David A. Jorgensen

A generalization of the Auslander–Buchsbaum formula

Let R be a local ring, and let M and N be nonzero nitely generated R-modules. In this paper we demonstrate the equality

Tate (co)homology via pinched complexes

For complexes of modules we study two new constructions, which we call the pinched tensor product and the pinched Hom. They provide new methods for computing Tate homology and Tate cohomology, which lead to conceptual proofs of balancedness of Tate (co)homology for modules over associative rings. Another application we consider is in local algebra. Under conditions of vanishing of Tate (co)homology, the pinched tensor product of two minimal complete resolutions yields a minimal complete resolution.

Oscillations in a model of repression with external control.

A mathematical model for control by repression by an extracellular substance is developed, including diffusion and time delays. The model examines how active transport of a nutrient can produce either oscillatory or stable responses depending on a variety of parameters, such as diffusivity, cell size, or nutrient concentration. The system of equations for the mathematical model is reduced to a system of delay differential equations and linear Volterra equations. After linearizing these equations and forming the limiting Volterra equations, the resulting linear system no longer has any spatial dependence. Local stability analysis of the radially symmetric model shows that the system of equations can undergo Hopf bifurcations for certain parameter values, while other ranges of the parameters guarantee asymptotic stability. One numerical study shows that the model can exhibit intracellular biochemical oscillations with increasing extracellular concentrations of the nutrient, which suggests a possible trigger mechanism for morphogenesis.

Asymmetric complete resolutions and vanishing of ext overGorenstein rings

We construct a class of Gorenstein local rings

VANISHING OF (CO)HOMOLOGY OVER COMMUTATIVE RINGS

R

Finite projective dimension and the vanishing of extR(M, M)

which admit minimal complete

Tate-Hochschild homology and cohomology of Frobenius algebras

R

Existence of unliftable modules

-free resolutions

On growth in minimal totally acyclic complexes

\bd C

The negative side of cohomology for Calabi-Yau categories

such that the sequence