The model of economic man is either empirically testable and false, or it is non-testable and always true. The fact that neither position is entirely satisfactory is called Macaulay’s problem. This paper first reviews and criticizes various attitudes toward this problem and then argues that Macaulay’s problem is a pseudoproblem, because it assumes that the explicandum of the economic man model ...

We consider a commutative ring R graded by an arbitrary abelian group G, and define the grade of a G-homogeneous ideal I on R in terms of vanishing of C̆ech cohomology. By defining the dimension in terms of chains of homogeneous prime ideals and supposing R satisfies a.c.c. on G-homogeneous ideals and has a unique G-homogeneous maximal ideal, we can define graded versions of the depth of R and C...

Inspired by the representation theory of rational Cherednik algebras, we consider whether certain rings corresponding to hyperplane arrangements in C satisfy the Cohen-Macaulay property. Specifically, for a partition λ = (λ1, . . . , λr) ` n we define a certain variety Xλ ⊂ C and study its coordinate ring C[Xλ]. The question which we work towards is: for which λ is Xλ CohenMacaulay? We consider...

The arithmetic degree, the smallest extended degree, and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not Cohen-Macaulay. We compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule. This leads to various bounds and to counterexamples to a conjecture of ...

Let G be a finite simple graph on a vertex set V (G) = {x11, . . . , xn1}. Also let m1, . . . , ,mn ≥ 2 be integers and G1, . . . , Gn be connected simple graphs on the vertex sets V (Gi) = {xi1, . . . , ximi}. In this paper, we provide necessary and sufficient conditions on G1, . . . , Gn for which the graph obtained by attaching the Gi to G is unmixed or vertex decomposable. Then we character...

We classify the tangential varieties of Segre–Veronese which are Cohen–Macaulay or Gorenstein.

To reduce to resolving Cohen–Macaulay singularities, Faltings initiated the program of “Macaulayfying” a given Noetherian scheme X. For wide class X, Kawasaki built sought-for modifications, with crucial drawback that his blowups did not preserve locus CM(X)?X, where X is already Cohen–Macaulay. We extend Kawasaki’s methods show every quasi-excellent, has X˜ proper map X˜?X an isomorphism over ...

The purpose of this paper is to present a family of CohenMacaulay monomial ideals such that their integral closures have embedded components and hence are not Cohen-Macaulay.

The method of shifted partial derivatives introduced in [9, 7] was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this method alone cannot prove that the padded permanent l perm m cannot be realized inside the GLn2 -orbit closure of the determinant detn when n > 2m 2 + 2m. Our proof relies on several simple degenerat...