1,721,155 research outputs found
Decomposition of local cohomology tables of modules with large E-depth
No full textWe introduce the notion of E-depth of graded modules over polynomial rings to measure the depth of certain Ext modules. First, we characterize graded modules over polynomial rings with (sufficiently) large E-depth as those modules whose (sufficiently) partial general initial submodules preserve the Hilbert function of local cohomology modules supported at the irrelevant maximal ideal, extending a result of Herzog and Sbarra on sequentially Cohen-Macaulay modules. Second, we describe the cone of local cohomology tables of modules with sufficiently high E-depth, building on previous work of the second author and Smirnov. Finally, we obtain a non-Artinian version of a socle-lemma proved by Kustin and Ulrich
Products of ideals may not be Golod
No full textWe exhibit an example of a product of two proper monomial ideals such that the residue class ring is not Golod. We also discuss the strongly Golod property for rational powers of monomial ideals, and introduce some sufficient conditions for weak Golodness of monomial ideals. Along the way, we ask some related questions. (C) 2015 Elsevier B.V. All rights reserved
A counterexample to a conjecture of Ding
No full textWe give a counterexample to a conjecture posed by S. Ding in [4] regarding the index of a Gorenstein local ring by exhibiting several examples of one dimensional local complete intersections of embedding dimension three with index 5 and generalised Loewy length 6. (C) 2016 Elsevier Inc. All rights reserved
ARTINIAN LEVEL ALGEBRAS OF LOW SOCLE DEGREE
No full textIn this article we study Hilbert functions and isomorphism classes of Artinian level local algebras via Macaulay's inverse system. Upper and lower bounds concerning numerical functions admissible for level algebras of fixed type and socle degree are known. For each value in this range we exhibit a level local algebra with that Hilbert function, provided that the socle degree is at most three. Furthermore, we prove that level local algebras of socle degree three and maximal Hilbert function are graded. In the graded case, the extremal strata have been parametrized by Cho and Iarrobino
Stability and deformation of F-singularities
No full textWe study the problem of m-adic stability of F-singularities, that is, whether the property that a quotient of a local ring (R,m) by a non-zero divisor x∈m has good F-singularities is preserved in a sufficiently small m-adic neighborhood of x. We show that m-adic stability holds for F-rationality in full generality, and for F-injectivity, F-purity and strong F-regularity under certain assumptions. We show that strong F-regularity and F-purity are not stable in general. Moreover, we exhibit strong connections between stability and deformation phenomena, which hold in great generality.La Caixa Postdoctoral Junior Leader,
Ramon y Cajal RYC2020-028976-
F-Stable Secondary Representations and Deformation of F-Injectivity
No full textWe prove that deformation of F-injectivity holds for local rings (R, m) that admit secondary representations of Hmi(R) which are stable under the natural Frobenius action. As a consequence, F-injectivity deforms when (R, m) is sequentially Cohen–Macaulay (or more generally when all the local cohomology modules Hmi(R) have no embedded attached primes). We obtain some additional cases if R/ m is perfect or if R is N-graded
A Cayley–Bacharach theorem for points in Pn
No full textWe prove a Cayley–Bacharach type theorem for points in projective space (Formula presented.) that lie on a complete intersection of (Formula presented.) hypersurfaces. This is made possible by new bounds on the growth of the Hilbert function of almost complete intersections
BOUNDS ON THE NUMBER OF GENERATORS OF PRIME IDEALS
No full textLet S be a polynomial ring over any field k, and let P ⊆ S be a non-degenerate homogeneous prime ideal of height h. When k is algebraically closed, a classical result attributed to Castelnuovo establishes an upper bound on the number of linearly independent quadrics contained in P which only depends on h. We significantly extend this result by proving that the number of minimal generators of P in any degree j can be bounded above by an explicit function that only depends on j and h. In addition to providing a bound for generators in any degree j, not just for quadrics, our techniques allow us to drop the assumption that k is algebraically closed. By means of standard techniques, we also obtain analogous upper bounds on higher graded Betti numbers of any radical ideal
On Monomial Golod Ideals
No full textWe study ideal-theoretic conditions for a monomial ideal to be Golod. For ideals in a polynomial ring in three variables, our criteria give a complete characterization. Over such rings, we show that the product of two monomial ideals is Golod
LINEARLY PRESENTED MODULES AND BOUNDS ON THE CASTELNUOVO-MUMFORD REGULARITY OF IDEALS
No full textWe estimate the Castelnuovo-Mumford regularity of ideals in a polynomial ring over a field by studying the regularity of certain modules generated in degree zero and with linear relations. In dimension one, this process gives a new type of upper bounds. By means of recursive techniques this also produces new upper bounds for ideals in any dimension
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