Modules, ideals and their

Rees algebras

Santiago Zarzuela

University of Barcelona

Conference on Commutative, Combinatorial

and Computational Algebra

In Honour to

Pilar Pison-Casares

Sevilla, February 11-16, 2008.

Joint work with Ana L. Branco Correia, Lisbon

1

1.- The Rees algebra of a module

• Let (R, m) be a commutative noetherian,

local ring of dimension d.

• G a finitely generated free R-module of rank

e > 0.

• E a submodule of G: E ⊂ G ' Re.

This embedding induces a natural morphism

of graded R-algebras between the symmetric

algebra of E and the symmetric algebra of

G, which is a polynomial ring over R in e

variables:

Φ : SymR(E) −→ SymR(G) ' R[t1, . . . , te]

2

Definition

The Rees algebra of E is the image of SymR(E)

by Φ:

R(E) := Φ(SymR(E))

Since Φ is a graded morphism we have that

R(E) =⊕

n≥0

Φn(SymnR(E))

Definition

The n-th Rees power of E is the homoge-

neous n-th component of the Rees algebra

of E

En := Φn(SymnR(E))

3

• R(E) =⊕

n≥0 En, and E = E1 generates

R(E) over R.

• En ⊂ Gn ' (R[t1, . . . , te])n ' R(n+e−1e−1 ).

Remark. This definition depends on the

chosen embedding of E into G:

Under slightly more general hypothesis, the

definition of the Rees algebra of a module

goes back to A. Micali, 1964 in the frame

of his study of the general properties of the

”universal algebras”.

A more recent discussion about ”what is the

Rees algebra of a module” has been done by

Eisenbud-Huneke-Ulrich, 2002.

4

Remark. If in addition E has rank, then

KerΦ = TR(Sym(E))

and so

R(E) ' SymR(E)/TR(SymR(E))

En ' SymnR(E)/TR(Symn

R(E))

So from now on we are going to assume

that

• E a finitely generated torsionfree R-module

having rank e > 0.

In this case, there exists an embedding

E ↪→ G ' Re

5

When G/E is of finite length the study of the

asymptotic behavior of the quotients Gn/En

is due to Buchsbaum-Rim, 1964:

They showed that for n À 0, the length

λR(Gn/En) assumes the values of a polyno-

mial in n of degree d + e− 1:

The Buchsbaum-Rim polynomial of E.

The normalized leading coefficient of this poly-

nomial is then known as the

Buchsbaum-Rim multiplicity of E: br (E).

6

2.- Integral closure and reductions of mod-

ules

Since R(E) ⊂ R[t1, . . . , te] we may consider

the integral closure R(E) of R(E) in R[t1, . . . , te]

which is a graded ring:

R(E) =⊕

n≥0

R(E)n

Definition

We call E := R(E)1 ⊂ G the integral closure

of E.

• Let U ⊂ E ⊂ G an R-submodule of E.

Definition

We say that U is a reduction of E if En+1 =

UEn for some n.

Equivalently, U is a reduction of E if, and

only if, U = E.

7

The theory of reductions and integral clo-

sure of modules was introduced by D. Rees

in 1987.

Later on, it was somehow rediscovered by T.

Gaffney in 1992 who used the Buchsbaum-

Rim multiplicity and the theory of integral

closure of modules in the study of isolated

complete intersection singularities (ICIS), ex-

tending B. Teissier’s work on Whitney’s reg-

ularity condition, 1973.

If U is a reduction of E and G/E is of finite

length then G/U is also of finite length and so

one can compute the Buchsbaum-Rim mul-

tiplicity of U .

One then can see that

br(U) = br(E)

8

The following result is the extension to mod-

ules of a well known criteria by D. Rees.

Theorem (Kirby-Rees 1994; Kleiman-Thorup,

1994)

Assume that R is quasi-unmixed. Let U ⊆E ⊆ G be such that G/U is of finite length.

Then, U is a reduction of E if, and only if,

br(U) = br(E).

This result may be extended by using the

notion of equimultiplicity.

9

• Let F(E) := R(E)/mR(E), the fiber cone

of E.

Definition

We call the dimension of the fiber cone of E

the analytic spread of E:

l(E) := dimF(E)

• Assume that in addition E has rank e.

Definition

We say that E is equimultiple if

l(E) = htFe(E) + e− 1

where Fe(E) is the e-th Fitting ideal of E.

10

With these definitions the following result

may be viewed as an extension to modules

of a result by E. Boger.

Theorem (D. Katz, 1995)

Let R be quasi-unmixed and

U ⊂ E ⊂ G ' Re

R-modules with rank e such that Fe(U) and

Fe(E) have the same radical. Assume that U

is equimultiple. The following conditions are

then equivalent:

(i) U is a reduction of E.

(ii) br(Up) = br(Ep) for all p ∈ MinFe(U).

11

3.- Minimal reductions

• Let U ⊂ E be a reduction of E.

Definition

The least integer r such that Er+1 = UEr

is called the reduction number of E with re-

spect to U , and it is denoted by rU(E).

Definition

U is said to be a minimal reduction of E if

it is minimal with respect to inclusion among

the reductions of E.

Minimal reductions always exist and they sat-

isfy good properties (similarly to the case of

ideals).

12

Proposition

Let U ⊂ E be a reduction of E. Then:

(a) There always exists V ⊂ U which is a

minimal reduction of E, and for any minimal

reduction V ⊂ U , µ(U) ≥ µ(V ) ≥ l(E).

(b) V ⊂ E is a reduction with µ(V ) = l(E) if,

and only if, any minimal system of generators

of V is a homogeneous system of parame-

ters of F(E) (after taking residue classes in

E/mE ⊂ F(E)). In this case, V is a minimal

reduction of E.

(c) If the residue field R/m is infinite and V ⊂E is a minimal reduction, then condition (b)

always holds, V n ∩ mEn = mV n for all n ≥ 0

and F(V ) ⊂ F(E) is a Noether normalization

13

Definition

The reduction number of E: r(E) is the min-

imum of rU(E) where U ranges over all min-

imal reductions of E.

- If E is a module of linear type, that is, if

R(E) = Sym(E) then r(E) = 0.

The folllowing lower and upper bounds for

l(E) were proven by Simis-Ulrich-Vasconcelos,

2003:

e ≤ l(E) ≤ d + e− 1

- l(E) = e if, and only if, any minimal reduc-

tion of E is a free R-module.

14

4.- Ideal modules

What can be said about SuppG/E?

We would like to realize this set as the variety

of some special ideal.

Observe first that any reduction U of E has

also rank e.

Proposition

Assume gradeG/E ≥ 2. Then

V (Fe(U)) = V (Fe(E)) = SuppG/E = SuppG/U

for any reduction U of E.

Definition

We call E an ideal module if gradeG/E ≥ 2

15

In fact, this is one of the various equivalent

conditions in Simis-Ulrich-Vasconcelos, 2003

to define ideal modules:

- E is an ideal module if E∗∗ is free.

We note that the definition of ideal module

is intrinsic, but the condition gradeG/E ≥ 2

is not and depends on the embedding of E

into G.

Ideal modules satisfy some good properties.

In particular the following lower bound for the

analytic spread:

Proposition

Let E be an ideal module. Then,

e + 1 ≤ htFe(E) + e− 1 ≤ l(E)

16

Modules with finite colength are ideal mod-

ules with maximal analytic spread.

Proposition

Assume that depthR ≥ 2. The following

conditions are then equivalent:

(i) dimG/E = 0;

(ii) E is free locally in the punctured spec-

trum and gradeG/E ≥ 2. In this case,

l(E) = d + e− 1 = htFe(E) + e− 1

For instance, if R is Cohen-Macaulay of di-

mension 2 any ideal module is locally free in

the punctured spectrum.

17

5.- Deviation and analytic deviation

Assume that E is an ideal module but not

free. We define:

- The deviation of E by

d(E) := µ(E)− e + 1− htFe(E)

- The analytic deviation of E by

ad(E) := l(E)− e + 1− htFe(E)

If E is an ideal module then

d(E) ≥ ad(E) ≥ 0

(These definitions slightly differ from similar

ones by Ulrich-Simis-Vasconcelos, 2003)

18

Definition

We say that E is

1. a complete intersection if d(E) = 0,

2. equimultiple if ad(E) = 0,

3. generically a complete intersection if

µ(Ep) = htFe(E) + e− 1

for all p ∈ MinR/Fe(E).

Complete intersection modules were defined

by Buchsbaum-Rim, 1962 in the case of finite

colength as parameter modules.

More in general, Katz-Naude, 1995 studied

them under the classical name of modules of

the principal class.

19

The following is a simple example of com-

plete intersection module of rank two and

not free:

• Let R = K[[x, y]].

• Let G = R2 = Re1 ⊕Re2.

Then,

E = 〈xe1, ye1 + xe2, ye2〉 ⊂ G

is a complete intersection module of rank 2.

In this case,

F2(E) = (0 :R G/E) = (x, y)2

20

There is a list of basic properties satisfied by

complete intersection and equimultiple mod-

ules. For instance,

(1) If E is a complete intersection then E is

equimultiple and generically a complete in-

tersection.

(2) If R/m is infinite, then E is equimultiple

if, and only if, every minimal reduction U of

E is a a complete intersection.

Now we may extend to modules some criteria

for an equimultiple module to be a complete

intersection. The first one extends a simi-

lar result for ideals by Eisenbud-Herrmann-

Vogel, 1977.

21

Theorem

Let R be a Cohen-Macaulay ring and E a

non-free ideal module having rank e > 0.

Suppose that E is generically a complete in-

tersection. Then E is a complete intersection

if and only if E is equimultiple.

We also have the following version of the fa-

mous result by A. Micali, 1964 who proved

that a local ring (R, m) is regular if and only

if S(m) is a domain.

Theorem

Let R be a Noetherian local ring and let E

be an ideal module. Then

a) E is a complete intersection if and only

if E is equimultiple and of linear type.

b) If S(E) is a domain then E is a complete

intersection if and only if E is equimultiple.

22

6.- Some examples with small reduction

number

Rees algebras of modules recover the so called

multi-Rees algebras.

Let I1, . . . , Ie be a family of ideals of R. The

multi-Rees algebra of I1, . . . , Ie is the graded

ring

R(I1, . . . , Ie) := R[I1t1, . . . , Iete]

Let E := I1 ⊕ · · · ⊕ Ie ⊂ G = Re. Then,

R(E) ' R(I1, . . . , Ie)

Multi-Rees algebras have been successfully

used in connection with the theory of mixed

multiplicities: J. Verma, 1991... or to study

the arithmetical properties of the blow up

rings of powers of ideals: Herrmann-Ribbe-

Hyry-Tang, 1997...

23

First we observe that:

Proposition

Let E = I1 ⊕ · · · ⊕ Ie with Ii ⊂ R ideals satis-

fying grade Ii ≥ 2. Then E is not a complete

intersection.

But:

Proposition

Assume R to be Cohen-Macaulay with in-

finite residue field. Let I be an equimulti-

ple ideal with ht I = 2 and r(I) ≤ 1. Write

E = I ⊕ · · · ⊕ I = I⊕e, e ≥ 2. Then,

(i) r(E) = 1, l(E) = e + 1.

(ii) E is equimultiple.

24

We may get examples of generically a com-

plete intersection modules in the following

way:

Proposition

Assume R to be Cohen-Macaulay with infi-

nite residue field and d ≥ 3. Let p1, . . . , pe be

pairwise distinct prime ideals which are per-

fect of grade 2. Write E = p1⊕· · ·⊕pe, e ≥ 2.

Then,

(1) E is generically a complete intersection.

(2) E is not equimultiple.

(3) l(E) ≥ e + 2, ad(E) ≥ 1 with equalities if

d = 3.

(4) If d = 3, e = 2 and p1, p2 are complete

intersection then r(E) = 0

25

We note that the direct sum of equimulti-

ple (even complete intersection) ideals is not

necessarily an equimultiple module, as the

following easy example shows:

Example

Let R = k[[X1, X2, X3]] with k an infinite field

and write E = (X1, X2)⊕ (X1, X3). Then,

- E is generically a complete intersection;

- l(E) = 4;

- ad(E) = 1;

- r(E) = 0.

26

7.- Arithemtical conditions

The following result is an extension to mod-

ules of the well known Burch’s inequality. It

holds more in general (F. Hayasaka, 2007 for

instance) but we only state for ideal modules:

Theorem

Let E ( G ' Re be an ideal module. Then,

l(E) ≤ d + e− 1− inf depthGn/En

In addition, equality holds if R(E) is Cohen-

Macaulay.

As a consequence, we have the following arith-

metical characterization for the equimultiplic-

ity of an ideal module, when its Rees algebra

is Cohen-Macaulay.

27

Proposition

Assume that R is Cohen-Macaulay and let

E ⊂ G ' Re be an ideal module with rank

e, but not free. If R(E) is Cohen-Macaulay

then the following are all equivalent:

(i) E is equimultiple;

(ii) depthGn/En = d− htFe(E) for all n > 0;

(iii) depthGn/En = d− htFe(E) for infinitely

many n.

Now, combining this with the previous char-

acterization of the complete intersection prop-

erty for equimultiple ideal modules we get the

following:

28

Proposition

Assume that R is Cohen-Macaulay and let

E ⊂ G ' Re be an ideal module with rank

e, but not free. Assume E is generically a

complete intersection. Then, the following

are equivalent:

(i) E is a complete intersection;

(ii) Gn/En are Cohen-Macaulay for all n > 0;

(iii) Gn/En are Cohen-Macaulay for infinitely

many n.

This is a version for ideal modules of an old

result by Cowsik-Nori, 1976 later on refined

by M. Brodmann, 1979.

(i) ⇒ (ii) was proven by Katz-Kodiyalam,

1997.

29

8.- The generic Bourbaki ideal of a mod-

ule

In order to get an ideal providing information

about the Rees algebra of E, Simis-Ulrich-

Vasconcelos, 2003 introduced the

- generic Bourbaki ideal of a module.

In general, an exact sequence of the form

0 → F → E → I → 0

where F is a free R-module and I is an R-

ideal is called a Bourbaki sequence. I is then

a Bourbaki ideal of E.

Roughly speaking, a generic Bourbaki ideal I

of E is a Bourbaki ideal of E, after a special

Nagata extension R′′ of R.

30

Under suitable hypothesis, the Rees algebra

of E is a isomorphic to the Rees algebra of I

modulo a regular sequence of homogeneous

elements of degree 1.

The construction is as follows:

Assume e ≥ 2 and let U =∑n

i=1 Rai be a

submodule of E such that E/U is a torsion

module (which holds if U is a reduction of

E). Further, let

Z = {zij | 1 ≤ i ≤ n,1 ≤ j ≤ e− 1}be a set of n× (e−1) indeterminates over R.

We fix the notation

R′ = R[Z] , R′′ = R′mR′ = R(Z) ,

U ′ = U ⊗R′ , E′ = E ⊗R′

U ′′ = U ⊗R′′ , E′′ = E ⊗R′′ .

31

Now, take the elements

xj =n∑

i=1

zijai ∈ U ′ ⊂ E′

and let

F =e−1∑

j=1

R′′xj.

Proposition (Simis-Ulrich-Vasconcelos, 2003)

F ⊂ E′′ is a free module over R′′ of rank e-1.

Consider now the exact sequence of R′′-modules

0 → F → E′′ → E′′/F → 0

If E′′/F is torsionfree then it is isomorphic to

an ideal of R′′:

IU(E)

that we call a generic Bourbaki ideal of E

with respect to U .

32

The above happens whenever

gradeFe(E) ≥ 2

in particular when E is an ideal module.

In this case, IU(E) may also be chosen with

grade IU(E) ≥ 2

Proposition

Assume that IU(E) is a generic Bourbaki ideal

of E with respect to U . Then:

a) l(IU(E)) = l(E)− e + 1.

b) If k is infinite, r(IU(E)) ≤ r(E).

c) µ(IU(E)) = µ(E)− e + 1.

33

Proposition

Furthermore to the above conditions, assume

that

(1) gradeR(E)+ = e or

(2) R(IU(E)) satisfies (S2).

Then, there exists a family of elements

x = x1, . . . , xe−1

such that x is regular sequence in R(E′′) and

R(IU(E)) ' R(E′′)/(x)

Moreover, r(IU(E)) ≥ r(E) and if U = E,

r(IU(E)) = r(E).

(In fact, these elements are homogeneous of

degree 1 and a basis of F ⊂ E′′.)

34

9.- Generic Bourbaki ideals as Fitting ide-

als

Sometimes, generic Bourbaki ideals can be

explicitly computed as a Fitting ideal.

The procedure is the following:

• Let {x1, . . . , xn} be a generating set of E′′containing the basis {x1, . . . , xe−1} of F .

• Let ϕ be a matrix presenting E′′ with re-

spect to the generators {x1, . . . , xn}.

Then, one can chose ϕ such that

ϕ =

[∗ ∗∗ ψ

]

where ψ be an (n− e+1)× (n− e) submatrix

of ϕ, with grade In−e(ψ) ≥ 1.

35

Proposition

Assume that E is an ideal module. Then,

any generic Bourbaki ideal IU(E) of E with

respect to U is isomorphic to In−e(ψ).

Moreover, if grade In−e(ψ) ≥ 2, then by Hilbert-

Burch theorem we have

- In−e(ψ) is perfect of grade 2;

- IU(E) has a finite free resolution of the form

0 → R′′n−e ψ→ R′′n−e+1 → IU(E) → 0

- IU(E) = aIn−e(ψ) for some a ∈ R′′ \ Z(R′′).

36

10.- Ideal modules with small reduction

number

Assume that R(E) is Cohen-Macaulay. As

a consequence of Burch’s inequality for ideal

modules (equality if the Rees algebra is Cohen-

Macaulay) we have that

l(E) ≥ d + e− depthE

The following is a partial converse:

Proposition

Let R be a Cohen-Macaulay ring with infinite

residue field and E an ideal module having

rank e > 0 with r(E) ≤ 1. Moreover, assume

that E is free locally in codimension l(E)− e.

Then, R(E) is Cohen-Macaulay if and only if

l(E) ≥ d + e− depthE.

37

Proof (sketch)

We may assume e ≥ 2. Let I ⊂ R′′ a generic

Bourbaki ideal of E with grade (I) ≥ 2. Then

l(I) = l(E)−e+1 and r(I) ≤ r(E) ≤ 1. More-

over, since E is free locally in codimension

l(E)− e then I satisfies conditions Gl(I) and

AN−l(I)−2 (Simis-Ulrich-Vasconcelos, 2003).

Therefore by (L. Ghezzi, 2002)

depthG(I) = min{d,depthR′′/I + l(I)}Then,

l(E) ≥ d + e− depthE ⇔

depthR′′/I + l(I) ≥ d ⇔ depthG(I) = d

On the other hand,

a(G(I)) = max{−ht I, r(I)− l(I)} < 0

and so G(I) is Cohen-Macaulay if and only

if R(I) is Cohen-Macaulay (by Ikeda-Trung).

The result, then, follows.

38

As a consequence we have the following:

Proposition

Let R be a Cohen-Macaulay ring with infinite

residue field and E an ideal module.

- If E is equimultiple with r(E) ≤ 1, then

R(E) is Cohen-Macaulay if and only if G/E

is Cohen-Macaulay.

- If E is a complete intersection then R(E)

is Cohen-Macaulay.

- If E is free locally on the punctured spec-

trum with r(E) ≤ 1 then R(E) is Cohen-

Macaulay.

- If dimR = 2, then R(E) is Cohen-Macaulay

if and only if r(E) ≤ 1.

39

As a final (revisited) example we have:

Proposition

Assume R to be Cohen-Macaulay with in-

finite residue field. Let I be an equimulti-

ple ideal with ht I = 2 and r(I) ≤ 1. Write

E = I ⊕ · · · ⊕ I = I⊕e, e ≥ 2. Then

(a) (E is equimultiple, r(E) = 1, and l(E) =

e + 1);

(b) R(E) is Cohen-Macaulay if and only if

R/I is Cohen-Macaulay.

40