Yoneda, Spec and Nilpotents

Let A be an abelian category.  Given an object C, we have \ h^C = \hom(C, ) and \ h_C = \hom( , C) are two functors from A into SETS.  And given \ C \xrightarrow{f} C' get \ h^C \xrightarrow{h^f} h^{C'} and a similar thing in the contravariant case.  So there is a functor \ h \colon A \ \to 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 \ X \colon A \ \to SETS a functor and C an object in A.  Then there is a map X(C) to \ \hom(h^C, X) which is bijective.

the map: given a set element  \ b \in X(C) we need a map \ h^C \to X, but this is exactly a map \ h^C(B) \to X(B) for every B.  So we reduce the problem to being given an \ C \xrightarrow{f} B and producing a set element of \ X(B).  But  X is a functor so we get \ X(C) \xrightarrow{X(f)} X(B) so take the set element to be the image of the original set element we started with.  

the inverse:  Given a morphism \ h^C \to X, apply this to C, to get a map \ h^C(C) \to X(C) 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 \ k \to k' be a homomorphism of fields.  Assume k’ is fin. gen. as k-alg. Then k’ is finite k-module.  

Cor. If \ E \subset k[x_1, \dotsc, x_n] TFAE

  1.  \ Z_E(k) = \emptyset
  2. \ Z_E(k') = \emptyset 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 \ A such that \ Z_E(A) \ne \emptyset.  Then a point in this set corresponds to \ k[x_1,\dotsc, x_n]/I \to A and so \ k[x_1, \dotsc, x_n]/I 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 \ Z_E(k'). QED

Cor. \ k = \bar{k} then \ Z_E(k) is in bijection with the maximal ideal of \ k[x_1, \dotsc, x_n]/I.

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

A proposition: for a ring R, and \ a \in R, then \ a is nilpotent iff \ a(p) = 0 for all \ p \in Spec R, where \ a(p) is the residue class of a in \ R_p/p.

pf(sketch): show \ R_a = 0 iff \ a is nilpotent.  Now if \ a is nilpotent, then \ a \in p for all p, so \ a(p) = 0 for all p.  If \ a not nilpotent, then \ R_a \ne 0, so it has a maximal ideal, this maximal ideal pulls back to a prime of \ R that doesn’t have \ a. QED.


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