There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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.

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Combinatorial Commutative Algebra

Introduction Polytopes Topology and combinatorics of simplicial complexes Commutative and homological algebra of simplicial complexes Cubical complexes Toric and quasitoric manifolds Moment-angle complexes Cohomology of moment-angle complexes and combinatorics of triangulated manifolds Cohomology rings of subspace arrangement complements Bibliography Index.

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Torus Actions and their Applications in Topology and Combinatorics

We study finitely generated modules has finite projective dimension. Comparison morphisms and link these functors. We give a self-contained treatment of modules of finite G-dimension, establish basic properties of relative and Tate cohomology, and embed the comparison morphisms into a canonical long exact sequence . We show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings. 2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.

Absolute, relative, and tate cohomology of modules of finite gorenstein dimension

Preface: Algebra and Geometry * Free Resolutions and Hilbert Functions * First Examples of Free Resolutions * Points in P^2 * Castelnuovo-Mumford Regularity * The Regularity of Projective Curves * Linear Series and 1-Generic Matrices * Linear Complexes and the Linear Syzygy Theorem * Curves of High Degree * Clifford Index and Canonical Embedding * Appendix 1: Introduction to Local Cohomology * Appendix 2: A Jog Through Commutative Algebra * References * Index

The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry

A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CI-dimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CI-dimension, providing the first class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions.

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Complete intersection dimension

Call a monomial ideal M "generic" if no variable appears with the same nonzero exponent in two distinct monomial generators Using a convex polytope first studied by Scarf, we obtain a minimal free resolution of M Any monomial ideal M can be made generic by deformation of its generating exponents Thus, the above construction yields a (usually nonminimal) resolution of M for arbitrary monomial ideals, bounding the Betti numbers of M in terms of the Upper Bound Theorem for Convex Polytopes We show that our resolutions are DG-algebras, and consider realizability questions and irreducible decompositions

Monomial Resolutions

Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were: • constructing the minimal free resolutions of special monomial ideals, cf., [AHH, BPS] • constructing non-minimal free resolutions; for example, Taylor’s resolution (cf., [Ei, p. 439]) and the cellular resolutions • the Stanley-Reisner theory for computing the Betti numbers of I by simplicial complexes; it has a long tradition and has led to important results in combinatorics and commutative algebra [St]. Working with special monomial ideals or with non-minimal resolutions simplifies the problem significantly because it removes the main difficulty (finding minimal generators of the homology in general). In this paper we obtain results on the minimal free resolution of an arbitrary monomial ideal. We introduce a new approach inspired by the topological theory of subspace arrangements. Some of the best results in this theory show that the cohomology algebra of the complement of a complex subspace arrangement is independent of the geometric position of the subspaces and is determined by the structure of a certain lattice. Inspired by this, we introduce the lcm-lattice of a monomial ideal and show how its structure relates to the Betti numbers, the maps in the minimal free resolution, and the structure of the Tor-algebra for the ideal. This is outlined more precisely below: The intersection lattice L of a complex subspace arrangement plays a significant role in describing the cohomology of the complementM of the arrangement: • the Goresky-MacPherson Formula [GM, III.1.5. Theorem A] expresses the dimensions of the cohomology groups ofM in terms of the dimensions of the homology groups of L. • L together with the additional data of the codimensions of the intersection spaces determine the algebra structure of the rational cohomology ofM [DP, Yu]. (Throughout we use the expression “the lattice of A determines the structure of X(A)” in the sense that objects A and B with isomorphic lattices

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1999/0006/0005/MRL-1999-0006-0005-a005.pdf

The LCM-lattice in monomial resolutions

The study of semigroup algebras has a long tradition in commutative algebra. Presentation ideals of semigroup algebras are called toric ideals, in reference to their prominent role in geometry. In this paper we consider the more general class of lattice ideals. Fix a polynomial ring S = k[x1, . . . , xn] over a field k and identify monomials x in S with vectors a ∈ N. Let L be any sublattice of Z. Then its associated lattice ideal in S is

Syzygies of codimension 2 lattice ideals

Let S = k[x1, . . . , xn] be a polynomial ring over a field k and I a homogeneous ideal in S. A basic problem in commutative algebra is to construct the minimal free resolution FI of S/I over S. The resolution is nicely structured and simple when I is a complete intersection: in this case FI is the Koszul complex. Complete intersections are ideals whose generators have sufficiently general coefficients, so they might be regarded as generic among all ideals. Yet there is another, entirely different, notion of genericity: ideals whose generators are generic with respect to their exponents – not their coefficients. This point of view was developed for monomial ideals in [BPS]. In the present work we introduce a notion of genericity for lattice ideals, which include ideals defining toric varieties. If L is any sublattice of Z, then its associated lattice ideal in S is IL := 〈xa − x : a,b ∈ N and a− b ∈ L 〉, where monomials are denoted x = x1 1 · · ·xan n for a = (a1, . . . , an). We call a lattice ideal IL generic if it is generated by binomials with full support, i.e.,

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Irena Peeva

Papers

Complete intersection dimension

Monomial Resolutions

The LCM-lattice in monomial resolutions

Syzygies of codimension 2 lattice ideals

Generic lattice ideals