π ψ

A

∞ j

i=’∞ Si , Si = {x f : j ∈ Z, f ∈ Ri’j }. (Though R may

S is graded again: S =

not be a subring of S, we do not distinguish notationally between elements in R and

their images under the homomorphism R ’ S. Furthermore we shall write I © R for the

preimage of an ideal I ‚ S in R.)

The structure of the graded ring S is particularly simple. Evidently:

(16.21) Proposition. (a) For every homogeneous ideal I ‚ S, in particular for

I = S, one has

∞

xi (I © S0 ).

I=

i=’∞

(b) The natural homomorphism S0 [X, X ’1 ] ’ S, X ’ x, is an isomorphism.

Furthermore A is not only a homomorphic image, but a subring of S, too:

(16.22) Proposition. The homomorphism ψ maps S0 isomorphically onto A.

Proof: Obviously the restriction of ψ to S0 is surjective. If, on the other hand,

ψ(f x’i ) = 0 for an element f ∈ Ri , then π(f ) = 0. So f = g(x ’ 1), g ∈ R. Since f is

homogeneous, gx = 0, f = ’g, and f = 0 in S. ”

In the following we shall identify A with S0 .

212 16. Appendix

(16.23) Proposition. If R is reduced (a (normal) domain), then A is reduced (a

(normal) domain), too.

Proof: S = R[x’1 ] inherits each of the listed properties from R, and A is a subring

of S. The only not completely obvious problem is whether normality carries over to A.

It is well known, that A is normal if and only if A[X] is normal, and the normality of

A[X] follows from the normality of S = A[X, X ’1 ] by the following lemma which will be

very useful several times:

(16.24) Lemma. Let T be a noetherian ring, and y ∈ T such that y is not a

zero-divisor, T /T y is reduced and T [y ’1 ] is normal. Then T is normal.

Proof: We use Serre™s normality criterion: T is normal if and only if TP is regular

for every prime ideal P of T such that depth TP ¤ 1. Let P be such a prime ideal. If

y ∈ P , TP is a localization of the normal ring T [y ’1 ], thus regular. Otherwise P is a

minimal prime of T y, and P TP = yTP , since T /T y is reduced. Having its maximal ideal

generated by an element which is not a zero-divisor, TP is a regular local ring. ”

Since the inversion of x may destroy pathologies of R, one cannot reverse (16.23)

in complete generality. For the rings of interest to us this is possible however, since the

additional hypotheses of the following proposition are satis¬ed.

(16.25) Proposition. Suppose additionally that x is not a zero-divisor. Then R

is reduced (a domain) if A is reduced (a domain). If furthermore R is noetherian and

R/Rx is reduced, then normality transfers from A to R.

This is immediate now. In the following we want to relate the ideals of R and A.

(16.26) Proposition. (a) One has π(I) = IS © A for a homogeneous ideal I ‚ R,

and J = π(JS © R) for every ideal J of A.

(b) By relating the ideals I and π(I) the homomorphism π sets up a bijective correspon-

dence between the homogeneous ideals of R, modulo which x is not a zero-divisor, and

all the ideals of A.

(c) This correspondence preserves set-theoretic inclusions and intersections. It further-

more preserves the properties of being a prime, primary or radical ideal in both directions.

Proof: (a) The ideal π(I) is generated by the images of the homogeneous elements

f ∈ I. If f has degree d, then π(f ) = f x’d ∈ IS © A. Conversely, let g ∈ IS © A. Then

g = xk h, k ∈ Z, h ∈ I, and g = ψ(g) = π(xk h) = π(h) ∈ π(I).

The ideal JS is homogeneous, so JS © R is homogeneous, and

π(JS © R) = (JS © R)S © A = JS © A = J.

(b) A (homogeneous) ideal I of R appears as the preimage of a (homogeneous) ideal

of S (namely of its own extension) if and only if x is not a zero-divisor modulo I. This

establishes a bijective correspondence between the ideals I under consideration and the

homogeneous ideals of S. The latter are in 1-1-correspondence with the ideals of A by

(16.21),(a), and the ¬rst equation in (a) shows that the desired correspondence is induced

by π.

(c) The ¬rst statement of (c) is completely obvious. The properties of being a prime,

primary or radical ideal are preserved in going from R to its ring of quotients S, and

213

E. How to Compare “Torsionfree”

also under taking preimages in A ‚ S. Conversely they cannot be destroyed by the

extensions A ’ A[X] and A[X] ’ A[X, X ’1 ] = S, from where we pass to R by taking

preimages. ”

One usally calls π(I) the dehomogenization of I, and JS © R the homogenization of

J. As a consequence of (16.26),(c) primary decompositions are preserved:

(16.27) Proposition. Let R be noetherian, I a homogeneous ideal modulo which

x is not a zero-divisor, and I = Qi an irredundant decomposition into homogeneous

primary ideals. Then π(I) = π(Qi ) is an irredundant primary decomposition of π(I).

The analogous statement holds for the process of homogenization.

Let P be a homogeneous prime ideal of R, x ∈ P , and Q its dehomogenization.

Then

RP = A[X]QA[X]

is a localization of AQ [X], and it is clear that RP and AQ share essentially all ring-

theoretic properties:

(16.28) Proposition. Suppose that R is noetherian. Let P be a homogeneous prime

ideal of R, x ∈ P , and Q its dehomogenization. Then RP and AQ coincide with respect

to the following quantities and properties: dimension, depth, being reduced, integrity,

normality, being Cohen-Macaulay , being Gorenstein, regularity.

In fact, the extension AQ ’ RP is faithfully ¬‚at. Its ¬ber is the ¬eld (AQ /QAQ )(X).

Thus (16.28) follows from the properties of ¬‚at extensions as given in [Mt], Sect. 21, and,

as far as the Gorenstein property is concerned, in [Wt]. (Of course one can give more

direct arguments in the special situation of (16.28).)

E. How to Compare “Torsionfree”

Since the notion “torsionfree” is fairly standard, we have used it without explanation:

An A-module M is torsionfree if every element of A which is not a zero-divisor of A, is

not a zero-divisor of M . In this subsection we introduce several notions which describe

higher degrees of being torsionfree, and give conditions under which they are equivalent.

Definition. Let A be a noetherian ring, M a ¬nitely generated A-module. M is

called n-torsionfree if every A-sequence of length at most n is an M -sequence, too.

There is a slightly stronger condition of Serre type:

(16.29) Proposition. Let A be a noetherian ring, M a ¬nitely generated A-module.

Then M is n-torsionfree if it satis¬es the condition

(Sn ) : depth MP ≥ min(n, depth AP ) for all prime ideals P.

It is an exercise on associated prime ideals to prove that M is n-torsionfree if and

only if

depth MP ≥ min(n, grade P ) for all prime ideals P,

and this inequality is obviously weaker than (Sn ). It is furthermore obvious that both

properties of (16.29) are equivalent if the localizations AP such that depth AP < n are

214 16. Appendix

Cohen-Macaulay rings, since grade P = depth AP then for all prime ideals P such that

depth AP ¤ n (cf. (16.13)).

Let A be an integral domain momentarily, M a torsionfree A-module, Q the ¬eld of

fractions of A. The natural map h : M ’ M —— becomes an isomorphism when tensored

with Q. Since M is torsionfree, the torsion module Ker h must be zero. An epimorphism

F ’ M — , F free, leads to an embedding M ’ M —— ’ F — : M is a submodule of a free

A-module, and therefore a ¬rst module of syzygies of an A-module.

Definition. Let A be a noetherian ring. An A-module M is called an n-th syzygy

if there is an exact sequence

0 ’’ M ’’ Fn ’’ · · · ’’ F1

with ¬nitely generated free A-modules Fi .

From the behaviour of depth along exact sequences one concludes immediately:

(16.30) Proposition. An n-th syzygy satis¬es (Sn ).

An A-module M for which the natural map h : M ’ M —— is injective, is called

torsionless. The argument above shows that a torsionless module is a ¬rst syzygy, and

conversely a ¬rst syzygy is torsionless: An embedding M ’ F extends to a commutative

diagram

M ’’’ F

’’

¦

¦

h

M —— ’ ’ ’ F —— .

’’

If h is an isomorphism, M is called re¬‚exive.

A natural idea how to make M an n-th syzygy, is to start with a free resolution of

the dual

Fn ’’ · · · ’’ F1 ’’ M — ’’ 0,

to dualize and to replace the embedding M —— ’ F1 by its composition with M ’ M —— .

—

This yields a zero-sequence

— —

0 ’’ M ’’ F1 ’’ · · · ’’ Fn .

Definition. M is called n-torsionless if the preceding sequence is exact.

Since it is irrelevant which resolution of M — has been chosen, this de¬nition is

justi¬ed.

(16.31) Proposition. Let A be a noetherian ring, M a ¬nitely generated A-module.

(a) If M is n-torsionless, then it is an n-th syzygy.

(b) M is 1-torsionless (2-torsionless) if and only if it is torsionless (re¬‚exive).

(c) M is k-torsionless for k ≥ 3 if and only if it is re¬‚exive and Exti (M — , A) = 0 for

A

i = 1, . . . , k ’ 2.

The proposition follows readily from the de¬nition of “n-torsionless”. A somewhat

smoother description of “n-torsionless” can be given by means of the Auslander-Bridger

dual of M : It is the cokernel of f — in a ¬nite free presentation

f

F ’’ G ’’ M ’’ 0.

Despite its non-uniqueness we denote it by D(M ). One has M = D(D(M )). (It is not

di¬cult to prove that D(M ) is unique up to projective direct summands.)

215

E. How to Compare “Torsionfree”

(16.32) Proposition. Let A be a noetherian ring, M a ¬nitely generated A-module,

h : M ’ M —— the natural map. Then:

(a) Ker h = Ext1 (D(M ), A),

A

(b) Coker h = Ext2 (D(M ), A), and

A

(c) M is n-torsionless if and only if Exti (D(M ), A) = 0 for i = 1, . . . , n.

A

Proof: Because of the preceding proposition and

Exti (D(M ), A) = Exti’2 (M — , A)

A A

for i ≥ 3 it is enough to prove (a) and (b). We choose a ¬nite free presentation of M —

K ’’ H ’’ M — ’’ 0

and splice its dual via the natural homomorphism h with a presentation

f

F ’’ G ’’ M ’’ 0.

Then D(M ) = Coker f — , and one has a commutative diagram

F ’ ’ ’ G ’ ’ ’ H— ’ ’ ’ K—

’’ ’’ ’’

¦

¦ ¦

¦

M ’ ’ ’ M ——

’’

whose upper row has homology Ker h at G and Coker h at H — . By construction

K ’’ H ’’ G— ’’ F — ’’ D(M ) ’’ 0

is the right end of a free resolution of D(M ). ”

The most elementary notion among the ones introduced is certainly the property

“n-th syzygy”. On the other hand it is the hardest to control, and the properties ( Sn ) and

“n-torsionless” should be regarded as a lower and an upper “homological” approximation.

Under certain hypotheses on M (or A) all the properties introduced are equivalent:

(16.33) Proposition. Let A be a noetherian ring, and M a ¬nitely generated A-

module such that pd MP < ∞ for all prime ideals P of A with depth AP < n. Then all

the properties

“n-torsionfree”, (Sn ), “n-th syzygy”, and “n-torsionless”

are equivalent.

As we shall see below “n-th syzygy” and “n-torsionless” are equivalent under a

slightly weaker hypothesis on M .

Proof: We ¬rst show that M satis¬es (Sn ) if it is n-torsionfree. For a prime ideal

P such that grade P ≥ n one clearly has depth MP ≥ n. Otherwise there is a prime ideal

Q ⊃ P such that depth AQ = grade Q = grade P < n. Then depth MQ = depth AQ , and

216 16. Appendix

MQ has to be a free AQ -module because of depth AQ + pd MQ = depth AQ . Even more

MP is a free AP -module.

Next one proves directly that (Sn ) implies “n-torsionless”. From the argument just

given it follows that a free resolution

f

· · · ’’ Fn · · · ’’ F1 ’’ M — ’’ 0

splits when localized with respect to prime ideals P such that depth AP < n, and fur-

——

thermore hP : MP ’ MP is an isomorphism. Therefore the cokernel N of the map g

in

g

— —

0 ’’ M ’’ F1 ’’ · · · ’’ Fn

(—)

has property (Sn’1 ) and pd NP < ∞ for all prime ideals P such that depth AP < n. M

is certainly torsionfree and (Ker g)P = Ker gP = 0 for all associated prime ideals P of A.

Since N — = Ker f , an inductive argument ¬nishes the proof. ”

As pointed out above, every ¬rst syzygy is 1-torsionless, and this fact signalizes that

“n-th syzygy” and “n-torsionless” should be equivalent under a weaker hypothesis.

(16.34) Proposition. With the remaining hypotheses of (16.33) suppose further-

more that pd MP < ∞ for all prime ideals P such that depth AP < n ’ 1. Then every

n-th syzygy is n-torsionless.

Proof: The case n = 1 being settled, we treat n = 2 as a separate case, too. There

is an exact sequence

0 ’’ M ’’ F ’’ N ’’ 0

in which N is a ¬rst syzygy and F is free. Then we have a commutative diagram

0 ’’’ M

’’ ’’’ F

’’ ’ ’ ’ N ’ ’ ’ 0.

’’ ’’

¦ ¦

¦ ¦g

h

f

M —— ’ ’ ’ F —— ’ ’ ’ N ——

’’ ’’

The kernel of f is (Ext1 (N, A))— , hence zero since Ext1 (N, A) is a torsion module: NP is

A A

free for all P ∈ Ass A because of pd NP < ∞. Since g is injective, h has to be surjective.

Let n > 2 now. We have an exact sequence 0 ’ M ’ F ’ N ’ 0 as above, in