8320notes4 - INTRODUCTION TO THE GEOMETRY OF SCHEMES PETE...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: INTRODUCTION TO THE GEOMETRY OF SCHEMES PETE L. CLARK 1. Brief reminders on sheaves It is a shame that one does not meet sheaves earlier in ones study of geometry: they are useful in differential and complex geometry as well, and their introduction in a basic course in one of these areas would cut down on the amount of vocabulary one has to acquire in an algebraic geometry course. I am not going to give much of an accout of sheaf theory either in my lectures or in my lecture notes; recall that on the first day I asked everyone if they knew what a sheaf was, and you all said yes! Lius book contains all the background on sheaves we will need. Moreover, the point of a sheaf is that all of its data is encapsulated in the stalks at every point. To be more precise, if F is any sheaf on X and U is an open subset of X , then F ( U ) can be recovered as the set of elements ( s x ) of the product group Q x U F x such that: for all x U , there exists an open neighborhood x V U and an element s F ( V ) such that for all x V , the stalk of s at x is equal to s x . In particular, if one starts with merely a presheaf F on X then the associated sheaf has the same stalks and is constructed from F via precisely the above recipe. As a good rule of thumb, if ever you find yourself beginning to be snowed un- der by the formalism of sheaves, just stop, take a breath, and ask yourself What is happening on the stalks? This is usually sufficient to clear up all confusion. Push forward and pullbacks : Let f : X Y be a continuous function, and F a sheaf on X . Then we can define a sheaf f * ( F ) on Y , the pushforward of F in a very simple way: for V open in Y , put f * ( F )( V ) := F ( f- 1 ( V ). Dually, if G is a sheaf on Y , then we define f * ( G ) to be the sheaf associated to the presheaf U 7 lim- V f ( U ) G ( V ). 1 Example (open restriction): let F be a sheaf on a space X , and let U X be an open subset. Writing : U , X for the natural inclusion map, consider the sheaf * ( F ) (actually no sheafification is required here). It is more transparently described as the restriction of F to U and denoted F| U . As we dont need the formalism of pullbacks to define F| U , this is sort of a trivial example. Example (arbitrary restriction): Now let F be a sheaf on X and Y X be any subset. Writing : Y X for the natural inclusion map, consider the sheaf * ( F ), 1 Note that Hartshorne uses f- 1 for the pullback and reserves f * for pullback on O X-modules. In my opinion, the context will make clear when this other meaning is intended. 1 2 PETE L. CLARK called the restriction of F to Y . This is more interesting: e.g. in case Y = { x } is a point, we get a sheaf on a one-point space i.e., just an abelian group which is nothing else than the stalk of F at x ....
View Full Document

This note was uploaded on 10/26/2011 for the course MATH 8320 taught by Professor Clark during the Fall '10 term at University of Georgia Athens.

Page1 / 8

8320notes4 - INTRODUCTION TO THE GEOMETRY OF SCHEMES PETE...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online