Sheafification of constant presheaf?

Presheaf; subsheaf; it’s just a functor defined from a “simple” category of opens (with hom set only the inclusion) to Ab. Require F(empty)=0, gluing condition and unique condition.

Lemma1.18 f.g k-alg has Nilradical = Jacobson radical.

The bridge are maximal ideals M=(Ti-ai) such that containing I, where A = k[Ti]/I. Either Taylor expand or k[Ti] to k; Ti to ai; we see P in M iff P(a)=0.

We assume k alg closed. Key point is using Cor 1.12, k[Ti]/m is iso to k.

Eg1.10 k[X,Y]/(XY); realize norther normalization; Prop1.11 is a “better” result of Norther normalization. It in particular implies that.

1) Statement of the result. 2) A F.g as a ring over B vs. A f.g as a module over B: latter iff feinter+integral over B; 3) integral: don’t forget monic poly

Parametrize by SpecZ: Union of affine lines over fields of different characteristics.

Spec f is an homeo between…

Radical ideal