Thomas Cassidy

PBW-deformation theory and regular central extensions

Abstract A deformation U, of a graded K-algebra A is said to be of PBW type if gr U is A. It has been shown for Koszul and N-Koszul algebras that the deformation is PBW if and only if the relations of U satisfy a Jacobi type condition. In particular, for these algebras the determination of the PBW property is a finite and explicitly determined linear algebra problem. We extend these results to an arbitrary graded K-algebra, using the notion of central extensions of algebras and a homological constant attached to A which we call the complexity of A.

NONCOMMUTATIVE KOSZUL ALGEBRAS FROM COMBINATORIAL TOPOLOGY

Abstract Associated to any uniform finite layered graph Γ there is a noncommutative graded quadratic algebra A(Γ) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW-complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups HX (n, k), generalizing the usual cohomology groups Hn (X). Along with several other results, our methods give a new and primarily topological proof of the main result of [Serconek and Wilson, J. Algebra 278: 473–493, 2004] and [Piontkovski, J. Alg. Comput. 15, 643–648, 2005].

Corrigendum. Generalizations of graded Clifford algebras and of complete intersections

A correction is provided for Proposition 3.5 in the article “Generalizations of Graded Clifford Algebras and of Complete Intersections”. The correction is: if S is a skew polynomial ring on finitely many generators of degree one that are normal elements in S, and if I is a homogeneous ideal of S that is generated by a normalizing sequence, then dimk(S/I) is finite if and only if S/I has no point modules and no fat point modules. A similar correction is provided for Corollary 3.6 of the same article. The proof of Proposition 3.5 in [4] contains an error, so that [4, Proposition 3.5 and Corollary 3.6] need to be modified (see Proposition 10 and Corollary 11 below). The authors would like to thank J. T. Stafford for alerting them to this issue, which occurs in the paragraph in [4] immediately preceding Proposition 3.5. The main result of [4], namely Theorem 4.2, is correct as stated, provided that the definitions of base point and base-point free are changed from those given in [4, Definition 1.7] to those given in Definition 2 below. The examples and other results in the remaining sections of [4] are unchanged. Additionally, the reader should note that the results in [7] are unchanged. We recall the notation of [4]: k denotes an algebraically closed field; k = k \ {0} and similarly for modules and other rings; Mc(k) is the ring of c× c matrices over k; μ = (μij) ∈ Mn(k ), where μijμji = 1 for all i, j; S = k〈z1, . . . , zn〉/〈U〉, where U = span{zjzi − μijzizj : 2010 Mathematics Subject Classification: . 16S38, 16S37, 16S36.

Homogenized Down-Up Algebras

Abstract This paper studies two homogenizations of the down-up algebras introduced in Benkart and Roby (Benkart, G., Roby, T. (1998). Down-up Algebras. J. Algebra 209:305–344). We show that in all cases the homogenizing variable is not a zero-divisor, and that when the parameter β is non-zero, the homogenized down-up algebra is a Noetherian domain and a maximal order, and also Artin-Schelter regular, Auslander regular, and Cohen-Macaulay. We show that all homogenized down-up algebras have global dimension 4 and Gelfand-Kirillov dimension 4, and with one exception all homogenized down-up algebras are prime rings. We also exhibit a basis for homogenized down-up algebras and provide a necessary condition for a Noetherian homogenized down-up algebra to be a Hopf algebra.

How slowing senescence translates into longer life expectancy

Mortality decline has historically been largely a result of reductions in the level of mortality at all ages. A number of leading researchers on ageing, however, suggest that the next revolution of longevity increase will be the result of slowing down the rate of ageing. In this paper, we show mathematically how varying the pace of senescence influences life expectancy. We provide a formula that holds for any baseline hazard function. Our result is analogous to Keyfitzs ‘entropy’ relationship for changing the level of mortality. Interestingly, the influence of the shape of the baseline schedule on the effect of senescence changes is the complement of that found for level changes. We also provide a generalized formulation that mixes level and slope effects. We illustrate the applicability of these models using recent mortality decline in Japan and the problem of period to cohort translation.

A Cohort Model of Fertility Postponement

We introduce a new formal model in which demographic behavior such as fertility is postponed by differing amounts depending only on cohort membership. The cohort-based model shows the effects of cohort shifts on period fertility measures and provides an accompanying tempo adjustment to determine the period fertility that would have occurred without postponement. Cohort-based postponement spans multiple periods and produces “fertility momentum,” with implications for future fertility rates. We illustrate several methods for model estimation and apply the model to fertility in several countries. We also compare the fit of period-based and cohort-based shift models to the recent Dutch fertility surface, showing how cohort- and period-based postponement can occur simultaneously.

Central Extensions of Stephenson's Algebras

Abstract This paper completes the classification of central extensions of three dimensional Artin-Schelter regular algebras to four dimensional Artin-Schelter regular algebras. Let A be an AS regular algebra of global dimension three and let D be an extension of A by a central graded element z, i.e., D/⟨z⟩ = A. If A is generated by elements of degree one, those algebras D which are again AS regular have been classified in Le Bruyn et al. (Le Bruyn L., Smith, S. P., Van den Bergh, M. (1996). Central extensions of three dimensional Artin-Schelter regular algebras. Math. Zeitschrift 222:171–212.) and Cassidy (Cassidy, T. (1999). Global dimension 4 extensions of Artin-Schelter regular algebras. J. Algebra 220:225–254.). If A is not generated by elements of degree one, then A falls under a classification due to Stephenson (Stephenson, D. R. (1996). Artin-Schelter regular algebras of global dimension three. J. Algebra 183(1):55–73 and Stephenson, D. R. (1997). Algebras associated to elliptic curves. Trans. Amer. Math. Soc. 349(6):2317–2340.). We classify the AS regular central extensions of Stephensons algebras by proving that the regularity of D and z is equivalent to the regularity of z in low degree and this is equivalent to easily verifiable conditions on the defining relations for D.

Modules with pure resolutions over Koszul algebras revisited

I construct a Koszul algebra A and a finitely generated graded A-module M that together form a counterexample to a recently published claim. M is generated in degree 0 and has a pure resolution, and the graded Jacobson radical of the Yoneda algebra of A does not annihilate the Ext module of M, but nonetheless M is not a Koszul module.

Quotients of Koszul Algebras and 2-d-Determined Algebras

Vatne [13] and Green and Marcos [9] have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green and Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.

The Yoneda Algebra of a 𝒦2 Algebra Need not be Another 𝒦2 Algebra

The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. 𝒦2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a 𝒦2 algebra would be another 𝒦2 algebra. We show that this is not necessarily the case by constructing a monomial 𝒦2 algebra for which the corresponding Yoneda algebra is not 𝒦2.