We relate the variance theory for Cousin complexes − developed by Lipman, Nayak and the author to Grothendieck duality for Cousin complexes. Specifically for a Cousin complex F on (Y, ∆)—with ∆ a codimension function on a formal scheme Y (noetherian, universally catenary)—and a pseudo-finite type map f : (X, ∆) → (Y, ∆) of such pairs of schemes with codimension functions, we show there is a der...

We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire’s splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bi...

Many years ago, in [2], a family of complexes was introduced which were to be considered generalizations of the usual Koszul complex, and in [3] they were studied in some detail, especially in relation to the generalized Cohen-Macaulay Theorem and a generalized multiplicity notion (see [8] and [9]). Given a commutative ring R, the idea was to take a map f : R → R (m ≥ n) and, for each integer k...

This research began as an effort to determine exactly which one-dimensional local rings have indecomposable finitely generated modules of arbitrarily large constant rank. The approach, which uses a new construction of indecomposable modules via the bimodule structure on certain Ext groups, turned out to be effective mainly for hypersurface singularities. The argument was eventually replaced by ...

For a simplicial complex on {1, 2, . . . , n} we define enriched homology and cohomology modules. They are graded modules over k[x1, . . . , xn] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein∗ complexes , and also orientable homology manifolds in terms of the enriched modules. We ...

This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its minimal resolution is acyclic. The results established track the change in this invariant under some standard operations in commutative algebra. As one of the ap...

In the work is constructed new acyclic resolutions of quasicoherent sheaves. These resolutions is connected with multidimensional local fields. Then the obtained resolutions is applied for a construction of generalization of the Krichever map to algebraic varieties of any dimension. This map gives in the canonical way two k -subspaces B ⊂ k((z1)) . . . ((zn)) and W ⊂ k((z1)) . . . ((zn)) ⊕r fro...

This paper examines the Macaulay, Hicks, and continuous time formulations of duration by analyzing the term structure dynamics that are necessary to derive these mathematical models. The results presented help to explain the true effects of interest rate risk upon the values of fixed-income securities, as well as provide a rationale for the development of duration models in a general equilibriu...

The main theorem says that any module-finite (but not necessarily commutative) algebra Λ over a commutative Noetherian universally catenary ring R is catenary. Hence the ring Λ is catenary if R is Cohen-Macaulay. When R is local and Λ is a Cohen-Macaulay R-module, we have that Λ is a catenary ring, dim Λ = dimΛ/Q + htΛQ for any Q ∈ Spec Λ, and the equality n = htΛQ− htΛP holds true for any pair...