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1.
Bhargav Bhatt 《Journal of Pure and Applied Algebra》2019,223(5):1940-1945
Given a commutative ring A and a finitely generated ideal I, we prove that -torsion A-modules that are also I-adically complete (or merely derived I-complete) must have bounded -torsion, i.e., they are killed by for some . 相似文献
2.
Sarah Mayes-Tang 《Journal of Pure and Applied Algebra》2019,223(2):571-579
Given an ideal I we investigate the decompositions of Betti diagrams of the graded family of ideals formed by taking powers of I. We prove conjectures of Engström from [5] and show that there is a stabilization in the Boij–Söderberg decompositions of for when I is a homogeneous ideal with generators in a single degree. In particular, the number of terms in the decompositions with positive coefficients remains constant for , the pure diagrams appearing in each decomposition have the same shape, and the coefficients of these diagrams are given by polynomials in k. We also show that a similar result holds for decompositions with arbitrary coefficients arising from other chains of pure diagrams. 相似文献
3.
For an ideal generated by all square-free monomials of degree m in a polynomial ring R with n variables, we obtain a specific embedding of a canonical module of to itself. The construction of this explicit embedding depends on a minimal free R-resolution of an ideal generated by . Using this embedding, we give a resolution of connected sums of several copies of certain Artin -algebras where is a field. 相似文献
4.
Federico Galetto Anthony V. Geramita Yong-Su Shin Adam Van Tuyl 《Journal of Pure and Applied Algebra》2019,223(6):2709-2731
Let I be a homogeneous ideal of . To compare , the m-th symbolic power of I, with , the regular m-th power, we introduce the m-th symbolic defect of I, denoted . Precisely, is the minimal number of generators of the R-module , or equivalently, the minimal number of generators one must add to to make . In this paper, we take the first step towards understanding the symbolic defect by considering the case that I is either the defining ideal of a star configuration or the ideal associated to a finite set of points in . We are specifically interested in identifying ideals I with . 相似文献
5.
Let R and S be standard graded algebras over a field k, and and homogeneous ideals. Denote by P the sum of the extensions of I and J to . We investigate several important homological invariants of powers of P based on the information about I and J, with focus on finding the exact formulas for these invariants. Our investigation exploits certain Tor vanishing property of natural inclusion maps between consecutive powers of I and J. As a consequence, we provide fairly complete information about the depth and regularity of powers of P given that R and S are polynomial rings and either or I and J are generated by monomials. 相似文献
6.
Raymond C. Heitmann 《Journal of Pure and Applied Algebra》2022,226(1):106809
It is shown that if is a complete local domain with and is its integral closure in an algebraic closure of the quotient field, then both the -adic and p-adic completions of are integral domains. More generally, this theorem remains true if the completeness assumption is relaxed to allow R to be an analytically irreducible Henselian local ring. It is also shown that these rings, which are Cohen-Macaulay R-modules (even balanced in the -adic case), will have dimension larger than the dimension of R unless . 相似文献
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Christian U. Jensen Søren Jøndrup Anders Thorup 《Journal of Pure and Applied Algebra》2019,223(5):1869-1896
Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion “almost slenderness” of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of “bounded type” (including countable rings, ‘small’ rings, and, for instance, rings that are countably generated as algebras over an Artinian ring).More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring over a von Neumann regular ring k.For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this ‘compact’ class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R-algebras vanishing on . 相似文献
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In this note we describe the minimal resolution of the ideal , the saturation of the Jacobian ideal of a nearly free plane curve . In particular, it follows that this ideal can be generated by at most 4 polynomials. Related general results by Hassanzadeh and Simis on the saturation of codimension 2 ideals are discussed in detail. Some applications to rational cuspidal plane curves and to line arrangements are also given. 相似文献
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Lukas Katthän 《Journal of Pure and Applied Algebra》2019,223(3):1227-1245
Let be a squarefree monomial ideal in a polynomial ring. In this paper we study multiplications on the minimal free resolution of . In particular, we characterize the possible vectors of total Betti numbers for such ideals which admit a differential graded algebra (DGA) structure on . We also show that under these assumptions the maximal shifts of the graded Betti numbers are subadditive.On the other hand, we present an example of a strongly generic monomial ideal which does not admit a DGA structure on its minimal free resolution. In particular, this demonstrates that the Hull resolution and the Lyubeznik resolution do not admit DGA structures in general.Finally, we show that it is enough to modify the last map of to ensure that it admits the structure of a DG algebra. 相似文献
13.
Let R be a polynomial ring over a field and I an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension of is at most 36, although the example with largest projective dimension he constructed has . Based on computational evidence, it had been conjectured that . In the present paper we prove this conjectured sharp bound. 相似文献
14.
Rasoul Ahangari Maleki 《Journal of Pure and Applied Algebra》2019,223(2):605-618
Let S be a regular local ring or a polynomial ring over a field and I be an ideal of S. Motivated by a recent result of Herzog and Huneke, we study the natural question of whether is a Golod ideal for all . We observe that the Golod property of an ideal can be detected through the vanishing of certain maps induced in homology. This observation leads us to generalize some known results from the graded case to local rings and obtain new classes of Golod ideals. 相似文献
15.
Let be a Noetherian local ring and M a finitely generated R-module. The invariants and of M were introduced in [3] and [17] in order to measure the non-Cohen–Macaulayness and the non-sequential-Cohen–Macaulayness of M, respectively. Let be the filtration of M such that is the largest submodule of M of dimension less than for all and . In this paper we prove that if , then there exists a constant c such that for all good parameter ideals of M with respect to this filtration. Here is the reducibility index of on M. This is an extension of the main results of [19], [20], [24]. 相似文献
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H.E.A. Campbell J. Chuai R.J. Shank D.L. Wehlau 《Journal of Pure and Applied Algebra》2019,223(5):2015-2035
Suppose is a field of prime characteristic p and E is a finite subgroup of the additive group . Then E is an elementary abelian p-group. We consider two such subgroups, say E and , to be equivalent if there is an such that . In this paper we show that rational functions can be used to distinguish equivalence classes of subgroups and, for subgroups of prime rank or rank less than twelve, we give explicit finite sets of separating invariants. 相似文献
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Charles Almeida Aline V. Andrade Rosa M. Miró-Roig 《Journal of Pure and Applied Algebra》2019,223(4):1817-1831
Let be a minimal monomial Togliatti system of forms of degree d. In [4], Mezzetti and Miró-Roig proved that the minimal number of generators of lies in the interval . In this paper, we prove that for and , the integer values in cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. We classify minimal monomial Togliatti systems of forms of degree d with or 3n (i.e. with the minimal number of generators reaching the border of the non-existence interval). Finally, we prove that for , and there exists a minimal monomial Togliatti system of forms of degree d with . 相似文献