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