From: Robin Chapman Subject: Re: Some Sheaf Theory Questions Date: Thu, 25 Jan 2001 09:24:28 GMT Newsgroups: sci.math Summary: Functoriality of sheaves on algebraic varieties In article <_UNb6.153$ZP4.186604@news.pacbell.net>, "Daniel Giaimo" wrote: > I've been taking a course in Algebraic Geometry this year, and we > just got to cohomology of sheaves, so we're beginning to really heavily > use some of the functors we've defined on them and a lot of their > properties that we haven't quite proven. In fact, some of the > properties that we need to do the exercises aren't even mentioned in the > text. (Hartshorne is the professor and he's using his > _Algebraic_Geometry_) So, first of all, I would like to know if anyone > here knows of a good detailed reference on sheaf theory. > > Most recently, the following problem has come up: > > Is it true that if f:X->Y, and g:Y->Z are continuous maps and F is a > sheaf on Z, then (gf)^-1(F) = f-1(g^-1(F))? > Yes. But there are two equivalent notions of sheaves. The functions on open sets notion used in Hartshorne's book, and the espace etale notion. For the first notion, direct images are easy to deal with and inverse images difficult, but for the espace etale it's the other way around. The espace etale of a sheaf F on X is a space A and a map A -> X which is locally homeomorphic (each point of A has a neighbourhood mapping homeomorphically to an open subset of X). It is produced by letting the fibre at x in X be the stalk of F at x and then topologizing the union of these suitably. On recovers the sheaf F by noting that F(U) is the set of sections of A -> X over U. Now in this paradigm, the inverse image of a sheaf under Y -> X simply corresponds to the fibre product A x_X Y, from which the functoriality of the inverse image is really obvious. -- Robin Chapman http://www.maths.ex.ac.uk/~rjc/rjc.html "His mind has been corrupted by colours, sounds and shapes." The League of Gentlemen Sent via Deja.com http://www.deja.com/