NOETHER-LASKER DECOMPOSITION OFCOHERENT ANALYTIC SUBSHEAVES

BYYUM-TONG SIU

In this paper we develop the theory of Noether-Lasker decomposition of co-herent analytic subsheaves as an analogue of the algebraic Noether-Lasker de-composition of ideals in Noetherian rings. The decomposition can be describedas follows : Suppose if is a coherent analytic subsheaf of a coherent analytic sheafJona complex space (X, 0) in the sense of Grauert. For every point x of X,yx as an CVsubmoduIe of 3~x has a Noether-Lasker decomposition into primaryCVsubmodules 0I" &~x- The radicals of these primary submodules are prime idealsof

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Suppose that Si and SF are analytic subsheaves of an analytic sheaf F and séis an ideal-sheaf on a complex space (X, 6). Denote by 0t;SF the ideal-sheafdefined as follows: for xeX, fe(@:F)x if and only if fe 6X and fSFx

1969] COHERENT ANALYTIC SUBSHEAVES 377

x in Jif and t e Y(U, .T) such that tx = s and ty e SPy for y e U-A. Let 0» = (SP\ U):t.E(SS, C\U)

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sheaf on the complex space (Y, Jf), where X = (G/0')) = {>'0,...., y'}, where j° = y, and F' be disjoint open neigh-borhoods of y\ OSiSh'm Y, such that F°

1969] COHERENT ANALYTIC SUBSHEAVES 379

Corollary. SPlíñ = SP\E"(SP, F)\

II. Noether-Lasker decomposition of subsheaves. Suppose y is a coherentanalytic subsheaf of a coherent analytic sheaf 3~ on a complex space. LetE"(SP¿F) = {JieIi0)Yf be the decomposition into irreducible branches. Then wecall each nonempty Yf, p^O, iel(p), an associated subvariety of SP in ST anddenote the set of all associated subvarieties of SP in ^~ by SC(SP, ¿7~). From thedefinition we see readily that 9£(Sf, 3~) is locally finite. SP is called a primary sub-sheaf of !T if SP has only one associated subvariety.

The following lemma is a well-known algebraic fact [5, Appendix, Chapter IV] :

Lemma 2. Suppose R is a Noetherian ring and N is an R-submodule of a finitelygenerated R-module M. A prime ideal P in R is an associated prime ideal in theNoether-Lasker decomposition of N as a submodule of M if and only ifP = \/(N:f)for some fe M.

Theorem 4. Suppose SP is a coherent analytic subsheaf of a coherent analyticsheaf SF on a complex space (X, =: 0, ieJ(p)} be the setof all associated subvarieties of SP passing through x, where dim Xf = p, ie J(p),and suppose (Id Xf)x = C\,eKip.n Bf, is the decomposition into prime ideals. Then{Pf, \p£0,ie J(p), j e K(p, i)}=P(SPx, fx).

Proof. Suppose PeP(SPx,^x) and dim F=p. Then P=vf(SPx:f) for somefe 9~x by Lemma 2. P defines a subvariety V of dimension p in an open neighbor-hood D of x in X. We can suppose after a shrinking of D that there existsg e Y(D, 3T) such that gx=fand Id V=y/((SP\D):g). This implies that

{y\yeD,gyeSPy}= D-V.

Hence V^Ep(SP,3r). Since dim V=P and dim E0(SP, $~) S p, P=Pf, for somei e J(p) and some j e K(p, i).

Fix p^O and ieJ(p). By definition Xf is an irreducible branch of Ea(SP,3r~)for some c^p. Let U be a Stein open neighborhood of x in X such that U n Xf=°Uie«p.« Xf, is the decomposition into irreducible branches and Pf, = (ld Xfj)x,

j e K(p, i). Fix j e K(p, i). Let Z1 be the union of irreducible branches ofE"(SP, T)c\U other than Xf, and let

Z = Z1KJ(Ep-l(SP,3T)rs U).

Take ye Xf,-Z.(y\U)[Xf,]y = SrWvf=Sr;.

Since (SP\U)[Xf,] is generated by global sections [1, Satz 4, §2], there existst e Y(U, (SP\ U)[Xf,]) such that ty $ SPy.

Let Y=E(SP\U,(SP\U) + (

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This theorem gives us a characterization of associated subvarieties and tells usthat the subvariety-germs defined by associated prime ideals in the Noether-Laskerdecomposition of the stalks of F can be pieced together to form globalsubvarieties.

Corollary 1. // YeF(F, F), then E(F, F[Y]) = Y.

Proof. Obviously E(F, F[Y])

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Proof. The first assertion follows from Theorems 4 and 5 and the second asser-tion follows from the first. Q.E.D.

Lemma 3. Suppose SPcz

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Lemma 5. Suppose F is a coherent analytic subsheaf of a coherent analytic sheafF on a complex space (X,

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J0={Xi | X1 is maximal in^(^ &~)}. By Lemma 5, (5) is true for d=0. Suppose(5) is proved for OSdSe. Since E(Ste, ^) = \JieKe X* and

E(0te,^) = \J{Y\ Year(3l„r)},

{X1 | iele + 1}={Ye&(&e,#~) | Y is maximal in 3C(0te, 5~)}. By Lemma 5 thereexist primary subsheaves ¿2{ of 3~, iele + x, and a coherent analytic subsheaf^of y such that (i) E(lt, ^) = X\ iele + x, (ii) E(SS, 9~) is thin in Uw.»i x'> and(iii) (fW, ^) n ^=^c. Hence (TWi.+1 âx)cM% = SP. Let

z = {F| Year&.r), ftUie,,tIn

By Corollary to Theorem 5, E(@[Z}s-, F) D (a,),. Q.E.D.

Remark. The decomposition S/'=f)ie,Sti is irredundant, i.e. SP=£(~\ie,_{n J,for any y e i; for otherwise by Theorem 4 we have

X(SP, ST)

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However, corresponding to the uniqueness of isolated ideal components in theusual Noether-Lasker decomposition in rings, we have the following:

A subset F of 2F(F, F) is called an isolated system of associated subvarieties ifYx c Y2, Y¡ £F(F, F),i=l,2, and Yx e L imply Y2 eL.lfL is an isolated systemof associated subvarieties, then f) {J, | X* eL} is unique, because it is equal toF[\J {X< | X' t L}] by Corollary to Theorem 5.

Theorem 7. Suppose F is a coherent analytic subsheaf of a coherent analyticsheaf F on a complex space (X, 6). If F is primary, then Y(X, F) is a primaryY(X,

1969] COHERENT ANALYTIC SUBSHEAVES 385

References1. H. Grauert, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer

Strukturen, Inst. Hautes Études Sei., No. 5, Paris, 1960.2. R. C. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-

Hall, Englewood Cliffs, N. J., 1965.3. O. Forster, Primärzelegung in Steinschen Algebren, Math. Ann. 154 (1964), 307-329.4. W. Thimm, Lückengarben von kohärenten analytischen Modulgarben, Math. Ann. 148

(1962), 372-394.5. O. Zariski and P. Samuel, Commutative algebra, Van Nostrand, Princeton, N. J., 1958.

Purdue University,Lafayette, Indiana

University of Notre Dame,Notre Dame, Indiana

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