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B ULLETIN DE LA S. M. F.

R. H ARTSHORNE A property of A-sequences

Bulletin de la S. M. F., tome 94 (1966), p. 61-65

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© Bulletin de la S. M. F., 1966, tous droits réservés.

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Bull. Soc. math. France, 94, 1966, p. 61 a 66.

A PROPERTY OF ^-SEQUENCES

ROBIN HARTSHORNE (*).

Let A be a noetherian local ring with maximal ideal m, containing a field k (not necessarily its residue field). Recall ([1]; [7]) that an A-sequence is a finite set Xi, . . . , Xr of elements of A, contained in the maximal ideal m, such that Xi is not a zero-divisor in A, and for each 1 = 2 , . . . , r, Xi is not a zero-divisor in A/(rci, .. ., rc;_i). We will show that for ma.ny purposes, the elements of an A-sequence behave just like the variables in a polynomial ring over a field. In particular, the sum, product, intersection and quotient of ideals generated by monomials in a given A-sequence are just what one would expect (see Corollary 1 below for a precise statement).,

PROPOSITION 1. — Let A be a noetherian local ring containing a field k, and let Xi, . . . , Xr be an A-sequence. Then the natural map

c p : T=:/c[Xi, . . . , X , ] - ^ A

of k-algebras, which sends X; into Xi for each i, is injective, and A is flat as a T-module.

Proof. — We show 9 is injective by induction on r, the case r == o being trivial. Let r > o be given. Then x^ . . . , X r is an (A/rciA)- sequence, so by the induction hypothesis, we may assume that

© : k[X,, ...,Xr]->AIXiA is injective. Now let f e T be given and write

t == ^^ X^ fn (Xs, ..., X/-),

(*) Junior Fellow, Harvard University.

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62 R. HARTSHORNE.

where each fn(X,, ..., X,:)^k[X,., ..., X,.]. Suppose that y(Q = o.

If t^ o, let /", be the first of the /"„ which is non-zero. Then

c?(.f)=X•l^Xrsfn(x,., ...,3-A

\n=s j

Since a-i is a non-zero-divisor in A, we have

^x^sfn(x„ ...,;r.)=o.

^==.y

Reducing modulo x,, we find /^, . . . , Xr) = o in A/r^A. Now since ^ is injective by the induction hypothesis, f,(X,, .. ., X,) = o, which is a contradiction. Hence t == o and cp is injective.

Now to show A is Hat over T, we use the local criterion of flatness ([3], chap. Ill, § 5, no 2, theorem 1, (iii)) applied to the ring T, the ideal J = (xi, . . . . Xr), and the T-module A. We must verify the four following statements :

(a) T is noetherian (well-known).

(b) A is separated for the J-adic topology, i. e. F^ J ^ A = o. This

is true since JA is contained in the radical m of A, and F\ m^== o by KrulFs theorem ([3], chap. Ill, § 3, n° 2).

(c) AfJA is flat over k = T/J. This is true since anything is flat over a field.

(d) Tor^T/J, A) == o. To calculate this Tor, we use the Koszul complex K.(X,, . . . . X,; T) ([4], EGA, III, 1.1) which is a resolution of T/J since Xi, . . . . X/. is a T-sequence. Tor^T/J, A) is the Ith homo-

^gy group of the complex

K.(X,, . . . , X , ; T ) ^ ) T A = K . ( X , , . . . , . r , ; A ) .

But since Xi, - . . . , Xr is an A-sequence, this homology is zero in degrees i > o ([4], EGA, III, 1.1.4). In particular TorT(T/J, A) = o, which completes the proof of the proposition.

COROLLARY 1. — With the notations of the proposition, let a and b be any two ideals in T. For any ideal c in T, denote by cA its extension to A. Then

(i) (a + b ) A = a A + b A ; (ii) (a.b)A = ( a A ) . ( b A ) ; (iii) ( a n b ) A = ( a A ) n ( b A ) ; (iv) ( a : b ) A = ( a A ) : (bA).

(Recall that for any two ideals a, b in a ring R, a : b = { XG. R \ rr.b C a j.

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A PROPERTY OF A-SEQUENCES. 63

Proof. — (i) and (ii) are trivially true for any ring extension and are repeated here for convenience, (iii) and (iv) are true for any flat ring extension, (iii) is proved in ([3], chap. I, § 2, n° 6, Prop. 6).

To prove (iv), let z/i, . . . , ^ be a set of generators for b. Then a : b = f \ (a : (yi)), and so using (iii) we are reduced to the case where b is generated by a single element y. Now a : (y) is characterized by the exact sequence of T-modules

o->a:(y)->T^TIa,

where the last map is multiplication by y. Tensoring with A we have an exact sequence of A-modules

o-^(a: 0/))A-^A-^A/aA

from which we deduce that (a : (y)) A == a A : y A (Note that for any ideal b in T, the natural map b (g)rA—^bA is an isomorphism, since A is flat over T, so we identify the two).

COROLLARY 2 (Theorem of Rees). — Let A be a noetherian local ring containing a field, and let J be an ideal generated by an A-sequence Xi, .... re,..

Then the images Xi, . . . , Xr of the Xi in the graded ring

go(A)=^J-/J^

n=0

are algebraically independent, so that gr./(A) is isomorphic to the poly- nomial ring A/J[Xi, . . . , Xr].

Proof (see also [7], Appendix 6, theorem 3). — It is sufficient to show that for each n, Jn|Jn+i is a free A/J-module, with the images of the monomials in Xi, . . . , Xr of degree n for basis. It is clear that these monomials generate Jn|Jn+i. To show they are linearly independent, let z be a monomial of degree n in Xi, .. ., Xr, and let J ' be the ideal gene- rated by all the other monomials of degree n and by J7^1. Then we must show that Jr : z = J , which follows from Corollary 1.

COROLLARY 3. — Let A be a noetherian local ring containing a field k, and let rci, . .., Xr be an A-sequence. Then any ideal of A generated by polynomials in the Xi, with coefficients in k, is of finite homological dimension over A.

Proof. — Using the notations of the proposition, any such ideal can be written as a A, where a is an ideal in the polynomial ring T == A'[Xi,.... X/.].

Over T, a has a finite projective resolution ([7], chap. VII, § 13, theorem 43)

o-> Ln—^.. .—^Li—^Lo—>- a —> o.

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64 R. HARTSHORNE.

Tensoring with A gives an exact sequence

o - ^ L , , ( g ) A — ^ . . . — ^ L i ( g ) A — ^ L o 0 A - > a A - > o which is a finite projective resolution of a A.

Remark. — A refinement of the proof of proposition 1 due to D. QUILLEN allows one to dispense with the hypothesis that A contains a field, provided that one is interested only in ideals of A generated by monic monomials in the Xi. In particular this is sufficient for the result of Corollary 2, and of Proposition 2 below.

As an application we give the following :

PROPOSITION 2. — Let A be a noetherian local ring containing a field.

Let I be a radical ideal in A (i. e. an ideal which is a finite intersection of prime ideals), and let J be any ideal generated by an A-sequence whose radical is I . Then, to within isomorphism, the A/I-module

M = Hom^(A/J, A/J) is independent of J.

Example. — An interesting case (already known [2]) is that of a local Cohen-Macaulay ring A, with I == m the maximal ideal. Then there are ideals J generated by an A-sequence with radical nx, so that M is defined. Its dimension as an A/m-vector space is an invariant of A, which is equal to i if and only if A is a Gorenstein ring. (See [2], where if n is the dimension of M, then A is called a MCn-ring. This number is also the (< vordere Loewysche Invariante " of A/J in [6], p. 28, and is the number e of the exercises in [5], § 4, p. 67.)

Proof of Proposition. — Let J be generated by the A-sequence X[, ..., Xr.

Then r is the height of J, and so is independent of J. We consider the r111 local cohomology group (see [5] for definition and methods of calculation)

H == H'^A) = limExt7 (A/J<^, A),

—>/t

where J^ = (x^, . . . , x'f). Using the Koszul complex K . ( x ' i , ..., x^\ A) to calculate the Ext, we find an isomorphism

cp, : Ext^A/J^, A) ^ A/J.^)

which transforms the maps of the direct system into the maps fn: A/J^-^A/J^1)

which are defined by multiplication by x'i - ' • Xr.

I claim that the maps fn are all injective. Indeed, it is sufficient to see that

J^' : (.Tr. .Xr)=J^.

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A PROPERTY OF ^4-SEQUENCES. 65

This follows from Corollary 1 and the fact that the analogous relation holds in a polynomial ring. Therefore we can write H as an increasing union

H=.\jEn,

where En is the isomorphic image of AfJW in H. Furthermore, I claim that for each n, En is the set of elements of H annihilated by J^. Indeed, we have only to observe that for each n, k > o,

J(n+k) ; JW=(x^"Xr)k

which follows from Corollary 1 and the analogous formula in a poly- nomial ring. Now since JCJ, anything in H annihilated by I is annihi- lated by J\ Hence

M = Hom^(A/J, A/J) == Hom^(A/J, E,) = Hom^(A/J, H).

But by definition, H depends only on the radical of J [5], so we are done.

BIBLIOGRAPHY.

[1] AUSLANDER (M.) et BUCHSBAUM (D.). — Codimension and multiplicity, Annals of Math., Series 2, t. 68, ig58, p. 626-657.

[2] BASS (H.). — Injective dimension in noetherian rings, Trans. Amer. math. Soc., t. 102, 1962, p. 18-29.

[3] BOURBAKI (N.). — Algebre commutative. Chap. 1-2, 3-4. — Paris, Hermann, 1961 (Act. scient. et ind., 1290-1293; Bourbaki, 27-28).

[4] GROTHENDIECK (A.). — Elements de geometric algebrique [EGA]. — Paris, Presses universitaires de France, 1960-1966 (Institut des Hautes Etudes Scientifiques, Publications mathematiques, n08 4, 8, 11, 17, 20, 24, 28).

[5] GROTHENDIECK (A.). — Local cohomology. Lecture notes by R. Hartshorne. — Harvard, Harvard University, 1961.

[6] KRULL (W.). — Idealtheorie. — Berlin, J. Springer, 1935 (Ergebnisse der Mathe- matik, Bd 4, Heft 3).

[7] ZARISKI (0.) et SAMUEL (P.). — Commutative algebra. Vol. 1 and 2. — Princeton, D. Van Nostrand, 1958-1960.

(Manuscrit recu Ie 8 decembre 1965.) R. HARTSHORNE,

Mathematics Department, 2, Divinity Avenue,

Cambridge, Mass. 021 38 (Etats-Unis).

BULL. SOC. MATH. — T. 94, PASO. 1. 5

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