Roy Skjelnes: Quotients by equivalence relations of schemes

The quotient of a scheme by an equivalence relation is a natural thing to consider, but unfortunately the quotient is not necessarily a scheme.

The usual trick to remedy this problem is to consider schemes as functors, and form the quotients in this larger category. However, functors in general are not determined by local data, and are therefore far from being schemes. Thus, when we form the quotient we also sheafify the quotient functor in some appropriate Grothendieck topology.

A quotient sheaf might be representable by a scheme in some topology, and not representable in other topologies. In many situations, however, the quotient sheaves are not representable by schemes. A natural question is then to determine which topology one should choose to sheafify the quotient within?

In the talk I will try to explain these notions. No a priori knowledge of these objects or concepts are needed to understand the talk, but some familiarity of schemes will be assumed.