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.
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