Let f : X —> B be a morphism of schemes. Suppose that for every open U of X we are given a category CU whose opposite is a subcategory of the category of surjections A —> OU of sheaves of f-1OB-algebras. Moreover, assume that these categories fit together to give a stack C over XZar with the usual notion of restriction of sheaves.
Since the purpose of this discussion is to study deformation theory, it make sense to assume the stalks of A are local rings, which means exactly that the localization of A as in the previous blog post doesn’t do anything. I will assume this from now on.
Example: Assume f locally of finite type. Given U let CU be the full subcategory of surjections i-1OT —> OU where i : U —> T is a closed immersion of U into a scheme T smooth over B. As maps we can take those maps that come from morphisms between smooth schemes over B. This does not form a stack over XZar but we can stackify.
Now suppose we have a third scheme S and a morphism of schemes g : S —> B. Then I claim there is a natural stack CS over (XS)Zar which can be called the base change of C. I will construct this by saying what the objects and morphisms look like locally on XS and you’ll have to stackify to get the real thing.
OK, suppose that V is an open of XS which maps into the open U of X. Denote p : V —> S and q : V —> U the projections and h : V —> B the structure morphism. Let A —> OU be an object of CU. Then the map
p-1OS ⊗h-1OB q-1A —-> OV
is a surjection (see previous post) and we can consider its localization A’ —> OV (as in previous post). This will be what our objects look like locally. Moreover, morphisms are maps which are locally the pullback of maps in C.
Here is how we can use this: The stack CS is naturally a ringed site with topology inherited from the Zariski topology on XS. Moreover, in the example above the rings A are all “smooth” over B thus the rings A’ in CS are all “smooth” over S. I think we can think of XS as a closed subspace of CS and use this to compute obstruction groups for deformations of modules, etc. I’ll come back to this later (and if not then it didn’t work).
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