Hi,(adsbygoogle = window.adsbygoogle || []).push({});

As is the case with functions, we can always define the inverse image of a subset. In the case of schemes I was wondering if there is something that could be taken as the inverse image of a subscheme?

Example:

Let f:X->Y be a scheme morphism. Then if U is an open subscheme of Y, we have that f^{-1}U is an open subset of X. The structure sheaf O_U of U can be taken to be a O_Y-module provided that we extend it to the space Y by

V -> O_U(V \cap U)

so this way we could define f^*O_U. For this to make any sense, we would need to have f^*O_U(V)=O_X(V) for any open V\subset X.

Thus the definition doesn't really give us an inverse image of a scheme, because it would have to an open subscheme of X. So is there any way of providing the kind of construction I'm looking at? I don't see any smart way of doing this for closed subschemes either. Does anybody know if there's a construction to take inverse images of subschemes?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Inverse images of schemes

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**