Research Interests of Sean Sather-Wagstaff

The first aspect of my research is in 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 behavior of intersections of symbolic 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 theorem of Serre. The techniques I use come from several classical regions of commutative algebra, mainly from the theory of multiplicities. My work in this direction is in items 1-4 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, for instance, new homological characterizations of interesting classes of rings and ring homomorphisms. The techniques employed in this inquiry are predominantly homological, but some of my work brings concrete methods from ideal theory and module theory to bear on problems in this area. My research on these topics is contained in items 5-23 below.

1. Multiplicities and a dimension inequality for unmixed modules , Journal of Algebra 238 (2001), 372-388.
2. A dimension inequality for Cohen-Macaulay rings , Transactions of the American Mathematical Society 354 (2002), 993-1005.
3. Intersections of symbolic powers of prime ideals , Journal of the London Mathematical Society (2) 65 no. 3, (2002), 560--574.
4. On symbolic powers of prime ideals , Commutative algebra. Interactions with Algebraic Geometry, Contemporary Math. 331, (2003), 329-342.
5. G-dimension over local homomorphisms. Applications to the Frobenius endomorphism , with Srikanth Iyengar, Illinois Journal of Mathematics 48 (2004) no.1, 241--272.
6. Complete intersection dimensions for complexes , Journal of Pure and Applied Algebra 190 (2004) no. 3, 267--290.
7. Characterizing local rings via homological dimensions and regular sequences , with Shokrollah Salarian and Siamak Yassemi. Journal of Pure and Applied Algebra, 207 (2006) no.1, 99--108.
8. The set of semidualizing complexes is a nontrivial metric space , with Anders Frankild, Journal of Algebra 308 (2007) no. 1, 124-143.
9. Reflexivity and ring homomorphisms of finite flat dimension , with Anders Frankild. Communications in Algebra, 35 (2007) no. 2, 461--500. arXiv
10. Semidualizing modules and the divisor class group . Illinois Journal of Mathematics, 51 (2007) no. 1, 255-285. offprint
11. Detecting completeness from Ext-vanishing , with Anders J. Frankild, 10 pages. Proceedings of the American Mathematical Society, 136 (2008), no. 7, 2303-2312. arXiv version.
12. Stability of Gorenstein categories , with Tirdad Sharif and Diana White, 22 pages. Journal of the London Mathematical Society 72 (2008), no. 2, 481-502. arXiv version.
13. An Euler characteristic for modules of finite G-dimension , with Diana White, Mathematica Scandinavica, 102 (2008), no. 2, 206-230.
14. Ascent of module structures, vanishing of Ext, and extended modules , with Anders J. Frankild and Roger A. Wiegand, 16 pages. Michigan Mathematical Journal, to appear.
15. A Cohen-Macaulay algebra has only finitely many semidualizing modules , with Lars Winther Christensen, 3 pages. Mathematical Proceedings of the Cambridge Philosophical Society, to appear.
16. Relations between semidualizing complexes , with Anders J. Frankild and Amelia Taylor, 28 pages. Journal of Commutative Algebra, to appear.
17. Complete intersection dimensions and Foxby classes , 24 pages. Journal of Pure and Applied Algebra, to appear.
18. Descent via Koszul extensions , with Lars Winther Christensen, 18 pages. Journal of Algebra, to appear.
19. Gorenstein cohomology in abelian categories , with Tirdad Sharif and Diana White, 22 pages. Journal of Mathematics of Kyoto University, to appear.
20. Comparison of relative cohomology theories with respect to semidualizing modules , with Tirdad Sharif and Diana White, 28 pages. Submitted.
21. On the number of semidualizing complexes , 5 pages. Submitted.
22. AB-contexts and stability for Gorenstein flat modules with respect to semidualizing modules, with Tirdad Sharif and Diana White, 22 pages. Submitted.
23. Transfer of Gorenstein dimensions along ring homomorphisms , with Lars Winther Christensen, 8 pages. preprint.