We show that the algebraic invariants multiplicity and depth of a graded ideal in the polynomial ring are closely connected to the fan structure of its generic tropical variety in the constant coefficient case. Generically the multiplicity of the ideal is shown to correspond directly to a natural definition of multiplicity of cones of tropical varieties. Moreover, we can recover information on ...

Let (R,m) be a commutative Noetherian local ring. Suppose that M and N are finitely generated modules over R such that M has finite projective dimension and such that TorRi (M,N) = 0 for all i > 0. The main result of this note gives a condition on M which is necessary and sufficient for the tensor product of M and N to be a Cohen–Macaulay module over R, provided N is itself a Cohen–Macaulay mod...

A certain squarefree monomial ideal HP arising from a finite partially ordered set P will be studied from viewpoints of both commutative algbera and combinatorics. First, it is proved that the defining ideal of the Rees algebra of HP possesses a quadratic Gröbner basis. Thus in particular all powers of HP have linear resolutions. Second, the minimal free graded resolution of HP will be construc...

A certain squarefree monomial ideal HP arising from a finite partially ordered set P will be studied from viewpoints of both commutative algebra and combinatorics. First, it is proved that the defining ideal of the Rees algebra of HP possesses a quadratic Gröbner basis. Thus in particular all powers of HP have linear resolutions. Second, the minimal free graded resolution of HP will be construc...

The goal of this note is to study the equations of a curve C ⊂ P3 = PK that is contained in some quadric Q. By a curve, we mean a pure one-dimensional locally Cohen-Macaulay subscheme (i.e. without zero-dimensional components.) We assume that the field K is algebraically closed. If C ⊂ Q is arithmetically Cohen-Macaulay, then, by Dubreil’s Theorem, C is defined by at most 3 equations. The conve...

Let X be a doubling of a smooth surface F in a smooth threefold and let C ⊂ X be a locally Cohen-Macaulay curve. Then C gives rise to two effective divisors on F , namely the curve part P of C ∩ F and the curve R residual to C ∩ F in C. We show that a general deformation of R on F lifts to a deformation of C on X when a certain cohomology group vanishes and give applications to the study of Hil...

Given a commutative noetherian non-positive DG-ring with bounded cohomology which has dualizing DG-module, we study its regular, Gorenstein and Cohen-Macaulay loci. We give sufficient condition for the regular locus to be open, show that is always open. However, both of these loci are often empty: no matter how nice H0(A) is, there examples where A empty. then DG-module contains dense open set....

A few years ago, Huneke and Leuschke proved a theorem which solved a conjecture of Schreyer. It asserts that an excellent Cohen-Macaulay local ring of countable Cohen-Macaulay type which is complete or has uncountable residue field has at most a one-dimensional singular locus. In this paper, it is verified that the assumption of the excellent property can be removed, and the theorem is consider...

These are lecture notes, in progress, on monomial ideals. The point of view is that monomial ideals are best understood by drawing them and looking at their corners, and that a combinatorial duality satisfied by these corners, Alexander duality, is key to understanding the more algebraic duality theories at play in algebraic geometry and commutative algebra. Sections written so far cover Alexan...

Conjecture, and the big and small Cohen Macaulay Conjectures. Many of these are shown to imply others. This document contains notes for a course taught by Tom Marley during the 2009 spring semester at the University of Nebraska-Lincoln. The notes loosely follow the treatment given in Chapters 8 and 9 of Cohen-Macaulay Rings, by W. Bruns and J. Herzog, although many other sources, including arti...