Modules, ideals and their. Rees algebras

Transcription

1 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, Joint work with Ana L. Branco Correia, Lisbon 1

2 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

3 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

4 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,

5 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

6 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

7 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

8 The theory of reductions and integral closure of modules was introduced by D. Rees in 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, 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

9 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

10 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

11 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

12 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

13 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

14 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

15 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

16 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

17 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

18 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

19 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

20 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

21 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,

22 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

23 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, or to study the arithmetical properties of the blow up rings of powers of ideals: Herrmann-Ribbe- Hyry-Tang,

24 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

25 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

26 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

27 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

28 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

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

30 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

31 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

32 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

33 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

34 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

35 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

36 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

37 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

38 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

39 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

40 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