Let C be a smooth projective curve over the complex number field C. We investigate the structure of the cohomology ring of the quot schemes Quot(r, n), i.e., the moduli scheme of the quotient sheaves of O r C with length n. We obtain a filtration on H(Quot(r, n)), whose associated graded ring has a quite simple structure. As a corollary, we obtain a small generator of the ring. We also obtain a...

Let S be a cancellative monoid with quotient group of torsion-free rank a. We show that the monoid ring R[S] is a Hilbert ring if and only if the polynomial ring R[{ X, },s/] is a Hilbert ring, where |/| = a. Assume that R is a commutative unitary ring and G is an abelian group. The first research problem listed in [K, Chapter 7] is that of determining equivalent conditions in order that the gr...

1. Introduction. We shall say that an integral domain1 R satisfies condition (M) if any two nonzero elements of R have a nonzero common right multiple. In this note it is proved that if 5 is an extension of a ring R such that S is, roughly speaking, a noncommutative polynomial ring in one variable with R as a coefficient ring, and if R has the property (M), then 5 has property (M). In case R is...

The skein algebra of an oriented three-manifold is a classical limit the Kauffman bracket module and gives coordinate ring [Formula: see text]-character variety. In this paper, we determine quotient polynomial which isomorphic to group with three generators two relators. As application, give explicit formula for Borromean rings complement in text].

Let R be a polynomial ring in r variables over an infinite field K, and denote by D a corresponding dual ring, upon which R acts as differential operators. We study type two graded level Artinian algebras A = R/I , having socle degree j. For each such algebra A, we consider the family of Artinian Gorenstein [AG] quotients of A having the same socle degree. By Macaulay duality, A corresponds to ...

We give a proof using so-called fusion rings and q-deformations of Brauer algebras that the representation ring of an orthogonal or symplectic group can be obtained as a quotient of a ring Gr(O(∞)). This is obtained here as a limiting case for analogous quotient maps for fusion categories, with the level going to ∞. This in turn allows a detailed description of the quotient map in terms of a re...

In this paper, we propose a weak version of quotient for the algebraic action group on variety, which shall call pseudo-quotient. They arise when focus purely topological properties good Geometric Invariant Theory (GIT) quotients regardless their properties. The flexibility granted by nature enables an easier identification in geometric constructions than classical GIT quotients. We obtain seve...

We prove that if $R$ is a commutative Noetherian ring, then every countably generated flat $R$-module quite flat, i.e., direct summand of transfinite extension localizations in countable multiplicative subsets. also show the spectrum cardinality less than $\kappa$, where $\kappa$ an uncountable regular cardinal, modules with generators. This provides alternative proof fact over ring spectrum, a...

We investigate the Cox ring of a normal complete variety X with algebraic torus action. Our first results relate the Cox ring of X to that of a maximal geometric quotient of X . As a consequence, we obtain a complete description of the Cox ring in terms of generators and relations for varieties with torus action of complexity one. Moreover, we provide a combinatorial approach to the Cox ring us...

Let X be a rational homogeneous space and let QH(X)× loc be the group of invertible elements in the small quantum cohomology ring of X localised in the quantum parameters. We generalise results of [ChMaPe06] and realise explicitly the map π1(Aut(X)) → QH (X)× loc described in [Se97]. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring QH T (X)× ...