The regularity of Tor and graded Betti Numbers
01 Jan 2006-American Journal of Mathematics (Johns Hopkins University Press)-Vol. 128, Iss: 3, pp 573-605
TL;DR: In this paper, the authors give bounds for the regularity of the local cohomology of Tork (A, B) in terms of the graded Betti numbers of A and B, under the assumption that dim Tor1 ≤ 1, and apply the results to syzygies, Grobner bases, products and powers of ideals.
Abstract: Let S = K(x1, ... , xn), let A, B be finitely generated graded S-modules, and let m = (x1, ... , xn) ⊂ S. We give bounds for the regularity of the local cohomology of Tork (A, B) in terms of the graded Betti numbers of A and B, under the assumption that dim Tor1 (A, B) ≤ 1. We apply the results to syzygies, Grobner bases, products and powers of ideals, and to the relationship of the Rees and symmetric algebras. For example we show that any homogeneous linearly presented m-primary ideal has some power equal to a power of m; and if the first � (n − 1)/2� steps of the resolution of I are linear, then I 2 is a power of m.
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TL;DR: In this paper, the authors studied the problem of containment of symbolic powers in a polynomial ring over an algebraically closed field, and showed that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.
Abstract: We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\subseteq I^r$. As corollaries, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in ${\bf P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.
233 citations
TL;DR: In this article, it was shown that if X ⊂ P r is a closed scheme in projective space whose homogeneous ideal is generated by quadrics, then a zero-dimensional or one-dimensional intersection of X with a plane of dimension p is 2-regular.
Abstract: Let X ⊂ P r be a closed scheme in projective space whose homogeneous ideal is generated by quadrics. We say that X (or its ideal IX) satisfies the condition N2,p if the syzygies of IX are linear for p steps. We show that if X satisfies N2,p then a zero-dimensional or one-dimensional intersection of X with a plane of dimension p is 2-regular. This extends a result of Green and Lazarsfeld. We give conditions when the syzygies of X restrict to the syzygies of the intersection. Many of our results also work for ideals generated by forms of higher degree. As applications, we bound the p for which some well-known projective varieties satisfy N2,p. Another application, carried out by us in a different paper, is a step
166 citations
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TL;DR: In this article, the authors studied the problem of containment of symbolic powers in a polynomial ring over an algebraically closed field, and showed that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.
Abstract: We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\subseteq I^r$. As corollaries, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in ${\bf P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.
142 citations
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TL;DR: In this paper, the authors derived geometric consequences from the presence of a long strand of linear syzygies in the minimal free resolution of a closed scheme in projective space whose homogeneous ideal is generated by quadrics.
Abstract: In this paper we derive geometric consequences from the presence of a long strand of linear syzygies in the minimal free resolution of a closed scheme in projective space whose homogeneous ideal is generated by quadrics. These consequences are given in terms of intersections with arbitrary linear subspaces. We use our results to bound homological invariants of some well-known projective varieties, to give a combinatorial characterization of quadratic monomial ideals with a long strand of linear syzygies, etc
110 citations
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References
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Book•
30 Mar 1995
TL;DR: In this article, the authors define basic constructions and dimension theory, and apply them to the problem of homological methods for combinatorial problem solving in the context of homology.
Abstract: Introduction.- Elementary Definitions.- I Basic Constructions.- II Dimension Theory.- III Homological Methods.- Appendices.- Hints and Solutions for Selected Exercises.- References.- Index of Notation.- Index.
5,674 citations
01 Jan 1985
TL;DR: This chapter discusses Brill-Noether theory on a moving curve, and some applications of that theory in elementary deformation theory and in tautological classes.
Abstract: Preface.- Guide to the Reader.- Chapter IX. The Hilbert Scheme.- Chapter X. Nodal curves.- Chapter XI. Elementary deformation theory and some applications.- Chapter XII. The moduli space of stable curves.- Chapter XIII. Line bundles on moduli.- Chapter XIV. The projectivity of the moduli space of stable curves.- Chapter XV. The Teichmuller point of view.- Chapter XVI. Smooth Galois covers of moduli spaces.- Chapter XVII. Cycles on the moduli spaces of stable curves.- Chapter XVIII. Cellular decomposition of moduli spaces.- Chapter XIX. First consequences of the cellular decomposition .- Chapter XX. Intersection theory of tautological classes.- Chapter XXI. Brill-Noether theory on a moving curve.- Bibliography.- Index.
2,597 citations
Book•
01 Jan 1970
TL;DR: The Grothendieck-lefschetz theorems of algebraic geometry and analytic geometry have been studied in this article, where they are extended to higher codimensions.
Abstract: Ample divisors.- Affine open subsets.- Generalization to higher codimensions.- The grothendieck-lefschetz theorems.- Formal-rational functions along a subvariety.- Algebraic geometry and analytic geometry.
803 citations
Book•
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TL;DR: In this article, the authors present a set of conditions générales d'utilisation of commercial or impression systématique, i.e., the copie ou impression de ce fichier doit contenir la présente mention de copyright.
Abstract: © Publications mathématiques de l’I.H.É.S., 1966, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
317 citations
TL;DR: In this paper, the asymptotic behavior of the Castelnuovo norm and the Mumford norm of the integral closure of the powers of a homogeneous ideal I is studied.
Abstract: In this paper the asymptotic behavior of the Castelnuovo$ndash;Mumford regularity of powers of a homogeneous ideal I is studied. It is shown that there is a linear bound for the regularity of the powers I whose slope is the maximum degree of a homogeneous generator of I, and that the regularity of I is a linear function for large n. Similar results hold for the integral closures of the powers of I. On the other hand we give examples of ideal for which the regularity of the saturated powers is asymptotically not a linear function, not even a linear function with periodic coefficients.
282 citations