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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 one’s 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! Liu’s 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 ι : U , → X for the natural inclusion map, consider the sheaf ι * ( 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 don’t 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 ι : Y → X for the natural inclusion map, consider the sheaf ι * ( F ), 1 Note that Hartshorne uses f 1 for the pullback and reserves f * for pullback on O Xmodules. 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 onepoint space – i.e., just an abelian group – which is nothing else than the stalk of F at x ....
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 Fall '10
 Clark
 Algebra, Geometry, Topology, Topological space, pete l. clark, locally ringed space

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