Cohen-Macaulay residual intersections and their Castelnuovo-Mumford regularity

Hassanzadeh, Seyed Hamid

doi:10.1090/s0002-9947-2012-05602-9

Cited by 9 publications

(27 citation statements)

References 31 publications

(27 reference statements)

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“…Proof. The proof of this fact is essentially the same as that of [Ha,Theorem 3.6] wherein the case of k = 0 is verified. The crucial point to repeat that proof is that we have to change the structure of the families { k Z + • } from the beginning by substituting the tails with a free complex.…”

Section: Residual Approximation Complexesmentioning

confidence: 68%

“…Assume that π is the projection fromR to R which sends x ij to c ij . By [Ha,Proposition 2.13], R/π(K) = H 0 ( 0 Z + • ) and by [HU,Theorem 4.7], π(J ) = J. Therefore R/K = R/π(K) = R/π(J) = R/J.…”

Section: It Then Follows Thatmentioning

confidence: 97%

“…Although the proof here is more involved than [Ha,2.8], we prefer to omit it to go faster to newer theorems. The Cohen-Macaulay hypothesis in this theorem is needed to show that if for an R-module M ,…”

Section: Residual Approximation Complexesmentioning

confidence: 99%

“…(4) To see this, we refer to the proof of [Ha,Theorem 2.11] wherein it is shown that in the spectral…”

Section: Residual Approximation Complexesmentioning

confidence: 99%

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Residual intersections and the annihilator of Koszul homologies

Hassanzadeh

Naéliton

2016

Algebra Number Theory

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Cohen-Macaulayness, unmixedness, the structure of the canonical module and the stability of the Hilbert function of algebraic residual intersections are studied in this paper. Some conjectures about these properties are established for large classes of residual intersections without restricting local number of generators of the ideals involved. A family of approximation complexes for residual intersections is constructed to determine the above properties. Moreover some general properties of the symmetric powers of quotient ideals are determined which were not known even for special ideals with a small number of generators. Acyclicity of a prime case of these complexes is shown to be equivalent to find a common annihilator for higher Koszul homologies. So that, a tight relation between residual intersections and the uniform annihilator of positive Koszul homologies is unveiled that sheds some light on their structure. a residual intersection of X if the number of equations needed to define X ∪ Y as a subscheme of Z is the smallest possible that is s. Precisely, if R is a Noetherian ring, I an ideal of height g and s ≥ g an integer, then • An (algebraic) s-residual intersection of I is a proper ideal J of R such that ht(J) ≥ s and J = (a : R I) for some ideal a ⊂ I generated by s elements. • A geometric s-residual intersection of I is an algebraic s-residual intersection J of I such that ht(I + J) ≥ s + 1.Based on a construction of Laksov for residual intersection, Fulton [Fu, Definition 9.2.2] presents a formulation for residual intersection that, locally, can be expressed as follows: Suppose that X = Spec(R)

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Section: Residual Approximation Complexesmentioning

confidence: 68%

Section: It Then Follows Thatmentioning

confidence: 97%

Section: Residual Approximation Complexesmentioning

confidence: 99%

“…(4) To see this, we refer to the proof of [Ha,Theorem 2.11] wherein it is shown that in the spectral…”

Section: Residual Approximation Complexesmentioning

confidence: 99%

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Residual intersections and the annihilator of Koszul homologies

Hassanzadeh

Naéliton

2016

Algebra Number Theory

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“…In Section 4, we prove one of the main results of this paper: the Cohen-Macaulayness and the description of the canonical module of residual intersections. Recall that in [8], Hassanzadeh proved that, under the sliding depth condition, H 0 ( 0 Z + • ) = R/K is Cohen-Macaulay of codimension s, with K ⊂ J, √ K = √ J, and further K = J whenever the residual is arithmetic. First, we consider the height two case and show that under the SD 1 condition, there exist an epimorphism ϕ :…”

Section: Introductionmentioning

confidence: 99%

Cohen–Macaulayness and canonical module of residual intersections

Chardin

Naéliton²,

Tran

2019

Trans. Amer. Math. Soc.

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We show the Cohen-Macaulayness and describe the canonical module of residual intersections J = a : R I in a Cohen-Macaulay local ring R, under sliding depth type hypotheses. For this purpose, we construct and study, using a recent article of Hassanzadeh and the second named author [11], a family of complexes that contains important informations on a residual intersection and its canonical module. We also determine several invariants of residual intersections as the graded canonical module, the Hilbert series, the Castelnuovo-Mumford regularity and the type. Finally, whenever I is strongly Cohen-Macaulay, we show duality results for residual intersections that are closely connected to results by Eisenbud and Ulrich [6]. It establishes some tight relations between the Hilbert series of some symmetric powers of I/a. We also provide closed formulas for the types and for the Bass numbers of some symmetric powers of I/a.

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Serre’s Condition R k for Sums of Geometric Links

Johnson

Ulrich

2015

Acta Math Vietnam

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Cohen-Macaulay residual intersections and their Castelnuovo-Mumford regularity

Cited by 9 publications

References 31 publications

Residual intersections and the annihilator of Koszul homologies

Residual intersections and the annihilator of Koszul homologies

Cohen–Macaulayness and canonical module of residual intersections

Serre’s Condition R k for Sums of Geometric Links

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