U⊂V⊂X and t(U^c)^c⊂t(V^c)^c⊂t(X) 3D comm.diagram

Sch.10

Speck-structure on t(U^c)^c (take 2)

Sch.10

Inverse morphism of ringed spaces

Sch.5, Appendix

X->t(X) being a functor

Sch.6

3D comm. diagram involving t(X), t(U), t(U^c)^c, U

Sch.10

Γ(t(U))=Γ(t(U^c)^c)

Sch.10

t(U), t(X), t(U^c)^c commuting diagram

Sch.6

U, t(U), and t(U^c)^c commuting diagram

Sch.5

How X->Y induces t(X)->t(Y)

Sch.5 (Affine)

Sch.6 (Any variety)

Hom(V,W)<->Hom(O(W),O(V)), W,V varieties, W affine

Sch.9

Spec k - structure of t(U^c)^c for affine U⊂V

Sch.10

Speck - scheme structure of t(V)', V affine variety

Sch.10

If V is affine variety, t(V) is an affine scheme

Scheme II

(x^-1A)/A=A/xA

Matsumura, p146

Domain Topology

Proj(P.P.)

V->W induces t(V)->t(W), any varieties V,W

Sch.6

Global section of t(V)->Spec k gives the canonical ring hom, V any variety

Sch.8

The morphism of γ : t(U^c)^c->t(U) and its global section being identity

Sch.6
In fact the morphism itself is identity (Sch.10)

Compatibility of t(V)-Spec k with restrictions to affine open subsets, V any variety

Sch.8

t(f) morphism definitioin

Sch.5

Spec k - structure of SpecA(V)

Sch.7

Spec k - structure of t(V) for general V

Sch.6

Spec k - structure of t(V) for affine V

Sch.5

Compatibility of Γ(Hom(SpecA,SpecB)) and Hom(A,B)

Scheme III

Spec k - structure of the iso between SpecA(V) and t(V)

Sch.7

ω map for t(V)

Sch.5

Definition of ω map

Sch.5

V,t(V),SpecA(V) comm diag, scheme version

Sch.6

Hom(V,W) <-> Hom(A(W),A(v)), A a domain

Sch.6, 7

Gluing morphisms (2)

Sch.7

Hom(X,Y)<->Hom(Γ(Y),Γ(X)), Y affine scheme

sch7

Global section functor

Sch.7

Equivalence of category: affine schemes, rings

Sch.5

(t(U),O)=(t(U^c)^c,F|)

Sch.5

t(Y^c)=t(Y)^c

Scheme II

α map revisited

Sch.5

Hom(A,G(X)) <-> Hom(X,SpecA)

Sch.5

α: X -> t(X)

continuity, dense image, homeo onto image... : Scheme II

Gluing morphisms

Scheme III