Symbolic powers of sums of ideals

Harbourne, Brian; Nguyen, Hop D.; Trung, Ngô Viêt; Trung, Trân Nam

doi:10.1007/s00209-019-02323-8

Cited by 40 publications

(38 citation statements)

References 28 publications

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“…in terms of sets of associated primes and minimal associated primes. By [4,Proposition 3.3], for any filtrations {I i } i≥0 and {J j } j≥0 in A and B, respectively, we have for any integer n ≥ 0 an isomorphism of C-modules deduced at the level of K-vector spaces:…”

Section: Proofs Of the Key Resultsmentioning

confidence: 99%

“…Thus for n ≥ 2 in the intersection of (I + J) n as in Lemma 2.5 we cannot omit any component involving P 2 . We certainly cannot omit the minimal component P 1 + Q 1 , and we cannot omit the component for P 1 + Q 2 because n i=0 p i1 q n−i,1 ∩ n i=0 p i2 q n−i,1 ∩ n i=0 p i2 q n−i,2 = n i=0 (x 1 , x 2 ) 4i q n−i,1 ∩ n i=0 p i2 J n−i = n i=0 (x 1 , x 2 ) 4i q n−i,1 ∩ J n + (x 4 1 , x 3 1 x 2 , x 1 x 3 2 , x 4 2 , x 3 ) contains x 3 q n1 and is thus not a subset of (I + J) n . This proves that for all n ≥ 2, (I + J) n has four associated primes.…”

Section: A Few Concluding Examplesmentioning

confidence: 99%

“…Remark 1.3. This definition differs from Hà-Nguyen-Trung-Trung [4,Section 3] only in that we do not require L 1 to be non-zero and proper -upon inspection this assumption is inessential for the proofs of [4, Proposition 3.3, Theorem 3.4]. We briefly recall this point at the start of Section 3.…”

Section: Introductionmentioning

confidence: 99%

“…Said ideal satisfies the persistence property by a result of Katz and Ratliff [7, (1.3) Theorem], and even if the other ideal does not, the corollary to follow indicates that their sum does satisfy it, indicating that the persistence property is remarkably persistent and robust under extension of scalars. [4]. Namely, they fix Noetherian commutative algebras A and B over a common field K, such that C = A ⊗ K B is Noetherian as well, along with non-zero ideals I ⊆ A and J ⊆ B.…”

Section: Introductionmentioning

confidence: 99%

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Tensor-multinomial sums of ideals: Primary decompositions and persistence of associated primes

Swanson

Walker

2019

Proc. Amer. Math. Soc.

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Given a polynomial ring C over a field and proper ideals I and J whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of I + J into a collection of primes described in terms of the associated primes of select powers of I and of J. We record two applications. First, in case the field is algebraically closed, we construct primary decompositions for powers of I + J from primary decompositions for powers of I and J. Separately, we attack the persistence problem for associated primes of powers of an ideal in case one of I or J is a non-zero normal ideal.

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Section: Proofs Of the Key Resultsmentioning

confidence: 99%

Section: A Few Concluding Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Tensor-multinomial sums of ideals: Primary decompositions and persistence of associated primes

Swanson

Walker

2019

Proc. Amer. Math. Soc.

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“…Random examples show that depth R/I t (respectively, reg R/I t ) tends to be a non-increasing (respectively, non-decreasing) function, though that is not the case in general [1,2,13,14]. Bandari, Herzog and Hibi [1] asked whether depth R/I t is a non-increasing function for squarefree monomial ideals.…”

Section: Depth and Regularity Functionsmentioning

confidence: 99%