Polynomial identities on superalgebras: Classifying linear growth

We classify, up to PI-equivalence, the superalgebras over a field of characteristic zero whose sequence of codimensions is linearly bounded. As a consequence we determine the linear functions describing the graded codimensions of a superalgebra.     

Algebraic properties of classes of path ideals of posets

We consider path ideals associated to special classes of posets such as tree posets and cycles. We express their property of being sequentially Cohen–Macaulay in terms of the underlying poset. Moreover, monomial ideals, which arise in algebraic statistics from the Luce-decomposable model and the ascending model, can be viewed as path ideals of certain posets. We study invariants of these so-called Luce-decomposable monomial ideals and ascending ideals for diamond posets and products of chains. In particular, for these classes of posets, we explicitly compute their Krull dimension, their projective dimension, their Castelnuovo–Mumford regularity and their Betti numbers.     

Mixed ladder determinantal varieties from two-sided ladders

We study the family of ideals defined by mixed size minors of two-sided ladders of indeterminates. We compute their Gröbner bases with respect to a skew-diagonal monomial order, then we use them to compute the height of the ideals. We show that these ideals correspond to a family of irreducible projective varieties, that we call mixed ladder determinantal varieties. We show that these varieties are arithmetically Cohen-Macaulay, and we characterize the arithmetically Gorenstein ones. Our main result consists in proving that mixed ladder determinantal varieties belong to the same G-biliaison class of a linear variety.     

Galois G-covering of quotients of linear categories

In this paper, we introduce the notion of G-liftable ideals, which extends the liftable ideas defined by Assem and Le Meur. We characterize the G-liftable ideals and construct the Galois G-coverings of quotients of categories associated to the G-liftable ideals. In particular, we study the behavior of G-liftable admissible ideals under Galois G-coverings. Furthermore, we show that the ideals generated by finite dimensional projective modules over a locally bounded linear categories are admissible G-liftable ideals. As an application, we provide a reduction technique for dealing with the existence of Serre functors in the stable categories of Gorenstein projective objects.     

Identities of *-superalgebras and almost polynomial growth

We study the growth of the codimensions of a *-superalgebra over a field of characteristic zero. We classify the ideals of identities of finite dimensional algebras whose corresponding codimensions are of almost polynomial growth. It turns out that these are the ideals of identities of two algebras with distinct involutions and gradings. Along the way, we also classify the finite dimensional simple *-superalgebras over an algebraically closed field of characteristic zero.     

A tight bound on the projective dimension of four quadrics

Motivated by Stillman's question, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring is at most 6; moreover, this bound is tight. We achieve this bound, in part, by giving a characterization of the low degree generators of ideals primary to height three primes of multiplicities one and two.     

IG-projective modules

This paper concerns finitely generated modules over Artin algebras. We introduce the notion of an IG-projective module and use this to prove that if, over such an algebra R$R$, each simple module is strongly Gorenstein projective, then any indecomposable R$R$-module is either projective or simple. We also prove that if R$R$ is local and the simple module is IG-projective, then 1-self-orthogonal modules are projective.     

Linearly representable codes over chain rings

In this paper, we consider linear codes over finite chain rings. We present a general mapping which produces codes over smaller alphabets. Under special conditions, these codes are linear over a finite field. We introduce the notion of a linearly representable code and prove that certain MacDonald codes are linearly representable. Finally, we give examples for good linear codes over finite fields obtained from special multisets in projective Hjelmslev planes.     

A going-up theorem for arbitrary chains of prime ideals

We prove that if an extension R ? T of commutative rings satisfies the going-up property (for instance, if T is an integral extension of R), then any increasing chain of prime ideals of R (indexed by an arbitrary linearly ordered set) is covered by some corresponding chain of prime ideals of T. As a corollary, we recover the recent result of Kang and Oh that any such chain of prime ideals of an integral domain D is covered by a corresponding chain in some valuation overring of D.     

Predicting syzygies over monomial relations algebras

We show that all projective resolutions over a monomial relations algebra Λ simplify drastically at the stage of the second syzygy; more precisely, we show that the kernel of any homomorphism between two projective left Λ-modules is isomorphic to a direct sum of principal left ideals generated by paths. As consequences, we obtain:

Higher codimension cycles on the Hilbert scheme of three points on the projective plane

In this paper, we study the higher codimensional cycle structure of the Hilbert scheme of three points in the projective plane. In particular, we compute all Chern/Segre classes of all tautological bundles on it and compute the nef (effective) cones of cycles in codimensions 2 and 3 (dimensions 2 and 3).     

Commutative rings in which every finitely generated ideal is quasi-projective

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. We provide some preliminaries on quasi-projective modules over commutative rings. Then we investigate the correlation with the well-known Prüfer conditions; that is, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky’s theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz’s related conjecture on Gaussian rings. We also establish an analogue of Bazzoni-Glaz results on the transfer of Prüfer conditions between a ring and its total ring of quotients. We then examine various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasi-projectivity, marking their distinction from related classes of Prüfer rings.     

Chains of prime ideals in tensor products of algebras

This paper investigates the length of particular chains of prime ideals in tensor products of algebras over a field k. As an application, we compute dim(A⊗_kA) for a new family of domains A that are k-algebras.     

Bounds on the regularity and projective dimension of ideals associated to graphs

In this paper, we give new upper bounds on the regularity of edge ideals whose resolutions are k-step linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We also discuss and connect these results to more classical topics in commutative algebra.     

Projectivity,prime ideals,and chain conditions of Stone algebras

Some properties of projective stone algebras are exhibited, which are connected with the ordered set of prime ideals. From this we derive a simple characterization of finite projective Stone algebras, and of those projective Stone algebras, whose centre is a projective Boolean algebra, and whose dense set is a projective Stone algebras, whose centre is a projective Boolean algebra, and whose dense set is a projective distributive lattice. Finally, we give some conditions under which a Stone algebra has no chains of type λ, where λ is an infinite regular cardinal. The results of this paper are part of the author's Ph.D. Thesis written under the direction of S. Koppelberg. The author wishes to express his gratitude to Prof. Koppelberg for her guidance and her patience. Presented by K. A. Baker.     

Projective bundle ideals and Poincaré duality algebras

The study of maximal-primary irreducible ideals in a commutative graded connected Noetherian algebra over a field is in principle equivalent to the study of the corresponding quotient algebras. Such algebras are Poincaré duality algebras. A prototype for such an algebra is the cohomology with field coefficients of a closed oriented manifold. Topological constructions on closed manifolds often lead to algebraic constructions on Poincaré duality algebras and therefore also on maximal-primary irreducible ideals. It is the purpose of this note to examine several of these and develop some of their basic properties.     

A remark on regularity of powers and products of ideals

We give a simple proof for the fact that the Castelnuovo–Mumford regularity and related invariants of products of powers of ideals are asymptotically linear in the exponents, provided that each ideal is generated by elements of constant degree. We provide examples showing that the asymptotic linearity is false in general. On the other hand, the regularity is always given by the maximum of finitely many linear functions whose coefficients belong to the set of the degrees of generators of the ideals.     

A generalized principal ideal theorem

The main theorem of this article is an extension of the generalized principal ideal theorem for ideals in Noetherian rings. Instead of requiring the rings to be Noetherian, some natural requirements are imposed on the chains of prime ideals under consideration. The standard (Noetherian) version of the generalized principal ideal theorem is deduced as a corollary and two other applications are presented.     

Asymptotic resurgences for ideals of positive dimensional subschemes of projective space

Recent work of Ein–Lazarsfeld–Smith and Hochster–Huneke raised the containment problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci–Harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on zero-dimensional subschemes of projective space. Here we take the first steps toward extending this work to higher dimensional subschemes. We introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals I$I$ of smooth subschemes, generalizing what is done in Bocci and Harbourne (2010)  [5]. We apply these bounds to ideals of unions of general lines in P^N${\mathbb{P}}^N$. We also pose a Nagata type conjecture for symbolic powers of ideals of lines in P³${\mathbb{P}}^3$.     

An example of an atomic pullback without the ACCP

We find necessary and sufficient conditions on a pullback diagram in order that every nonzero nonunit in its pullback ring admits a finite factorization into irreducible elements. As a result, we can describe a method of easily producing atomic domains that do not satisfy the ascending chain condition on principal ideals.