Algebra retracts and Stanley–Reisner rings

In a paper from 2002, Bruns and Gubeladze conjectured that graded algebra retracts of polytopal algebras over a field k are again polytopal algebras. Motivated by this conjecture, we prove that graded algebra retracts of Stanley–Reisner rings over a field k are again Stanley–Reisner rings. Extending this result further, we give partial evidence for a conjecture saying that monomial quotients of standard graded polynomial rings over k descend along graded algebra retracts.     

Vinberg’s θ-groups in positive characteristic and Kostant–Weierstrass slices

We generalize the basic results of Vinberg’s θ-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify the relationship between the little Weyl group and the (standard) Weyl group. We deduce that the ring of invariants associated to the grading is a polynomial ring. This approach allows us to prove the existence of a KW-section for a classical graded Lie algebra (in zero or odd positive characteristic), confirming a conjecture of Popov in this case.     

Whittaker Modules for Graded Lie Algebras

In this paper, we study Whittaker modules for graded Lie algebras over ℂ. We define Whittaker modules for a class of graded Lie algebras and obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra. As a consequence of this, we obtain a classification of simple Whittaker modules for such algebras. Also, we discuss some concrete algebras as examples.     

Hilbert schemes and maximal Betti numbers over veronese rings

Macaulay??s Theorem (Macaulay in Proc. Lond Math Soc 26:531?C555, 1927) characterizes the Hilbert functions of graded ideals in a polynomial ring over a field. We characterize the Hilbert functions of graded ideals in a Veronese ring R (the coordinate ring of a Veronese embedding of P ^r-1). We also prove that the Hilbert scheme, which parametrizes all graded ideals in R with a fixed Hilbert function, is connected; this is an analogue of Hartshorne??s Theorem (Hartshorne in Math. IHES 29:5?C48, 1966) that Hilbert schemes over a polynomial ring are connected. Furthermore, we prove that each lex ideal in R has the greatest Betti numbers among all graded ideals with the same Hilbert function.     

Monomial modules and graded Betti numbers

Let K be a field, S = K[x ₁,…, x _n ], the polynomial ring over K, and let F be a finitely generated graded free S-module with homogeneous basis. Certain invariants, such as the Castelnuovo-Mumford regularity and the graded Betti numbers of submodules of F, are studied; in particular, the componentwise linear submodules of F are characterized in terms of their graded Betti numbers.     

Hochschild Cohomology of <Emphasis Type="Italic">q</Emphasis>-Schur Algebras

We compute the Hochschild cohomology of any block of q-Schur algebras. We focus on the even part of this Hochschild cohomology ring. To compute the Hochschild cohomology of q-Schur algebras, we prove the following two results: first, we construct two graded algebra surjections between the Hochschild cohomologies of quasi-hereditary algebras because all q-Schur algebras over a field are quasi-hereditary. Second, we give the graded algebra isomorphism of Hochschild cohomologies by using a certain derive equivalence.     

On the Image of the Totaling Functor

Let A be a differential graded (DG) algebra with a trivial differential over a commutative unital ring. This paper investigates the image of the totaling functor, defined from the category of complexes of graded A-modules to the category of DG A-modules. Specifically, we exhibit a special class of semifree DG A-modules which can always be expressed as the totaling of some complex of graded free A-modules. As a corollary, we also provide results concerning the image of the totaling functor when A is a polynomial ring over a field.     

Differents in modular invariant theory

Let be a graded polynomial algebra over a field k, such that each variable is homogeneous of positive degree. No restrictions are made with respect to the field. Let the finite group G act on A by graded algebra automorphisms and denote the subalgebra of invariants by B. In this paper the various "different ideals" of the extension are studied that define the ramification locus. We prove, for example, that the subring of invariants is itself a polynomial ring if and only if the ramification locus is pure of height one. Here the ramification locus is defined by either the Kahler different, the Noether different or the Galois different. As a consequence we prove that the invariant ring is itself a polynomial ring if and only if there are invariants whose Jacobian determinant does not vanish and is of degree δ, where δ is the degree of the Dedekind different. Using this criterion we give a quick proof of Serre's result that if the invariant ring is a polynomial algebra, then the group is generated by generalized reflections.     

Some algebras similar to the 2×2 Jordanian matrix algebra

The impetus for this study is the work of Dumas and Rigal on the Jordanian deformation of the ring of coordinate functions on 2×2 matrices. We are also motivated by current interest in birational equivalence of noncommutative rings. Recognizing the construction of the Jordanian matrix algebra as a skew polynomial ring, we construct a family of algebras relative to differential operator rings over a polynomial ring in one variable which are birationally equivalent to the Weyl algebra over a polynomial ring in two variables.     

Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the n-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defined using the relation complex of the Coxeter arrangement and fiber polytopes of the dual Coxeter zonotope. They generalize the algebra of piecewise polynomial functions on the original arrangement.     

The Castelnuovo regularity of the Rees algebra and the associated graded ring

It is shown that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology and the Koszul homology of the Rees algebra and the associated graded ring of an ideal. From this it follows that these graded rings share the same Castelnuovo regularity and the same relation type. The main result of this paper is however a simple characterization of the Castenuovo regularity of these graded rings in terms of any reduction of the ideal. This characterization brings new insights into the theory of -sequences.

     

The Cohen–Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non-Cohen–Macaulay actually imply that no graded separating algebra is Cohen–Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen–Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen–Macaulay graded separating algebra implies the group is generated by bireections. Additionally, we give an example which shows that Cohen–Macaulay separating algebras can occur when the ring of invariants is not Cohen–Macaulay.     

A monomial cycle basis on Koszul homology modules

We give a class of p-Borel principal ideals of a polynomial algebra over a field K for which the graded Betti numbers do not depend on the characteristic of K and the Koszul homology modules have a monomial cyclic basis.     

Polynomial invariants for a semisimple and cosemisimple Hopf algebra of finite dimension

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. The coefficients of polynomial invariants are integers if is a finite Galois extension of Q, and A is a scalar extension of some finite-dimensional semisimple Hopf algebra over Q. Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of A, and recognize the difference between the representation category and the representation ring of A. Actually, by computing and comparing polynomial invariants, we find new examples of pairs of Hopf algebras whose representation rings are isomorphic, but whose representation categories are distinct.     

Graded algebras with cyclotomic Hilbert series

Let R be a positively graded algebra over a field $\mathbb{k}$. We say that R is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all of its roots on the unit circle. Such rings arise naturally in commutative algebra, numerical semigroup theory and Ehrhart theory. If R is standard graded, we prove that, under the additional hypothesis that R is Koszul or has an irreducible h-polynomial, Hilbert-cyclotomic algebras coincide with complete intersections. In the Koszul case, this is a consequence of some classical results about the vanishing of deviations of a graded algebra.     

Borel-plus-powers monomial ideals

Let S=K[x₁,…,x_n] be a standard graded polynomial ring over a field K. In this paper, we show that the lex-plus-powers ideal has the largest graded Betti numbers among all Borel-plus-powers monomial ideals with the same Hilbert function. In addition in the case of characteristic 0, by using this result, we prove the lex-plus-powers conjecture for graded ideals containing , where p is a prime number.     

Graph products of spheres, associative graded algebras and Hilbert series

Given a finite, simple, vertex-weighted graph, we construct a graded associative (noncommutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Noncommutative Gr?bner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment?Cangle complex. We apply our result to the loop-space homology of this space.     

Generic Flatness and the Jacobson Conjecture

A result of Artin, Small, and Zhang is used to show that a Noetherian algebra over a commutative, Noetherian Jacobson ring will be Jacobson if the algebra possesses a locally finite, Noetherian associated graded ring. This result is extended to show that if an algebra over a commutative Noetherian ring has a locally finite, Noetherian associated graded ring, then the intersection of the powers of the Jacobson radical is nilpotent. The proofs rely on a weak generalization of generic flatness and some observations about G-rings.