# 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.