My research focuses on two sides of commutative algebra, the first of which sits under the umbrella of ideal theory and touches upon several areas of algebra and algebraic geometry. The specific questions I am investigating arise from intersection theory and concern the fundamental behavior of intersections of powers of prime ideals. The properties exhibited by these ideals have unexpected applications including a new lower bound for the orders of vanishing of functions on intersections of algebraic varieties and a dimension inequality that generalizes a celebrated theorem of Serre. The techniques I use to attack these questions come from several classical regions of commutative algebra, with the unifying theme being the theory of multiplicities. My work in this direction is in items 0-4 below.

The second aspect of my research is homological in nature, with a primary focus on understanding certain dualities that expand upon work of Auslander and Bridger. These dualities manifest themselves as homological dimensions that parallel the classical projective dimension. My current work in this area is centered upon extending and strengthening these parallels and determining the structure of the objects that give rise to these dualities. These ideas have yielded several surprising consequences, for instance, new homological characterizations of interesting classes of rings and ring homomorphisms. The techniques employed in this inquiry are predominantly homological, but my most recent work brings concrete methods from ideal theory to bear on problems in this area. My research on these topics is contained in items 5-11 below.

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10. Reflexivity and ring homomorphisms of finite flat dimension, with Anders Frankild, 38 pages. Communications in Algebra, to appear. This item can be downloaded from the arXiv.

9. Semidualizing modules and the divisor class group, 23 pages. This item can be downloaded from the arXiv.

8. The set of semidualizing complexes is a nontrivial metric space, with Anders Frankild, 25 pages. This item can be downloaded from the arXiv.

7. Characterizing local rings via homological dimensions and regular sequences, with Shokrollah Salarian and Siamak Yassemi. Journal of Pure and Applied Algebra, to appear. 9 pages. This item can be downloaded from the arXiv.

6. Complete intersection dimensions for complexes, Journal of Pure and Applied Algebra 190 (2004) no. 3, 267--290. This item can be downloaded from the arXiv.

5. G-dimension over local homomorphisms. Applications to the Frobenius endomorphism, with Srikanth Iyengar, Illinois Journal of Mathematics 48 (2004) no.1, 241--272. This item can be downloaded from the arXiv.

4. On symbolic powers of prime ideals, Commutative algebra. Interactions with Algebraic Geometry, Contemporary Math. 331, (2003), 329-342. This item can be downloaded from the arXiv.

3. Intersections of symbolic powers of prime ideals, Journal of the London Mathematical Society (2) 65 no. 3, (2002), 560--574. This item can be downloaded from the arXiv.

2. Multiplicities and a dimension inequality for unmixed modules, Journal of Algebra 238 (2001), 372-388, doi:10.1006/jabr.2000.8630. This item can be downloaded from the arXiv.

1. A dimension inequality for Cohen-Macaulay rings, Transactions of the American Mathematical Society 354 (2002), 993-1005. This item can be downloaded from the Transactions.

Dissertation: "A Dimension Inequality for Excellent, Cohen-Macaulay Rings Related to the Positivity of Serre's Intersection Multiplicity" .ps, .pdf, .dvi