✬ ✩ IDEALS AND THEIR INTEGRAL CLOSURE ALBERTO CORSO (joint work with C. Huneke and W.V. Vasconcelos) Department of Mathematics Purdue University West Lafayette, 47907 San Diego — January 8, 1997 ✫ ✪ 1
✬ ✩ SETTING Let R be a Noetherian ring a and I one of its ideals. • The integral closure of I is the ideal I of all elements of R that satisfy an equation of the form X n + a 1 X n − 1 + · · · + a n − 1 X + a n = 0 , a i ∈ I i . • The radical √ I consists of all the solutions in R of equations of the form X m − b = 0 , b ∈ I. I ⊆ I ⊆ √ I . • In particular: • I is integrally closed (resp. normal ) if I = I (resp. I n = I n for all n ). a Later we may even want to assume the ring to have other additional prop- ✫ ✪ erties (such as: Cohen–Macaulay, with infinite residue field, etc. 2
✬ ✩ PROBLEMS For an ideal I of R we would like to give an answer to questions of the following kind: • Find ‘efficient’ and ‘global’ criteria to test whether or not I is integrally closed. Comment: The criteria should involve natural objects associated with I : e.g. the powers of I , the radical of I , modules of syzygies, etc. • If I fails the ‘above’ tests, find a procedure to compute (part of) the integral closure of I . Comment: It is not an easy task. For example, if R is a polynomial ring over a field and I is a monomial ideal, I is then the monomial ideal defined by the integral convex hull of the exponent vectors of I . However, if I is binomial then I need not be binomial. • When I integrally closed implies I normal. ✫ ✪ 3
✬ ✩ POSSIBLE APPROACHES • A consequence of the determinant trick is that for every finitely generated faithful R -module M then I ⊆ IM : M ⊆ I . In particular if I is integrally closed, for any such M , IM : M = I. We then need appropriate ‘test modules’ for a given ideal I . • An alternative approach is through the Rees algebra R [ It ] = R + It + I 2 t 2 + · · · + I n t n + · · · of I ; one then looks for its integral closure inside R [ t ]: R + It + I 2 t 2 + · · · ⊂ R [ t ] . This is obviously wasteful of resources since the integral closure of all powers of I will be computed. Of course this could be ✫ ✪ profitably taken if it turns out that I is normal. 4
✬ ✩ CRITERION Let I be a height unmixed ideal in a Cohen–Macaulay ring R . Suppose that I is generically a complete intersection. Then the following conditions are equivalent: • I is integrally closed; I = IL : L , where L = I : √ I . • Remarks: • An earlier version of this criterion had the condition I = IL : L replaced by √ I = IL : L 2 . • The proof in both cases is essentially the same, and it is based on the following two ‘local criteria.’ ✫ ✪ 5
✬ ✩ A ‘LOCAL’ RESULT OF GOTO Theorem a : Let I be an ideal in a Noetherian ring R and assume that µ R ( I ) = height R ( I ) = g . Then the following conditions are equivalent: • I = I , i.e. I is integrally closed; • I n = I n , i.e. I is normal; • for each p ∈ Ass R ( R/I ), the local ring R p is regular and � IR p + p 2 R p � λ R p ≥ g − 1 . p 2 R p When this is the case, Ass R ( R/I ) = Min R ( R/I ) and I is generated by an R -regular sequence. a S. Goto: Integral closedness of complete-intersection ideals , J. Alge- ✫ bra 108 (1987), 151–160. ✪ 6
✬ ✩ LINKAGE AND REDUCTION NUMBER Theorem a + b : Let ( R, m ) be a Cohen–Macaulay local ring and let I = ( z 1 , . . . , z g ) be an ideal generated by a regular sequence inside a prime ideal p of height g . If we set L = I : p then L 2 = IL if one of the following two conditions holds: (l 1 ) R p is not a regular local ring; (l 2 ) R p is a regular local ring with dimension at least 2 and two of the z i ’s in p (2) . a A. Corso, C. Polini and W. V. Vasconcelos: Links of prime ideals , Math. Proc. Camb. Phil. Soc. 115 (1994), 431–436. b A. Corso and C. Polini: Links of prime ideals and their Rees algebras , ✫ J. Algebra 178 (1995), 224–238. ✪ 7
✬ ✩ ANOTHER USEFUL CRITERION Let R be a Gorenstein ring and let I be a Gorenstein ideal of codimension 3. Then the following conditions are equivalent: • I is generically a complete intersection; I 2 : I = I . • Remarks: • In particular, if I is integrally closed then it is generically a complete intersection. • Furthermore, one could combine this and the previous criterion to get a condition for a perfect Gorenstein ideal of codimension 3 to be integrally closed. However, this is not necessary as the next result shows. ✫ ✪ 8
✬ ✩ GORENSTEIN IDEALS Let I be a Gorenstein ideal of codimension g ≥ 2 of a regular local ring R . Then the following conditions are equivalent: • I is an integrally closed ideal; • I = IL : L , where L = I : √ I . Comments: • As a consequence of the proof, I is generically a complete intersection of the kind described in Goto’s paper. • Is it possible to find an easy, global criterion to characterize integrally closed ideals with type 2 (3, etc...)? ✫ ✪ 9
✬ ✩ INTEGRAL CLOSEDNESS AND NORMALITY: ideals with linear presentation Theorem: Let k be a field of characteristic zero and let I ⊂ R = k [ x 1 , . . . , x d ] be a Gorenstein ideal defined by the Pfaffians of a n × n skew–symmetric matrix ϕ with linear forms as entries. Suppose n = d + 1 (hence d is even) and that I is a complete intersection on the punctured spectrum. If I is integrally closed it is also normal. Remark: The proof is based on the ‘Jacobian criterion.’ Corollary: Let k be a field of characteristic zero and let I ⊂ R = k [ x 1 , x 2 , x 3 , x 4 ] be a Gorenstein ideal defined by the Pfaffians of a five by five skew–symmetric matrix ϕ with linear ✫ forms as entries. If I is integrally closed it is also normal. ✪ 10
✬ ✩ (COUNTER)EXAMPLE Let R = k [ a, b, c, d ] with a, b, c, d be variables and char( k ) = 0. The Pfaffians of the 5 × 5 matrix ϕ − a 2 − b 2 − c 2 − d 2 0 a 2 − d 2 − c 2 0 − ab b 2 d 2 − a 2 ϕ = , 0 − ab c 2 a 2 − b 2 ab 0 d 2 c 2 b 2 ab 0 define an height 3 Gorenstein ideal I such that I 2 : I = I and I = IL : L, where L = I : √ I . Hence I is integrally closed BUT it is not normal. Note that I 2 is integrally closed as well. Is I 3 ‘BAD’? ✫ ✪ 11
✬ ✩ INTEGRAL CLOSEDNESS AND NORMALITY: complete intersections of codimension 2 Theorem: Let R be a regular local ring and I = ( a, b ) a complete intersection of codimension 2. Then I is normal, i.e., I n = ( I ) n for all n ≥ 1. Remark: For dimension 2 this is a very well known result of Zariski. ✫ ✪ 12
✬ ✩ A METHOD TO COMPUTE THE INTEGRAL CLOSURE If I = ( a 1 , . . . , a n ) then one can represent its Rees algebra as R [ It ] = R [ T 1 , . . . , T n ] / P , where P is the kernel of the map ϕ : R [ T 1 , . . . , T n ] − → R [ It ] that sends T i to a i t . If R [ It ] is an affine domain over a field of characteristic zero and Jac denotes its Jacobian ideal, then the ring Hom R [ It ] (Jac − 1 , Jac − 1 ) = · · · = (Jac Jac − 1 ) − 1 , is guaranteed to be larger than R [ It ] if the ring is not already normal. Naturally, this process can be repeated several times until the integral closure of R [ It ] has been reached. The degree one component of the final output gives the desired integral closure of I . ✫ ✪ 13
✬ ✩ EXAMPLE Let k be a field of characteristic zero and let I ⊆ R = k [ x, y ] be the codimension 2 complete intersection I = ( x 3 + y 6 , xy 3 − y 5 ) . Iterating three times the method outlined before we can compute I . To be precise, the three outputs are: J 1 = ( x 3 + y 6 , xy 3 − y 5 , y 8 ) , J 2 = ( x 3 + y 6 , xy 3 − y 5 , x 2 y 2 − y 6 , y 7 ) , J 3 = I = ( xy 3 − y 5 , y 6 , x 3 , x 2 y 2 ) . Note that I is also a normal ideal. For the records, despite the fact that the original setting for the problem is a polynomial ring in 2 variables over a field of characteristic zero, overall we had to make use of 18 additional ✫ ✪ variables: quite a waste! 14
✬ ✩ FINAL COMMENT If I is a complete intersection, several initial iterations of the process may be avoided (see [CP2] and [PU]). If I is an m -primary complete intersection of a Gorenstein local ring ( R, m ) and I ⊂ m s but I �⊂ m s +1 then one has an increasing sequence of ideals I k = I : m k satisfying I 2 k = II k for k = 1 , . . . , s if dim( R ) ≥ 3 or for k = 1 , . . . s − 1 if R is a regular local ring and dim( R ) = 2. This says that the I k ’s are contained in the integral closure of I . Hence, instead of computing the integral closure of R [ It ] one may start directly from R [ I s t ] (or R [ I s − 1 t ] if R is a regular local ring and dim( R ) = 2). ✫ ✪ 15
✬ ✩ If I is not primary to the maximal ideal one may use instead the sequence of ideals I k = I : ( √ I ) ( k ) = I : ( √ I ) k , provided that at each localization at the associated primes of I the conditions in [CP2] and [PU] are satisfied so that I 2 k = II k for all the k ’s in the appropriate range. In the case of the example: R = k [ x, y ] and I = ( x 3 + y 6 , xy 3 − y 5 ) one has that I ⊂ ( x, y ) 3 . Since dim( R ) = 2 one can consider only the cases k = 1 , 2 = s − 1. We can check that J 2 = I : m 2 . J 1 = I : m ✫ ✪ 16
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