Let f : X —> B be a morphism of schemes. Suppose that for every open U of X we are given a category *C*_{U} whose opposite is a subcategory of the category of surjections A —> O_{U} of sheaves of f^{-1}O_{B}-algebras. Moreover, assume that these categories fit together to give a stack *C* over X_{Zar} 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 *C*_{U} be the full subcategory of surjections i^{-1}O_{T} —> O_{U} 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 X_{Zar} 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 *C*_{S} over (X_{S})_{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 X_{S} and you’ll have to stackify to get the real thing.

OK, suppose that V is an open of X_{S} 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 —> O_{U} be an object of *C*_{U}. Then the map

p

^{-1}O_{S}⊗_{h-1OB}q^{-1}A —-> O_{V}

is a surjection (see previous post) and we can consider its localization A’ —> O_{V} (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 *C*_{S} is naturally a ringed site with topology inherited from the Zariski topology on X_{S}. Moreover, in the example above the rings A are all “smooth” over B thus the rings A’ in *C*_{S} are all “smooth” over S. I think we can think of X_{S} as a closed subspace of *C*_{S} 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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