What is the construction for taking inverse images of subschemes?

[eof]

Hi,

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?

 

[fresh_42]

As schemes build a category, we also have pre-images of scheme morphisms. So the key lies in the proof that schemes build a category. This means especially that the function $f$ and the corresponding ring homomorphism must be considered together. You cannot separate the two, as they build the morphisms.