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.