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.

Theoretical and Experimental Studies in Nuclear Magnetic Resonance

Nuclear magnetic resonance (NMR) is a tool used to probe the physical and chemical environments of specific atoms in molecules. This research explored small molecule analogues to biological materials to determine NMR parameters using ab initio computations, comparing the results with solid-state NMR measurements. Models, such as dimethyl phosphate (DMP) for oligonucleotides or CuCl for the active site of the protein azurin, represented computationally unwieldy macromolecules. 31P chemical shielding tensors were calculated for DMP as a function of torsion angles, as well as for the phosphate salts, ammonium dihydrogen phosphate (ADHP), diammonium hydrogen phosphate, and magnesium dihydrogen phosphate. The computational DMP work indicated a problem with the current standard 31P reference of 85% H3PO4(aq.). Comparison of the calculations and experimental spectra for the phosphate salts indicated ADHP might be a preferable alternative as a solid state NMR reference for 31P. Experimental work included magic angle spinning experiments on powder samples using the UNL chemistry department’s Bruker Avance 600 MHz NMR to collect data to determine chemical shielding anisotropies. For the quadrupolar nuclei of copper and scandium, the electric field gradient was calculated in diatomic univalent metal halides, allowing determination of the minimal level of theory necessary to compute NMR parameters for these nuclei.

Online Dictionary of Invertebrate Zoology: T

tabula tabular tachyauxesis tachyblastic tachygen tachygenesis tachytelic tactic tactile tactoreceptors taenia taeniate taenidium taenioglossate tagma tagmata tagmosis tail tailfan Takakura's talon talus tandem tangent tangoreceptor tanylobous tapetal tapetum tapinoma-odor Tardigrada tardigrades tarsal tarsation tarsite tarsomere tarsungulus tarsus taste tautonomy tautonym taxa taxes taxis taxis taxodont taxometrics taxon taxonomic taxonomist taxonomy tectiform tectostracum tectum teeth teges tegillum tegmen tegmentum tegula tegular tegulum tegumen tegument tegumentary tela telaform telamon telegonic teleiochrysalis telenchium teleoconch teleodont teleology teleotrocha telepod telescope telescopic teletrophic telioderma teliophan telmophage telocentric telodendria telofemur telogonic telolecithal telomitic telophase telophragma telopod telopodite telorhabdions telosonic telostome telosynapsis telosyndesis telotarsus telotaxis telotroch telson template temporal tenacipeds tenaculum tenent teneral tensor tentacle tentacular tentaculocyst tentaculozooid tentilla tentorial tentorium tenuous teratocyte teratogen teratogenesis teratogyne teratology terebella terebra terebrant terebrate teres terete terga tergal tergite tergolateral tergopleural tergopore tergum tergum termen terminal terminalia termitarium termitophile terranes terrestrial terricolous territory tertiary tertibrach tertibrachial tessellate test testaceology testaceous test-cross testes testis testisac testudinate tetanus tetany tetractinal tetractine tetrad tetradelphic tetramerous tetramorphic tetraploid tetrapod tetrapterous tetrasomic tetrathyridial tetrathyridium tetraxon tetraxonid thalassophilous thallus thamnophilous thanatocoenosis thanatosis theca thecae thecal thecate thelycum thelygenesis thelygenous thelyotokous thelyotoky theory thermocline thermophile thermophobe thermoreceptor thermotaxis thickness thigmotactic thigmotaxis thigmotropism third-form thoraces thoracic thoracomere thoracopod(ite) thorax thoraxes thread thylacium thylacogen thyridial thyridium thyroid thysanuriform tibia tibial tibiotarsal tibiotarsus Tiedemann's tiled timbal tinctorial tine tissue tissue titilla titillae titillator tocopherol tocospermal tocospermia tocostome tokostome tomentose tomentum Tomosvary tone tonic tonofibrillae tonus topochemical topogamodeme topomorph topomorphic toponym topotype tori torma tormogen tornote tornus torose torpid torqueate torsion tortuose torulose torus totipotent totomount toxa toxicognath toxicology toxin toxinosis toxoglossate toxoid trabecula trabeculate trabeculated trachea tracheae tracheal tracheate tracheoblast tracheolar tracheoles trachychromatic tract Tragardh's tragus transad transcoxa transcurrent transect transection transformation transient transitional translocation translucent transmission transposed transscutal transstadial transtilla transverse trapeziform trapezium trapezoid trema tremata Trematoda trenchant trepan triact triactinal triad triaene triage triangle triangular triangulate triaulic triaxial triaxon tribe tribocytic trichite trichobothrium trichobranchia trichobranchiate trichocerous trichodes trichodeum trichodragmata trichogen trichoid trichomes trichophore trichopore trichosors trichostichal trichotomous trichroism tricolumella tricomes tricostate tricrepid tricuspid tricuspidate tridactyl trident tridentate trifid trifurcate triglycerides trignathan trigonal trigoneutism trilabiate trilateral trilobate trilocular trimorphic trimorphism Trinominal triordinal tripartite tripectinate triplet triploblastic triploid triquetral triquetrous triradiate triradiates tritocerebral tritocerebrum tritocerebrum tritonymph tritosternum triturate triungulin triungulinid trivial trivium trivoltine trixenic troch trochal trochalopodous trochantellus trochanter trochanteral trochantin trochi trochiform trochlea trocholophous trochophore trochosphere trochus troglobiont troglodytic troglophile trogloxene tropeic trophal trophallactic trophallaxis trophamnion trophi trophic trophidium trophobiont trophobiont trophobiosis trophobiotic trophocytes trophodisc trophogeny trophoporic trophorhinium trophosome trophotaxis trophothylax trophozooid trophus tropis tropism tropotaxis trumpet truncate truncation trunk trypsin tryptic tryptophan tryptophane T-tubule tube tube-feet tubercle tubercula tuberculate tuberculose tuberiferous tubicolous tubifacient tubule tubulus tubus tuft Tullgren tumefaction tumescence tumid tumulus tunic tunica tunicary tunicate turbinate turgid turreted turriculate tychoparthenogenesis tylasters tylenchoid tyli tyloid tyloides tylosis tylostyle tylote tylus tymbal tympanal tympanal tympanic tympanum Tyndall type typhlosole typologist typolysis typostasis