Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let be the blow-up of a projective scheme along the ideal sheaf of . It is known that there are embeddings for , where denotes the maximal generating degree of , and that there exists a Cohen-Macaulay ring of the form (which gives an arithmetic Macaulayfication of ) if and only if , for , and is equidimensional and Cohen-Macaulay. We show that under these conditions, there are well-determined invariants and such that is Cohen-Macaulay for all d(I)e + \varepsilon$"> and e_0$">, and that these bounds are the best possible. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form . If has negative -invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if , for 0$">, and is equidimensional and Cohen-Macaulay. Moreover, these conditions imply the Cohen-Macaulayness of for all d(I)e + \varepsilon$"> and e_0$">.

     

Dominions of universal algebras and projective properties

Let A be a universal algebra and H its subalgebra. The dominion of H in A (in a class {ie304-01}) is the set of all elements a ∈ A such that every pair of homomorphisms f, g: A → ∈ {ie304-02} satisfies the following: if f and g coincide on H, then f(a) = g(a). A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras H whose dominions coincide with H. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup H is closed in each group 〈H, a〉 generated by one element modulo H. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 541–557, September–October, 2008.     

Uniform modules over serial rings with krull dimension

We study the symbolic blow-up ring of a prime ideal defining a monomial curve in the power series ring in 3 variables over a field. We characterize the conditions required to have the symbolic blow-up generated in degree 4 when the monomial curve is non-self-linked. When this is the case we also find that the symbolic blow-up cannot be Cohen–Macaulay.     

Realizing enveloping algebras via varieties of modules

By using the Ringel-Hall algebra approach, we investigate the structure of the Lie algebra L（A） generated by indecomposable constructible sets in the varieties of modules for any finite- dimensional C-algebra A. We obtain a geometric realization of the universal enveloping algebra R（A） of L（A）, this generalizes the main result of Riedtmann. We also obtain Green＇s formula in a geometric form for any finite-dimensional C-algebra A and use it to give the comultiplication formula in R（A）.     

Examples of rational homotopy types of blow-ups

We give an example of two homotopic embeddings of manifolds with isomorphic complex normal bundles but such that the blow-ups of along and along have different rational homotopy types.

     

Constraints for the normality of monomial subrings and birationality

Let be a field and let be a finite set of monomials whose exponents lie on a positive hyperplane. We give necessary conditions for the normality of both the Rees algebra and the subring . If the monomials in have the same degree, one of the consequences is a criterion for the -rational map defined by to be birational onto its image.

     

Commutativity of the centralizer of a subalgebra in a universal enveloping algebra

Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let \(\mathfrak{h}\) be an algebraic subalgebra of the tangent Lie algebra \(\mathfrak{g}\) of G. We find all subalgebras \(\mathfrak{h}\) that have no nontrivial characters and whose centralizers \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) and \(P(\mathfrak{g})^\mathfrak{h} \) in the universal enveloping algebra \(\mathfrak{U}(\mathfrak{g})\) and in the associated graded algebra \(P(\mathfrak{g})\), respectively, are commutative. For all these subalgebras, we prove that \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} = \mathfrak{U}(\mathfrak{h})^\mathfrak{h} \otimes \mathfrak{U}(\mathfrak{g})^\mathfrak{g} \) and \(P(\mathfrak{g})^\mathfrak{h} = P(\mathfrak{h})^\mathfrak{h} \otimes P(\mathfrak{g})^\mathfrak{g} \). Furthermore, we obtain a criterion for the commutativity of \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) in terms of representation theory.     

On stem covers and the universal central extensions of Lie crossed modules

In this article, we investigate stem covers and the universal central extensions of Lie crossed modules. In particular, a cover for the direct sum of two Lie crossed modules in term of given covers of them will be constructed. Using the results obtained, we give some properties of covers and the universal relative central extensions of pairs of Lie algebras.     

Notes on Hilbert-Kunz Multiplicity of Rees Algebras

Abstract

In this paper, we estimate the Hilbert-Kunz multiplicity of the (extended) Rees algebras in terms of some invariants of the base ring. Also, we give an explicit formula for the Hilbert-Kunz multiplicities of Rees algebras over Veronese subrings.     

Many associated primes of powers of primes

We construct families of prime ideals in polynomial rings for which the number of associated primes of the second power (or higher powers) is exponential in the number of variables in the ring.     

Bounds of divided universal Bernoulli numbers and universal Kummer congruences

Let be a prime. We obtain good bounds for the -adic sizes of the coefficients of the divided universal Bernoulli number when is divisible by . As an application, we give a simple proof of Clarke's 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo for the divided universal Bernoulli numbers for the case , which is a new result.

     

对偶扩张代数的Frobenius态射和固定点代数

设A是由箭图Q和关系I所确定的代数,D(A)是代数A的对偶扩张代数, 对应的箭图Q*和关系I*由Q和I决定．本文证明:带关系箭图(Q*,I*)的自同构由带关系箭图(Q,I)的自同构决定;D(A)的Frobenius态射由A的Frobenius态射完全决定;代数D(A)的固定点代数同构于相应的代数A的固定点代数与A°P的固定点代数的张量积,特别地,当Q为单的箭图时,代数D(A)的固定点代数同构于代数A的固定点代数的对偶扩张代数．     

Posets of infinite prinjective type and embeddings of Kronecker modules into the category of prinjective peak I-spaces

For a restricted Lie algebra L over a field of characteristic p > 0 we study the Lie nilpotency index t _L (u(L)) of its restricted universal enveloping algebra u(L). In particular, we determine an upper and a lower bound for t _L (u(L)). Finally, under the assumption that L is p-nilpotent and finite-dimensional, we establish when the Lie nilpotency index of u(L) is maximal.

Communicated by I. Shestakov.     

On the Comparison of Rees's Valuations Over a Quasi-Unmixed Local Ring

In this paper, we will show that if (R, 𝔪) is a quasi-unmixed local ring, I an 𝔪-primary ideal of R and ?𝒱(I) is the set of Rees valuations of I, then the number of minimal prime ideals in the 𝔪-adic completion of R equals exactly the number of equivalence classes on the set ?𝒱(I) under the equivalence relation ～defined by: ν₁ ～ ν₂ if there exist a constant c ≥ 1 such that for all x ∈ R, ν₁(x) ≤ cν₂(x) and ν₂(x) ≤ cν₁(x).     

Torsion freeness of symmetric powers of ideals

Let be an ideal in a Noetherian commutative ring with unit, let be an integer, and let be the canonical surjective -module homomorphism from the th symmetric power of to the th power of . When or when is a perfect Gorenstein ideal of grade , we provide a necessary and sufficient condition for to be an isomorphism in terms of upper bounds for the minimal number of generators of the localisations of . When is a maximal ideal of we show that is an isomorphism if and only if is a regular local ring. In all three cases for our results yield that if is an isomorphism, then is also an isomorphism for each .

     

On Burch's inequality and a reduction system of a filtration

Let be a multiplicative filtration of a local ring such that the Rees algebra is Noetherian. We recall Burch's inequality for and give an upper bound of the a-invariant of the associated graded ring using a reduction system of . Applying those results, we study the symbolic Rees algebra of certain ideals of dimension .