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