10

CHAPTER 0. PRELIMINARIES

nngen(^) denotes the category of all finitely generated modules over a ring O.

fmlen(Q) denotes the category of all finite-length ^-modules.

fingenoo(0) denotes the category of all modules in fingen(Q) that have no

nonzero direct summands in fmlen(fJ). In particular, {0} G fingenoo(0). Sec-

tion 7 discusses the unique decomposition of M G fmgen(Q) into a direct sum

M = M^ 0 M0 where M ^ G fingenoo(0) and M0 G finlen(fi).

maxspec(Q) and minspec(17) denote, respectively, the sets of maximal ideals

and minimal prime ideals of a ring ft.

singspec(^) — when ft is reduced and noetherian of Krull dimension 1, with

normalization T — denotes the set of singular maximal ideals of f2, that is, the set

of m G maxspec(fJ) such that ftm C Tm (proper inclusion). If ftm = Tm we call m

nonsingular.

maxsuppA(i^) denotes the set of maximal ideals m of A in the support of the

A-module # , that is, such that Hm ^ 0.

X(m) denotes X/mX where X is any A-module and m G maxspec(A). If X is

a T-module, so is X(m). When A is a local ring, we often shorten X(m) to X.

pm

denotes the natural homomorphism X—» X(m).

Ctm and Mm denote the tn-adic completions of a ring ft and a module M G

fingen(f2) respectively. See Section 5 for a discussion of this. See Remarks 5.3 for

a discussion of Tm (m G maxspec(Sl)). When ft is local we often shorten ftm to ft.

(... )m denotes the m-adic completion of an expression consisting of more than

a single letter.

(aVm) denotes for almost all m, that is, "for all except finitely many m."

(... )

x

denotes the group of units of (...).

4. Q-Localization and Minimal Primes

LEMMA 4.1. Let F = (Bhen^h be the normalization of a reduced noetherian

ring Q7 where each Th is an integral domain. Then:

(i) The finite set of minimal prime ideals ph offl, is in one-to-one correspon-

dence with the set of coordinate rings Th ofY, via p^ = ker(Q — I \ ) .

(ii) The set of zero-divisors of ft is the union of the minimal prime ideals of

n.

(hi) Each ker(r — Th) is the unique minimal prime ideal ofT lying over ph-

(iv) Every element of ft that is a zero-divisor in T is already a zero-divisor in

n

PROOF, (i) We claim that there are no inclusion relations among the ideals p^.

To keep the notation simple, select two indices h, calling them 1 and 2, and choose

a nonzero element of T2. Since T is contained in the total quotient ring of A, this

element has the form x/d where x, d G A and d is a regular element of A. Moreover,

coordinate 2 of x is nonzero, while x\ — 0. Therefore x G pi — p2- Thus the claim

is proved.

Since the product of the prime ideals p^ is zero, every minimal prime ideal

is among them. Since, in addition, there are no inclusion relations between these

kernels, each is a minimal prime.

(iv) This holds since every element of T has the form x/d with x, d G Q and d

regular in ft.