Author
Jürgen Herzog
Other affiliations: Bulgarian Academy of Sciences, Purdue University, University of Regensburg ...read more
Bio: Jürgen Herzog is an academic researcher from University of Duisburg-Essen. The author has contributed to research in topics: Monomial & Monomial ideal. The author has an hindex of 54, co-authored 305 publications receiving 9453 citations. Previous affiliations of Jürgen Herzog include Bulgarian Academy of Sciences & Purdue University.
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TL;DR: In this paper, the authors studied the relations of finitely generated abelian semigroups and showed that the number of relations defining a semigroup is greater than or equal to the least number of generators of S minus the rank of the associated group of S.
Abstract: The object of this paper is the study of the relations of finitely generated abelian semigroups. We give a new proof of the fact that each such semigroup S is finitely presented. Moreover, we show that the number of relations defining S is greater than or equal to the least number of generators of S minus the rank of the associated group of S. If equality holds, we say S is a complete intersection. The main part of this study is devoted to semigroups of natural numbers generated by 3 elements. These semigroups are complete intersections if and only if they are symmetric in the sense of R. Apery [1]. This result applies to algebraic geometry: An affine space-curve C with the parametric equations x=ta, y=tb, z=tc, a, b, c natural numbers with greatest common divisor 1, is a global idealtheoretic complete intersection, if and only if the semigroup S generated by a, b, c is symmetric.
494 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
TL;DR: It follows that all binomial edge ideals are radical ideals, and the results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones.
244 citations
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Book•
01 Jan 2004
TL;DR: In this paper, the authors present a set of monomial ideals for three-dimensional staircases and cellular resolutions, including two-dimensional lattice ideals, and a threedimensional staircase with cellular resolutions.
Abstract: Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.
1,476 citations
Book•
01 Mar 1998TL;DR: In this article, a detailed algebraic introduction to Grothendieck's local cohomology theory is provided, with many illustrations of applications of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties.
Abstract: This book provides a careful and detailed algebraic introduction to Grothendieck’s local cohomology theory, and provides many illustrations of applications of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Castelnuovo–Mumford regularity, the Fulton–Hansen connectedness theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. It is designed for graduate students who have some experience of basic commutative algebra and homological algebra, and also for experts in commutative algebra and algebraic geometry.
1,104 citations
TL;DR: In this article, the additive group of R has a direct-sum decomposition R = R + R, +..., where RiRi C R,+j and 1 E R,.
948 citations
Book•
01 Jan 2006
TL;DR: In this paper, the authors define the integral closure of rings and define a table of basic properties including separation, separationability, separation of rings, and normal homomorphisms, and the Briancon-Skoda theorem.
Abstract: Table of basic properties Notation and basic definitions Preface 1. What is the integral closure 2. Integral closure of rings 3. Separability 4. Noetherian rings 5. Rees algebras 6. Valuations 7. Derivations 8. Reductions 9. Analytically unramified rings 10. Rees valuations 11. Multiplicity and integral closure 12. The conductor 13. The Briancon-Skoda theorem 14. Two-dimensional regular local rings 15. Computing the integral closure 16. Integral dependence of modules 17. Joint reductions 18. Adjoints of ideals 19. Normal homomorphisms Appendix A. Some background material Appendix B. Height and dimension formulas References Index.
826 citations
TL;DR: Khovanov et al. as mentioned in this paper constructed a doubly-graded homology theory of links with the Euler characteristic, which is based on matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.
Abstract: Author(s): Khovanov, Mikhail; Rozansky, Lev | Abstract: For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.
715 citations