Thanks to Bhargav and some editing by yours truly we now have a section on formal glueing in the stacks project. In fact it is in a new chapter entitled “More Algebra”. The main results are Proposition Tag 05ER, Theorem Tag 05ES, and Proposition Tag 05ET (look up tags here). The original more self-contained version can be found on Bhargav’s home page.

What can you do with this? Well, the simplest application is perhaps the following. Suppose that you have a curve C over a field k and a closed point p ∈ C. Denote D the spectrum of the completion of the local ring of C at p, and denote D* the punctured spectrum. Then there exists an equivalence of categories between quasi-coherent sheaves on C and triples (F_U, F_D, φ) where F_U is a quasi-coherent sheaf on on U = C – {p} and F_D is a quasi-coherent sheaf on D and φ : F_U|_{D*} —> F_D|_{D*} is an isomorphism of quasi-coherent sheaves on D*.

An interesting special case occurs when considering vector bundles with trivial determinant, i.e., finite locally free sheaves with trivial determinant. Namely, in this case the sheaves F_U and F_D are automatically free(!) and we can think of φ as an invertible matrix with coefficients in O(D^*). In other words, the set of isomorphism classes of vector bundles of rank n with trivial determinant on C is given by the double coset space

SL_n(O(U)) \ SL_n(O(D*)) / SL_n(O(D))

Another interesting application concerns the study of “models” of schemes over C. Namely, instead of considering quasi-coherent sheaves we could consider triples (X_U, X_D, φ) where X_U is a scheme over U, and so on. In this generality it is probably not the case that such triples correspond to schemes over C (counter example anybody?). But if X_U, resp. X_D is affine over U, resp. D or if they are endowed with compatible (via φ) relatively ample invertible sheaves, then the result above implies in a straightforward manner that the triple (X_U, X_D, φ) arises from a scheme X over C.

For the failure of the glueing theorem for models of schemes, one can give Raynaud-type examples over 2-dimensional (normal, local) base schemes; presumably one can get similar examples over non-normal one-dimensional base schemes. I’ll write down the 2-dimensional one here in case someone’s curious.

Say (S,s) is the Zariski local ring of the affine cone on an elliptic curve E. Let S’ be a degree 2 finite etale cover with S’ connected; these exist because S is essentially a global object. Then, after fixing an infinite order point ‘x’ on E, there is an obvious E-torsor one can write down on S’, essentially by putting a copy of E at each closed point of S’, and glueing using x. This torsor has infinite order in H^1(S’,E) by construction. Taking the restriction of scalars along S’ -> S gives an infinite order element of H^1(S,A) where A is some abelian surface over S. Raynaud showed that the representable A-torsors on S are precisely the finite order ones (as S is local and normal). So this torsor is an fppf-sheaf X -> S that is is not representable. On the other hand, it’s pretty clear that away from the closed point this sheaf is representable, and that the completion at ‘s’ is also representable (as it can be done at the S’ level). So we get non-effective glueing data associated to the sheaf X.

Its also interesting to try to see what happens in higher dimensions. I worked on this with Michael Temkin, see this preprint: http://arxiv.org/abs/1201.4227

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