Homology of perfect complexes

It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective dimension, obstructions to realizing R as a closed fiber of some flat local homomorphism. Other applications include, as special cases, uniform proofs of known results on free actions of elementary abelian groups and of tori on finite CW complexes. The arguments use numerical invariants of objects in general triangulated categories, introduced here and called levels. They allow one to track, through changes of triangulated categories, homological invariants like projective dimension, as well as structural invariants like Loewy length. An intermediate result sharpens, with a new proof, the New Intersection Theorem for commutative algebras over fields. Under additional hypotheses on the ring $R$ stronger estimates are proved for Loewy lengths of modules of finite projective dimension.Comment: This version corrects an error in the statement (and proof) of Theorem 7.4 in the published version of the paper [Adv. Math. 223 (2010) 1731--1781]. These changes do not affect any other results or proofs in the paper. A corrigendum has been submitted

Shapes of free resolutions over a local ring

We classify the possible shapes of minimal free resolutions over a regular local ring. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. We also give an asymptotic characterization of the possible shapes of minimal free resolutions over hypersurface rings. Our key new technique uses asymptotic arguments to study formal Q-Betti sequences.Comment: 14 pages, 1 figure; v2: sections have been reorganized substantially and exposition has been streamline

GG-prime and GG-primary GG-ideals on GG-schemes

Let $G$ be a flat finite-type group scheme over a scheme $S$, and $X$ a noetherian $S$-scheme on which $G$-acts. We define and study $G$-prime and $G$-primary $G$-ideals on $X$ and study their basic properties. In particular, we prove the existence of minimal $G$-primary decomposition and the well-definedness of $G$-associated $G$-primes. We also prove a generalization of Matijevic-Roberts type theorem. In particular, we prove Matijevic-Roberts type theorem on graded rings for $F$-regular and $F$-rational properties.Comment: 54pages, added Example 6.16 and the reference [8]. The final versio

From Heun to Painlev\'e on Sasaki-Einstein Spaces and Their Confluent Limits

The aim of this paper is to study the effect of isomonodromic deformations of the evolution of scalar fields in Sasaki-Einstein spaces in the context of holography. Here we analyze the monodromy data of the general Heun equation, resulting from a scalar on Y$^{p,q}$, thus obtaining the corresponding Painlev\'e VI equation. Furthermore we have considered limits leading to a coalescence of singularities, which in turn transform the original Painlev\'e VI equation, to one of lower rank. The confluent limits we have considered are Y$^{p,p}$, T$^{1,1} / \mathbb{Z}_2$ and Y$^{\infty, q}$.Comment: 32 pages, three figure

Class and rank of differential modules

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a substitute for the length of a free complex--and on the rank of a differential module in terms of invariants of its homology. These results specialize to basic theorems in commutative algebra and algebraic topology. One instance is a common generalization of the equicharacteristic case of the New Intersection Theorem of Hochster, Peskine, P. Roberts, and Szpiro, concerning complexes over noetherian commutative rings, and of a theorem of G. Carlsson on differential graded modules over graded polynomial rings.Comment: 27 pages. Minor changes; mainly stylistic. To appear in Inventiones Mathematica

An equation for the description of volume and temperature dependences of the dynamics of supercooled liquids and polymer melts

A recently proposed expression to describe the temperature and volume dependences of the structural (or alpha) relaxation time is discussed. This equation satisfies the scaling law for the relaxation times, tau = f(TV^g), where T is temperature, V the specific volume, and g a material-dependent constant. The expression for the function f is shown to accurately fit experimental data for several glass-forming liquids and polymers over an extended range encompassing the dynamic crossover, providing a description of the dynamics with a minimal number of parameters. The results herein can be reconciled with previously found correlations of the isochoric fragility with both the isobaric fragility at atmospheric pressure and the scaling exponent g.Comment: to be published in the special edition of J. Non-Crystalline Solids honoring K.L. Nga

Brauer-Thrall for totally reflexive modules over local rings of higher dimension

Let $R$ be a commutative Noetherian local ring. Assume that $R$ has a pair $\{x,y\}$ of exact zerodivisors such that $\dim R/(x,y)\ge2$ and all totally reflexive $R/(x)$-modules are free. We show that the first and second Brauer--Thrall type theorems hold for the category of totally reflexive $R$-modules. More precisely, we prove that, for infinitely many integers $n$, there exists an indecomposable totally reflexive $R$-module of multiplicity $n$. Moreover, if the residue field of $R$ is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexive $R$-modules of multiplicity $n$.Comment: to appear in Algebras and Representation Theor

Elektrooksidacija smeše formaldehida i 2-propanola na monokristalima zlata orijentacije (100) i (111) u alkalnoj sredini

The effect of formaldehyde on the oxidation of 2-propanol and vice versa on gold single crystal planes (100 and 111) was studied. An activating effect in the reaction of the simultaneous oxidation of 2-propanol and formaldehyde was obtained on a gold (100) plane. In the case of a gold (111) electrode, the activation effect was not obtained. It was concluded that the adsorption of formaldehyde on the electrode surface prevents the adsorption of poisoning species formed during the electro-oxidation of 2-propanol on the Au(100) plane, while this is not the case on the Au(111) plane. The different behavior is caused by the difference in the symmetry of the surface atoms of these two Au single crystal planes. .Ispitivan je uticaj formaldehida na oksidaciju 2-propanola i vice versa na monokristalima zlata orijentacija (100) i (111). U reakciji simultane oksidacije 2-propanola i formaldehida na monokristalu zlata orijentacije (100) dobijen je aktivacioni efekat. Ovaj efekat nije dobijen na elektrodi od zlata orijentacije (111). Zaključeno je da prisustvo adsorbovanog formaldehida onemogućava adsorpciju čvrsto vezanih intermedijera formiranih tokom elektrooksidacije 2-propanola, koji se ponašaju kao otrovi za elektrodu, na ravni (100). Ovo nije slučaj na ravni (111). Različito ponašanje je uzrokovano razlikama u simetriji i energijama površinskih atoma na ove dve monokristalne ravni.

EPIDEMIOLOGICAL, CLINICA.L AND VIROLOGICAL STUDIES ON INFLUENZA MORBIDITY RATES FOR ТНЕ PERIOD APRIL 1962 – МАY 1963 IN ТНЕ СIТУ OF VARNA

Тhe epidemic process of influenza acquires vaгious forms in its couгse: pandemics, epidemics аnd sporadic morbidity inbetween epidemics. Wе dispose of mоrе precise data оn its course in Bulgaria since 1952. Мorе scanty аrе the data which characteгize influenza epidemics in the city of Vагnа up to 1955 which is seen on Diagr. 1. From the latter it becomes evident that influenza morbidity rates in the city of Vагnа run parallel to morbidity rates in the whole country, being occasionally morе intensive - 1959, 1962. Тlie diаgrаm reveals high level for influenza morbldity rates fог the countгy and the city of Vаrnа in 1956 and аn epidemic peak in 1957, 1959 аnd 1962. Тhe epidemic peak in 1962 is due to аn influenza outbreak caused bу virus А2 which started at the end of January and continued till Магсh, the same уеаr.After the epidemics а period was established, characteгized bу а low level of influenza morbldity rates in the city of Varna. Stress should bе laid оn the fact that determination of actual levels of influenza morbidity rates inbetween epidemics is particularly difficult owing to the imperfect diagnosis and the diversity of clinical forms of influenza