Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's paqb Theorem; Injective modules.

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.

Avian Use of Harvested Crop Fields in North Dakota During Spring Migration

Broad-spectrum herbicide applications and improved harvesting efficiency of crops have reduced the availability of weed seeds and waste grains for game and nongame wildlife. Over the last decade, corn and soybean plantings have steadily increased in the Prairie Pothole Region (PPR) of North Dakota, while sunflower plantings have declined. The PPR is an important corridor for migratory birds, and changes in food availabilities at stopover habitats may affect how food resources are used. In early spring 2003 and 2004, we compared bird use of harvested fields of sunflower, soybeans, small grains, and corn in the PPR of North Dakota. Across both years and all crop types, we observed 20,400 birds comprising 29 species. Flocks of Lapland Longspurs (Calcarius lapponicus) and Horned Larks (Eremophila alpestris) and flocks of Red-winged Blackbirds (Agelaius phoeniceus) made up 60% and 15%, respectively, of the bird counts. We found that species richness and bird densities were higher in harvested sunflower fields and cornfields than in harvested small-grain and soybean fields, with soybean fields harboring the fewest species and lowest bird density. Blackbird densities tended to be lower in fields tilled after fall harvest than in fields not tilled. These results suggest that some granivorous bird populations in the Northern Great Plains could be positively affected by planting of row crops with postharvest vertical structure (e.g., sunflower, corn) and use of no-till land management practices.