Modules, ideals and their Rees algebras Santiago Zarzuela University of Barcelona Conference on Commutative, Combinatorial and Computational Algebra In Honour to Pilar Pisón-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 R e. 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 : Sym R (E) Sym R (G) R[t 1,..., t e ] 2
Definition The Rees algebra of E is the image of Sym R (E) by Φ: R(E) := Φ(Sym R (E)) Since Φ is a graded morphism we have that R(E) = n 0 Φ n (Sym n R (E)) Definition The n-th Rees power of E is the homogeneous n-th component of the Rees algebra of E E n := Φ n (Sym n R (E)) 3
R(E) = n 0 E n, and E = E 1 generates R(E) over R. E n G n (R[t 1,..., t e ]) n R (n+e 1 e 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 Ker Φ = T R (Sym(E)) and so R(E) Sym R (E)/T R (Sym R (E)) E n Sym n R (E)/T R(Sym n 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 R e 5
When G/E is of finite length the study of the asymptotic behavior of the quotients G n /E n is due to Buchsbaum-Rim, 1964: They showed that for n 0, the length λ R (G n /E n ) assumes the values of a polynomial in n of degree d + e 1: The Buchsbaum-Rim polynomial of E. The normalized leading coefficient of this polynomial is then known as the Buchsbaum-Rim multiplicity of E: br (E). 6
2.- Integral closure and reductions of modules Since R(E) R[t 1,..., t e ] we may consider the integral closure R(E) of R(E) in R[t 1,..., t e ] which is a graded ring: Definition R(E) = n 0 R(E) n 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 E n+1 = UE n for some n. Equivalently, U is a reduction of E if, and only if, U = E. 7
The theory of reductions and integral closure 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), extending B. Teissier s work on Whitney s regularity 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 multiplicity of U. One then can see that br(u) = br(e) 8
The following result is the extension to modules 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) := dim F(E) Assume that in addition E has rank e. Definition We say that E is equimultiple if l(e) = ht F e (E) + e 1 where F e (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. Böger. Theorem (D. Katz, 1995) Let R be quasi-unmixed and U E G R e R-modules with rank e such that F e (U) and F e (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(u p ) = br(e p ) for all p Min F e (U). 11
3.- Minimal reductions Let U E be a reduction of E. Definition The least integer r such that E r+1 = UE r is called the reduction number of E with respect to U, and it is denoted by r U (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 satisfy 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 parameters 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 me n = 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 minimum of r U (E) where U ranges over all minimal 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 reduction of E is a free R-module. 14
4.- Ideal modules What can be said about Supp G/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 grade G/E 2. Then V (F e (U)) = V (F e (E)) = Supp G/E = Supp G/U for any reduction U of E. Definition We call E an ideal module if grade G/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 grade G/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 ht F e (E) + e 1 l(e) 16
Modules with finite colength are ideal modules with maximal analytic spread. Proposition Assume that depth R 2. The following conditions are then equivalent: (i) dim G/E = 0; (ii) E is free locally in the punctured spectrum and grade G/E 2. In this case, l(e) = d + e 1 = ht F e (E) + e 1 For instance, if R is Cohen-Macaulay of dimension 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 ht F e (E) - The analytic deviation of E by ad(e) := l(e) e + 1 ht F e (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(e p ) = ht F e (E) + e 1 for all p Min R/F e (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 complete intersection module of rank two and not free: Let R = K[[x, y]]. Let G = R 2 = Re 1 Re 2. Then, E = xe 1, ye 1 + xe 2, ye 2 G is a complete intersection module of rank 2. In this case, F 2 (E) = (0 : R G/E) = (x, y) 2 20
There is a list of basic properties satisfied by complete intersection and equimultiple modules. For instance, (1) If E is a complete intersection then E is equimultiple and generically a complete intersection. (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 similar 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 intersection. Then E is a complete intersection if and only if E is equimultiple. We also have the following version of the famous 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 I 1,..., I e be a family of ideals of R. The multi-rees algebra of I 1,..., I e is the graded ring R(I 1,..., I e ) := R[I 1 t 1,..., I e t e ] Let E := I 1 I e G = R e. Then, R(E) R(I 1,..., I e ) 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 = I 1 I e with I i R ideals satisfying grade I i 2. Then E is not a complete intersection. But: Proposition Assume R to be Cohen-Macaulay with infinite residue field. Let I be an equimultiple 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 complete intersection modules in the following way: Proposition Assume R to be Cohen-Macaulay with infinite residue field and d 3. Let p 1,..., p e be pairwise distinct prime ideals which are perfect of grade 2. Write E = p 1 p e, 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 p 1, p 2 are complete intersection then r(e) = 0 25
We note that the direct sum of equimultiple (even complete intersection) ideals is not necessarily an equimultiple module, as the following easy example shows: Example Let R = k[[x 1, X 2, X 3 ]] with k an infinite field and write E = (X 1, X 2 ) (X 1, X 3 ). 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 modules 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 R e be an ideal module. Then, l(e) d + e 1 inf depth G n /E n In addition, equality holds if R(E) is Cohen- Macaulay. As a consequence, we have the following arithmetical characterization for the equimultiplicity of an ideal module, when its Rees algebra is Cohen-Macaulay. 27
Proposition Assume that R is Cohen-Macaulay and let E G R e 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) depth G n /E n = d ht F e (E) for all n > 0; (iii) depth G n /E n = d ht F e (E) for infinitely many n. Now, combining this with the previous characterization of the complete intersection property for equimultiple ideal modules we get the following: 28
Proposition Assume that R is Cohen-Macaulay and let E G R e 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) G n /E n are Cohen-Macaulay for all n > 0; (iii) G n /E n 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 module 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 Ra i 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 = {z ij 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 and let x j = n i=1 F = z ij a i U E e 1 j=1 R x j. 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 : I U (E) that we call a generic Bourbaki ideal of E with respect to U. 32
The above happens whenever grade F e (E) 2 in particular when E is an ideal module. In this case, I U (E) may also be chosen with grade I U (E) 2 Proposition Assume that I U (E) is a generic Bourbaki ideal of E with respect to U. Then: a) l(i U (E)) = l(e) e + 1. b) If k is infinite, r(i U (E)) r(e). c) µ(i U (E)) = µ(e) e + 1. 33
Proposition Furthermore to the above conditions, assume that (1) grade R(E) + = e or (2) R(I U (E)) satisfies (S 2 ). Then, there exists a family of elements x = x 1,..., x e 1 such that x is regular sequence in R(E ) and R(I U (E)) R(E )/(x) Moreover, r(i U (E)) r(e) and if U = E, r(i U (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 ideals Sometimes, generic Bourbaki ideals can be explicitly computed as a Fitting ideal. The procedure is the following: Let {x 1,..., x n } be a generating set of E containing the basis {x 1,..., x e 1 } of F. Let ϕ be a matrix presenting E with respect to the generators {x 1,..., x n }. Then, one can chose ϕ such that [ ] ϕ = ψ where ψ be an (n e + 1) (n e) submatrix of ϕ, with grade I n e (ψ) 1. 35
Proposition Assume that E is an ideal module. Then, any generic Bourbaki ideal I U (E) of E with respect to U is isomorphic to I n e (ψ). Moreover, if grade I n e (ψ) 2, then by Hilbert- Burch theorem we have - I n e (ψ) is perfect of grade 2; - I U (E) has a finite free resolution of the form 0 R n e ψ R n e+1 IU (E) 0 - I U (E) = ai n 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 depth E 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 depth E. 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. Moreover, since E is free locally in codimension l(e) e then I satisfies conditions G l(i) and AN (Simis-Ulrich-Vasconcelos, 2003). l(i) 2 Therefore by (L. Ghezzi, 2002) Then, depth G(I) = min{d, depth R /I + l(i)} l(e) d + e depth E depth R /I + l(i) d depth G(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 spectrum with r(e) 1 then R(E) is Cohen- Macaulay. - If dim R = 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 infinite residue field. Let I be an equimultiple 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