# Yoneda, Spec and Nilpotents

Let **A** be an abelian category. Given an object C, we have and are two functors from **A** into **SETS**. And given get and a similar thing in the contravariant case. So there is a functor **A** **FUN(A)** where the latter is the category of functors from **A** to **SETS**.

Ogus’s version of Yoneda…

Let **A** be a category and **A** **SETS** a functor and C an object in **A**. Then there is a map X(C) to which is bijective.

the map: given a set element we need a map , but this is exactly a map for every B. So we reduce the problem to being given an and producing a set element of . But X is a functor so we get so take the set element to be the image of the original set element we started with.

the inverse: Given a morphism , apply this to C, to get a map and take the desired set element to be the image of the identity.

After a few more generalities on representable functors, comment about the Grothendieck topology, and something about algebraic spaces, he then talked some commutative algebra.

Thm (Hilbert) Let be a homomorphism of fields. Assume k’ is fin. gen. as k-alg. Then k’ is finite k-module.

Cor. If TFAE

- for all finite extensions k’ of k

pf: 1 imples 2 is clear. For the other direction, show not 1 implies not 2. In this case there is a k-algebra such that . Then a point in this set corresponds to and so is not zero, so its has a maximal ideal m. Then modding out by m we get a fin. alg. k’ over k, so its finite a module so its a finite extension of k, and there is a point in . QED

Cor. then is in bijection with the maximal ideal of .

He then talked about the nullstellensatz, the zariski topology, the functor spec, and distinguished affines.

A proposition: for a ring R, and , then is nilpotent iff for all , where is the residue class of a in .

pf(sketch): show iff is nilpotent. Now if is nilpotent, then for all p, so for all p. If not nilpotent, then , so it has a maximal ideal, this maximal ideal pulls back to a prime of that doesn’t have . QED.

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- May 21, 2009 / 12:41 am

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- Ogus Excerpts

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