The Hilbert function of a module over a positively graded algebra is of quasi-polynomial type (Hilbert–Serre). We derive an upper bound for its grade, i. e. the index from which on its coefficients are constant. As an application, we give a purely algebraic proof of an old combinatorial result (due to Ehrhart, McMullen and Stanley). 1. HILBERT QUASIPOLYNOMIALS Let K be a field, and R a positive...

Throughout this note all rings are commutative, and A is a fixed ring. If S, T are graded A-algebras then the tensor product S ⊗A T becomes a graded A-algebra in a canonical way with the grading given by (TES,Lemma 13). That is, S ⊗A T is the coproduct of the morphisms of A-modules Sd ⊗A Te −→ S ⊗A T for d, e ≥ 0. The canonical morphisms p1 : S −→ S ⊗A T, p2 : T −→ S ⊗A T are then morphisms of ...

Our goal in this paper is to define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called “module algebras” (Definition 2.1). Our motivation lies in the following problem: for an algebra A which admits a module structure over an arbitrary bialgebra B compatible with its product structure, the Hochschild or the cyclic ...

There are well-known constructions of the exceptional simple Lie algebras of type E8 and F4 which go back to Witt [Wit41], as Z2-graded algebras g = g0̄⊕g1̄ with even part the orthogonal Lie algebras so16 and so9 respectively, and odd part given by their spin representations (see [Ada96]). Brown [Bro82] found a new simple finite dimensional Lie algebra over fields of characteristic 3 which presen...

Let v be a valuation of the quotient field of a noetherian local domain R. Assume that v is centered at R. This paper studies the structure of the value semigroup of v, S. Ideals defining toric varieties can be defined from the graded algebra K[T ] of cancellative commutative finitely generated semigroups such that T ∩ (−T ) = {0}. The value semigroup of a valuation S need not be finitely gener...

In this article, we are interested in some subspaces of the exterior algebra a simple Lie $\mathfrak {g}$ . particular, prove that graded degree d generate ${\mathfrak {g}}$ -module $\bigwedge ^{d}{\mathfrak for integers d.

We show that there exists a natural non-degenerate pairing of the homomorphism space between two neighbor standard modules over a quasi-hereditary algebra with the first extension space between the corresponding costandard modules and vise versa. Investigation of this phenomenon leads to a family of pairings involving standard, costandard and tilting modules. In the graded case, under some ”Kos...

In this paper we study subalgebras A of the algebra D(X) of differential operators on a smooth variety X which are big in the following sense: using the order of a differential operator, the ring D(X) is equipped with a filtration. Its associated graded algebra D(X) is commutative and can be regarded as the set of regular functions on the cotangent bundle ofX . The subalgebra A inherits a filtr...

In this paper, we present several algorithms for dealing with graded components of Laurent polynomial rings. To be more precise, let S be the Laurent polynomial ring k[x1, . . . , xr, x ±1 r+1, . . . , x ±1 n ], k algebraicaly closed field of characteristic 0. We define the multigrading of S by an arbitrary finitely generated abelian group A. We construct a set of fans compatible with the multi...

Let $ \pi:X \rightarrow X_{0}$ be a projective morphism of schemes, such that $ X_{0}$ is Noetherian and essentially of finite type over a field $ K$. Let $ i \in \mathbb{N}_{0}$, let $ {\mathcal{F}}$ be a coherent sheaf of $ {\mathcal{O}}_{X}$-modules and let $ {\mathcal{L}}$ be an ample invertible sheaf over $ X$. Let $ Z_{0} \subseteq X_{0}$ be a closed set. We show that the depth of the hig...