Definitions for CM, etc

CM ring - CM
CM-mod over local ring - Regular Seq.
Regular local ring - Reg. Seq.

(A,M) flat then (B,M*B) flat

new CM

Chase(Over a Noetherian ring, direct product of flat modules is flat.)

CM

M->S^-1M being injective or surjective iff s is M-regular

Post-c.i.

Extension & Contraction of scalars for modules again

Including localization

post-c.i.

Localization, universal property for modules

DVR & Dedekind

Also see Jordan-Holder

Irreducible ideal being primary (commutative ring)

Ass(Redux), dim(M)

0->M->M->M->0 ACC,DCC, f.g. module over Noeth is Noeth

NC-IV

ann(m), noncommutative

NC-III

Injective module redux

Restriction & extension of scalars

Projectivity of module M and Ext^1(M,N)=0 for all f.g. N

Ext,Tor, pd, id

Also Eisenbud's projective characterization from appendix

simple module

Injective modules

Jordan-Holder

Isomorphic to R/m (NC-III, Jacobson)

Socle review

Injective Modules

Maximal subset of direct sum

Injective Modules

Matrix ring example

QLOCAL

Submodule definition

NON-COMM-PROP

(r)_L, (r)_R, (r)

NON-COMM-PREL

If I is a left ideal, so is IS for any subset S

NON-COMM-PROP

aM, Im, IM, ann(M), etc. (commutative)

Commutative - Injective modules, under APPENDIX

Non-commutative - NON-COMM-II

Localization again (S,T, transitivity, p followed by q)

Regular rings (Also localization of A[x])

Jordan-Holder

Saturated mult subset

Module Category & Rank & Matsumura

Module category revisited (II) (Localization, extension, etc.)

Jordan-Holder

First module category -- Matsu&

Filtration of modules, support, Ass

Ext, Tor, pd, id

Also Ass(Redux)

Every module has a maximal submodule.

Injective modules

E(A/p) is p-primary

Matlis

Injective embedding

Restriction & Extension of scalars

UFD, irreducible elt review

UFD, C.I.

gcd, lcm

Matsumura book, p.164,
Res(new)-M

UFD iff every prime of height 1 is principal, ACCPI, properties of principal primes

Eisenbud, Ex.3.11 (Old)
DVR & Dedekind Folder (New)

Nagata CM ring

Res(new)-M

Direct sum of CM modules may not be CM

Res(new)-M

Category of modules revisited

Res(new)-M

CM module definition

Res(new)-M

CM module & modding out by an ideal (A/I, I ∈ann(M))

Gor
Res(new)-M

Regular sequence and flat extension

CM,
Bruns

n-th syzygy of projective resolution being projective

Ext,Tor,pd,id

Cor. f.g. proj. module over a Noetherian ring which has a finite proj. dim has a finite proj res. consisting of f.g. projective modules. -- UFD, c.i.

M^* -> M^***

Bruns

Ass(Hom(M,N))=Supp(M)∩Ass(N)

BrunIII

Transitivity of inverse sheaf revisited, restriction of t(V) sheaf is O_V

Sch.10

\mathcal{F} \simeq (j_{*}\mathcal{F})|_{X}

j: X -> Y inclusion

Equivalence of categories (II)

C(Affine Varieties) = C(f.g.k-algebraic domains) = C(integral affine schemes over Spec k)

The last part is a conjecture.

Ringed spaces (or schemes) being a category

Scheme III

Bijection between Hom(V,W) and Hom_Speck(t(V),t(W)), V,W any variety

Sch.10

If W∈t(X), W is a generic point of t(W).

Sch.10

α map being closed

Scheme.II

α(V) = the set of closed points in t(V)

Sch.10

Spec-k compatibility of the following morphisms

t(U^c)^c=t(U), U⊂V
t(V)'->t(V), X affine variety

Sch.10

Restriction morphism of the second kind

Sch.10

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

Localizing graded ring, 1-1 correspondence of primes

Proj (P.P.)

Graded associated primes

Modules & Rings: Proj(P.P.) folder

Graded by Monoid: Ezra (Rs.)

Change of rings for Exterieure (Matsu)

Ci.

Regular local ring is integral domain

C.i.

Analytic Independence

C.i.

Regular local ring, the system of parameters being prime

CM, CI

Regular local ring (Atiyah), algebraic independence, system of parameters

Regular Sequences

Regular local ring examples, almost integral

Regular Rings

ann(a), socle

Gor

CM examples

Bruns

self injective gorenstein

Kaplansky, Rotman

maximal CM of type 1 with finite id(M) iff dim Ext = delta

Matlis

Prime Avoidance revisited (Eisenbud)

Brun III

Minimal free resolution revisited

Ext, Tor, pd, id folder

McCoy's Rank

Atiyah soln project 2.11

A module over PID is torsion-free if and only if flat

Brun folder

M flat over A -> M*B flat over B

CM

New CM

Localization stuffs

are in Regular Rings.

Hom commuting with tensor

Hom(M,N)*B=Hom(M_B,N_B) was first proved in Free,...

It was then used in Regular Sequences to prove Ext and Tor versions.

It was proved again in Extensions along with naturalness.

Hom_B(M*N,P)=Hom_A(M,Hom_B(N,P)) was first proved in Extensions in the context of bimodule theory, but (not its naturalness). It will be proved again in Bruns, along with naturalness in bimodule theory.

Edit: naturalness seems to be proved in Extension, not in Bruns.

Hom & Direct, Inverse limits

Kaplansky, Rotman folder