Vanishing of Ext and Tor over complete intersections

Let (R,m) be a local complete intersection, that is, a local ring whose m-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of Tor(M, N) and Ext(M, N). In this context, M satisfies Serre's condition (S_{n}) if and only if M is an nth syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r-1 for all sufficiently large n. We use this notion of Serre's condition and complexity to study the vanishing of Tor_{i}(M, N). In particular, building on results of C. Huneke, D. Jorgensen and R. Wiegand [32], and H. Dao [21], we obtain new results showing that good depth properties on the R-modules M, N and MtensorN force the vanishing of Tor_{i}(M, N) for all i>0. We give examples showing that our results are sharp. We also show that if R is a one-dimensional domain and M and MtensorHom(M,R) are torsion-free, then M is free if and only if M has complexity at most one. If R is a hypersurface and Ext^{i}(M, N) has finite length for all i>>0, then the Herbrand difference [18] is defined as length(Ext^{2n}(M, N))-(Ext^{2n-1}(M, N)) for some (equivalently, every) sufficiently large integer n. In joint work with Hailong Dao, we generalize and study the Herbrand difference. Using the Grothendieck group of finitely generated R-modules, we also examined the number of consecutive vanishing of Ext^{i}(M, N) needed to ensure that Ext^{i}(M, N) = 0 for all i>>0. Our results recover and improve on most of the known bounds in the literature, especially when R has dimension two.

Class Notes for Math 918: Local Cohomology, Instructor Tom Marley

Topics include: Injective Module, Basic Properties of Local Cohomology Modules, Local Cohomology as a Cech Complex, Long exact sequences on Local Cohomology, Arithmetic Rank, Change of Rings Principle, Local Cohomology as a direct limit of Ext modules, Local Duality, Chevelley’s Theorem, Hartshorne- Lichtenbaum Vanishing Theorem, Falting’s Theorem.

Thermal Imagery Applied to Reducing Bird Hazards to Aircraft at Airports

Airports worldwide are at a disadvantage when it comes to being able to spot birds and warn aircrews about the location of flocks either on the ground or close to the airfield. Birds simply cannot be easily seen during the day and are nearly invisible targets for planes at night or during low visibility. Thermal imaging (infrared) devices can be used to allow ground and tower personnel to pinpoint bird locations day or night, thus giving the airport operators the ability to launch countermeasures or simply warn the aircrews. This technology is available now, though it has been predominately isolated to medical and military system modifications. The cost of these devices has dropped significantly in recent years as technology, capability, and availability have continued to increase. Davison Army Airfield (DAAF), which is located about 20 miles south of Ronald Reagan National Airport in Washington, DC, is the transient home to many bird species including an abundance of ducks, seagulls, pigeons, and migrating Canadian geese. Over the past few years, DAAF implemented a variety of measures in an attempt to control the bird hazards on the airfield. Unfortunately, when it came to controlling these birds on or near our runways and aircraft movement areas we were more reactive than proactive. We would do airfield checks several times an hour to detect and deter any birds in these areas. The deterrents used included vehicle/human presence, pyrotechnics, and the periodic use of a trained border collie. At the time, we felt like we were doing all we could to reduce the threat to aircraft and human life. It was not until a near fatal accident in October 1998, when we truly realized how dangerous our operating environment really was to aircraft at or near the airfield. It was at this time, we had a C-12 (twin-engine passenger plane) land on our primary runway at night. The tower cleared the aircraft to land, and upon touchdown to the runway the aircraft collided with a flock of geese. Neither the tower nor the crew of the aircraft saw the geese because they were obscured in the darkness. The end result was 12 dead geese and $374,000 damage to the C-12. Fortunately, there were no human fatalities, but it was painfully clear we needed to improve our method of clearing the runway at night and during low visibility conditions. It was through this realization that we ventured to the U.S. Army Communications and Electronics Command for ideas on ways to deal with our threat. It was through a sub-organization within this command, Night Vision Labs, that we realized the possibilities of modifying thermal imagery and infrared technology to detecting wildlife on airports.