Let n, k, and r be nonnegative integers and let Sn be the symmetric group. We introduce a quotient Rn,k,r of the polynomial ring Q[x1, . . . , xn] in n variables which carries the structure of a graded Sn-module. When r > n or k = 0 the quotient Rn,k,r reduces to the classical coinvariant algebra Rn attached to the symmetric group. Just as algebraic properties of Rn are controlled by combinator...

In this paper, we introduce a suitable algebraic structure for efficient computation of the parametric normal form of Hopf singularity based on a notion of formal decompositions. Our parametric state and time spaces are respectively graded parametric Lie algebra and graded ring. As a consequence, the parametric state space is also a graded module. Parameter space is observed as an integral doma...

We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomol-ogy. We establish the key properties of regularity: its connection with the ...

We show that any quasi-Gorenstein deformation of a 3-dimensional Buchsbaum local ring with I-invariant 1 admits maximal Cohen-Macaulay module, provided it is quotient Gorenstein ring. Such class rings includes two instances unique factorization domains constructed by Marcel-Schenzel and Imtiaz-Schenzel, respectively. Apart from this result, motivated the small conjecture in prime characteristic...

Given a graph whose edges are labeled by ideals of commutative ring R with identity, generalized spline is vertex labeling the elements such that difference labels on adjacent vertices lies in ideal associated to edge. The set splines has and an R-module structure. We study module structure where base greatest common divisor domain. give basis criteria for cycles, diamond graphs trees using det...

We present a new class of graded irreducible representations Leavitt path algebra. This is in the sense that its representation space not isomorphic to any existing simple Chen modules. The corresponding modules complete list which are graded, creating an exhaustive class: annihilator module equal either or this type. Our characterization primitive ideals algebra terms properties underlying gra...

Let $F$ be a non-negatively graded free module over polynomial ring $\mathbb{K}[x_1,\dots,x_n]$ generated by $m$ basis elements. $M$ submodule of elements in with degrees bounded $D$ and dim $F/M$=$r$. We prove that if is graded, the degree reduced Gr\"{o}bner for any term order $2\left[1/2((Dm)^{n-r}m+D) \right]^{2^{r-1}}$. If not bound $2\left[1/2((Dm)^{(n-r)^2}m+D) \right]^{2^{r}}$. This gen...

Let G be a connected reductive algebraic group with Lie algebra g defined over an algebraically closed field, k, with char k = 0. Fix a parabolic subgroup of G with Levi decomposition P = LU where U is the unipotent radical of P . Let u = Lie(U) and let z denote the center of Lie(L). Let T be a maximal torus in L with Lie algebra t. Then the root system of (g, t) is a subset of t∗ and by restri...

This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its minimal resolution is acyclic. The results established track the change in this invariant under some standard operations in commutative algebra. As one of the ap...

Let [Formula: see text] where be a complete regular local ring of dimension text], for some and an MCM text]-module with then we prove that text]. If is hypersurface infinite residue field let or Our paper the first systematic study depth associated graded modules over rings.