IDEALS AND THEIR INTEGRAL CLOSURE ALBERTO CORSO (joint work with C. - - PowerPoint PPT Presentation

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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

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SETTING

Let R be a Noetherian ringa 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 Xn + a1Xn−1 + · · · + an−1X + an = 0, ai ∈ Ii.

• The radical √I consists of all the solutions in R of equations
• f the form

Xm − b = 0, b ∈ I.

• In particular:

I ⊆ I ⊆ √I.

• I is integrally closed (resp. normal) if I = I (resp. I n = In

for all n).

aLater we may even want to assume the ring to have other additional prop-

erties (such as: Cohen–Macaulay, with infinite residue field, etc.

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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
• f) the integral closure of I.

Comment: It is not an easy task. For example, if R is a polynomial ring

• ver 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.

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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 + I2t2 + · · · + Intn + · · · of I;

• ne then looks for its integral closure inside R[t]:

R + It + I2t2 + · · · ⊂ R[t]. This is obviously wasteful of resources since the integral closure

• f all powers of I will be computed. Of course this could be

profitably taken if it turns out that I is normal.

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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: L2.

• The proof in both cases is essentially the same, and it is based
• n the following two ‘local criteria.’

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A ‘LOCAL’ RESULT OF GOTO Theorema:

Let I be an ideal in a Noetherian ring R and assume that µR(I) = heightR(I) = g. Then the following conditions are equivalent:

• I = I, i.e. I is integrally closed;
• In = In, i.e. I is normal;
• for each p ∈ AssR(R/I), the local ring Rp is regular and

λRp IRp + p2Rp p2Rp

• ≥ g − 1.

When this is the case, AssR(R/I) = MinR(R/I) and I is generated by an R-regular sequence.

• aS. Goto: Integral closedness of complete-intersection ideals, J. Alge-

bra 108 (1987), 151–160.

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LINKAGE AND REDUCTION NUMBER Theorema+b:

Let (R, m) be a Cohen–Macaulay local ring and let I = (z1, . . . , zg) be an ideal generated by a regular sequence inside a prime ideal p

• f height g. If we set L = I: p then L2 = IL if one of the following

two conditions holds: (l1) Rp is not a regular local ring; (l2) Rp is a regular local ring with dimension at least 2 and two of the zi’s in p(2).

• aA. Corso, C. Polini and W. V. Vasconcelos: Links of prime ideals, Math.
• Proc. Camb. Phil. Soc. 115 (1994), 431–436.
• bA. Corso and C. Polini: Links of prime ideals and their Rees algebras,
• J. Algebra 178 (1995), 224–238.

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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;
• I2: 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.

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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...)?

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INTEGRAL CLOSEDNESS AND NORMALITY: ideals with linear presentation Theorem: Let k be a field of characteristic zero and let

I ⊂ R = k[x1, . . . , xd] be a Gorenstein ideal defined by the Pfaffians

• f 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[x1, x2, x3, x4] 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.

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(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 ϕ =           −a2 −b2 −c2 −d2 a2 −d2 −ab −c2 b2 d2 −a2 −ab c2 ab a2 −b2 d2 c2 ab b2           , define an height 3 Gorenstein ideal I such that I2: I = I and I = IL: L, where L = I: √I. Hence I is integrally closed BUT it is not normal. Note that I2 is integrally closed as well. Is I3 ‘BAD’?

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INTEGRAL CLOSEDNESS AND NORMALITY: complete intersections of codimension 2 Theorem:

Let R be a regular local ring and I = (a, b) a complete intersection

• f codimension 2. Then I is normal, i.e.,

In = (I)n for all n ≥ 1.

Remark: For dimension 2 this is a very well known result of

Zariski.

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A METHOD TO COMPUTE THE INTEGRAL CLOSURE

If I = (a1, . . . , an) then one can represent its Rees algebra as R[It] = R[T1, . . . , Tn]/P, where P is the kernel of the map ϕ : R[T1, . . . , Tn] − → R[It] that sends Ti to ait. If R[It] is an affine domain over a field of characteristic zero and Jac denotes its Jacobian ideal, then the ring HomR[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.

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EXAMPLE

Let k be a field of characteristic zero and let I ⊆ R = k[x, y] be the codimension 2 complete intersection I = (x3 + y6, xy3 − y5). Iterating three times the method outlined before we can compute I. To be precise, the three outputs are: J1 = (x3 + y6, xy3 − y5, y8), J2 = (x3 + y6, xy3 − y5, x2y2 − y6, y7), J3 = I = (xy3 − y5, y6, x3, x2y2). 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!

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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 ⊂ ms but I ⊂ ms+1 then one has an increasing sequence of ideals Ik = I: mk satisfying I2

k = IIk 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 Ik’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[Ist] (or R[Is−1t] if R is a regular local ring and dim(R) = 2).

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✬ ✫ ✩ ✪ If I is not primary to the maximal ideal one may use instead the sequence of ideals Ik = 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 I2

k = IIk for all

the k’s in the appropriate range. In the case of the example: R = k[x, y] and I = (x3 + y6, xy3 − y5)

• ne 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 J1 = I: m J2 = I: m2.

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