Deformations and topoi

Let k be a field and let D = k[epsilon] be the ring of dual numbers. Suppose V is some geometric object over k. A geometric object U over D is called a deformation of V if it  (1) is flat over D and (2) has special fibre U_k = U \otimes_D k isomorphic to V. (This is intentionally vague.)

When V is a scheme, then U can be conveniently thought of as a locally ringed space whose underlying topological space is identical with the underlying space of V. In other words, you just change the sheaf of rings, and not the actual space.

However, some types of deformations in the literature do cause the underlying space or rather topos to change! And this is just one of the reasons why deformations of algebraic stacks are just a little more confusing than the case of schemes.

Here is a silly example: Let’s look at the lisse-etale site of k, call it C_1, and the lisse-etale site of D, call it C_2. For simplicity (and because it doesn’t matter for the associated topoi), let’s assume we only look at affine schemes. So an object of C_1 is an smooth affine scheme V over k and an object of C_2 is a smooth affine scheme U over D. In fact the sets of isomorphism classes of objects of C_1 and C_2 are naturally bijective, via the rules V —> V \otimes_k D and U —> U_k (Hartshorne, Exercise II 8.7). Moreover, if U is such an object, then the categories of etale coverings of U and U_k are canonically identified (by topological invariance of etale morphisms, see Theorem Tag 039R). For every isomorphism class of objects pick a particular object U_i of C_2 and let V_i be the corresponding object of C_1. Then we can try to match a sheaf F on C_1 with a sheaf G on C_2 by the rule F(V_i) = G(U_i). Does this work?

It doesn’t! Given two objects U_i, U_j of C_2 the collection of morphisms in C_2 between U_i and U_j is drastically different from the collection of morphisms between V_i and V_j in C_1. For example the value G(Spec(k[epsilon, x])) is acted upon by all automorphisms of k[epsilon, x] not just the automorphisms of k[x]. And in fact there is no way of identifying the categories of sheaves on C_1 and C_2 in any reasonable way. (I have several ways of saying this precisely, but none that is completely satisfactory. If you have one, please leave a comment. In fact, I would love a direct argument showing that Sh(C_1) and Sh(C_2) are not isomorphic as abstract topoi.)

Maybe this is just another reason for thinking that the lisse-etale site was a bad idea in the first place?

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