $d$. Then the graph of $s$ could be considered both as a finite correspondence from $Y$ to $X$ and as a finite correspondence from $X$ to $Y$ (in the sense of Voevodsky, or in the sense of the 'classical' intersection theory). Now compose these correspondences to get a correspondence from $X$ to $X$ (using intersection theory); is this true that the result is $d\cdot id_{X}$? I would say yes, since the composite correspondce seems to have only the diagonal component as a cycle in $X\times X$, and it would be very strange if its multiplicity would be anything else but d. Yet I would be glad both for any explanations/references here as well as for pointing out any subtle points. Does one require any extra conditions here? It would be ok for me to replace $X$ and $Y$ by their open subsets, but I don't want to demand $s$ to be generically etale.