THE STRUCTURE OF THE TANGENT CONE: AN INTERESTING BIJECTION#

ABSTRACT

Let X = Spec(R) be a reduced equidimensional algebraic variety over an algebraically closed field k. Let Y = Spec(R/𝔮) be a codimension one ordinary multiple subvariety, where 𝔮 is a prime ideal of height 1 of R. If U is a nonempty open subset of Y and 𝔪 a closed point of U, we denote by A ? R _𝔪 its local ring in X, by 𝔭 the extension of 𝔮 in A, and by K the algebraic closure of the residue field k(𝔭).

Then there exists a bijection γ^𝔪:Proj(G _𝔭(A) ?  _A/𝔭 k) → Proj(G(A _𝔭) ?  _k(𝔭)K) such that for every subset Σ of Proj(G _𝔭(A) ?  _A/𝔭 k), the Hilbert function of Σ coincides with the Hilbert function of γ^𝔪(Σ). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety.     

A Family of Complex Irreducible Characters Possessed by Unitary and Special Unitary Groups Defined over Local Rings

Abstract

Let 𝒪 be a discrete valuation ring whose residue field 𝒪/𝔭 is finite and has odd characteristic. Let l be a positive integer. Set R = 𝒪/𝔭 ^l and let R = R[θ] be the ring obtained by adjoining to R a square root of a non-square unit. Consider the involution σ of R that fixes R elementwise and sends θ to ? θ. Let V be a free R-module of rank n > 0 endowed with a non-degenerate hermitian form ( , ) relative to σ. Let U _n (R) be the subgroup of GL(V) that preserves ( , ). Let SU _n (R) be the subgroup of all g ∈ U _n (R) whose determinant is equal to one. Let Ψ be the Weil character of U _n (R).

All irreducible constituents of Ψ are determined. An explicit character formula is given for each of them. In particular, all character degrees are computed. For n > 2 the corresponding results are also obtained for the restriction of Ψ to SU _n (R).     

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).     

Non-abelian Tensor Product of Leibniz Algebras and an Exact Sequence in Leibniz Homology

Abstract

Let 𝔪 and 𝔫 be two-sided ideals of a Leibniz algebra 𝔤 such that 𝔤 = 𝔪 + 𝔫. The goal of the paper is to achieve the exact sequence Ker(𝔪  𝔫 + 𝔫  𝔪 → 𝔤) → HL ₂(𝔤) → HL ₂(𝔤/𝔪) ⊕ HL ₂(𝔤/𝔫) → 𝔪 ∩ 𝔫/ [𝔪,𝔫] → HL ₁(𝔤) → HL ₁(𝔤/𝔪) ⊕ HL ₁(𝔤/𝔫) → 0, where HL denotes the Leibniz homology with trivial coefficients of a Leibniz algebra and denotes a non-abelian tensor product of Leibniz algebras.     

Second Homology Groups and Universal Coverings of Steinberg Leibniz Algebras of Small Characteristic

It is known that the second Leibniz homology group HL ₂ (𝔰𝔱𝔩 _n (R)) of the Steinberg Leibniz algebra 𝔰𝔱𝔩 _n (R) is trivial for n ≥ 5. In this article, we determine HL ₂(𝔰𝔱𝔩 _n (R)) explicitly (which are shown to be not necessarily trivial) for n = 3, 4 without any assumption on the base ring.     

Cohen-Macaulayness of Special Fiber Rings

Abstract

Let (R, 𝔪) be a Noetherian local ring and let Ibe an R-ideal. Inspired by the work of Hübl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ? = ?/𝔪? of I, where ? denotes the Rees algebra of I. Our key idea is to require ‘good’ intersection properties as well as ‘few’ homogeneous generating relations in low degrees. In particular, if Iis a strongly Cohen-Macaulay R-ideal with G _?and the expected reduction number, we conclude that ? is always Cohen-Macaulay. We also obtain a characterization of the Cohen-Macaulayness of ?/K? for any 𝔪-primary ideal K. This result recovers a well-known criterion of Valabrega and Valla whenever K = I. Furthermore, we study the relationship between the Cohen-Macaulay property of the special fiber ring ? and the Cohen-Macaulay property of the Rees algebra ? and the associated graded ring 𝒢 of I. Finally, we focus on the integral closedness of 𝔪I. The latter question is motivated by the theory of evolutions.     

Tensor Product Representations of the Lie Superalgebra 𝔭(n) and Their Centralizers

Abstract

We construct an associative algebra A _k and show that there is a representation of A _k on V ^?k , where V is the natural 2n-dimensional representation of the Lie superalgebra 𝔭(n). We prove that A _k is the full centralizer of 𝔭(n) on V ^?k , thereby obtaining a “Schur-Weyl duality” for the Lie superalgebra 𝔭(n). This result is used to understand the representation theory of the Lie superalgebra 𝔭(n). In particular, using A _k we decompose the tensor space V ^?k , for k = 2 or 3, and show that V ^?k is not completely reducible for any k ≥ 2.     

Relative Completions of Linear Groups Over Coordinate Rings of Curves

We compute the completion of the special linear group over the coordinate ring A of a curve over a number field k relative to its representation in SL _n (A/𝔪), for 𝔪 a maximal ideal.     

Representations of Nonstandard Poincaré Hopf Algebras

Let 𝔭 _q (1 + 1) be a nonstandard Poincaré Hopf algebra, we characterize all finite dimensional completely E-semisimple modules of 𝔭 _q (1 + 1). We also classify all finite dimensional E-semisimple modules of 𝔭 _q for a special quotient algebra of 𝔭 _q (1 + 1). Moreover, the decomposition of tensor product of two finite dimensional E-semisimple indecomposable modules is obtained.     

Remarks on torsionfreeness and its applications

In this article, we shall characterize torsionfreeness of modules with respect to a semidualizing module in terms of the Serre’s condition (S_n). As its applications, we give a characterization of Cohen-Macaulay rings R such that R_𝔭 is Gorenstein for all prime ideals 𝔭 of height less than n, and we will give a partial answer of Tachikawa conjecture and Auslander-Reiten conjecture.     

Algebraic K-theory of non-linear projective spaces

In the spirit of “The Fundamental Theorem for the algebraic K-theory of spaces: I” (J. Pure Appl. Algebra 160 (2001) 21–52) we introduce a category of sheaves of topological spaces on n-dimensional projective space and present a calculation of its K-theory, a “non-linear” analogue of Quillen's isomorphism K_i(P_Rⁿ)₀ⁿK_i(R).     

The Annihilating-Ideal Graph of a Commutative Ring with Respect to an Ideal

For a commutative ring R with identity, the annihilating-ideal graph of R, denoted 𝔸𝔾(R), is the graph whose vertices are the nonzero annihilating ideals of R with two distinct vertices joined by an edge when the product of the vertices is the zero ideal. We will generalize this notion for an ideal I of R by replacing nonzero ideals whose product is zero with ideals that are not contained in I and their product lies in I and call it the annihilating-ideal graph of R with respect to I, denoted 𝔸𝔾 _I (R). We discuss when 𝔸𝔾 _I (R) is bipartite. We also give some results on the subgraphs and the parameters of 𝔸𝔾 _I (R).     

On the depth of certain graded rings associated to an ideal

Let (R,m) be a local ring and Ian ideal of R. In this work we find conditions on Ithat allow us to describe simple relations among depth R(It), depth gr_I(R), depth S(I) and depth S(I/I ²). These relations are useful also from a practical point, of view since it is usually difficult to evaluate depth gr_I(R) and depth S(I/I ²) even with the help of a computer. Furthermore we study the class of ideals that satisfy one of the required conditions and we show that ideals generated by quadratic sequences are in this class     

The 𝒥-Radical and the Pierce Radicals of a Lattice-Ordered Ring

Abstract

The 𝒥-radical of a lattice-ordered ring Ris the ?-ring analogue of the Jacobson radical of a ring. It is shown that if the matrix ring R _n has the usual lattice order, then 𝒥(R _n ) = 𝒥(R) _n . The connection between an element abeing right ?-quasi-regular and the inequality a ○ x ≤ 0 is also investigated. For squares in an f-ring the connection is an equivalence. In general it is still an equivalence provided xis the sum of elements from a larger f-ring whose absolute values lie in R. It is also shown that the vanishing of annihilators in an f-ring is inherited by enough totally ordered homomorphic images to give a subdirect product decomposition.     

𝔏-Rickart Modules

It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M _R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M _R is a finitely generated 𝔏-Rickart module, we prove that End _R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End _R (M) is a left extending ring iff M is a Baer module and End _R (M) is left cononsingular.     

Graded Polynomial Identities for Tensor Products by the Grassmann Algebra

Abstract

Let 𝕂 be a field of characteristic zero, and R be a G-graded 𝕂-algebra. We consider the algebra R ? E, then deduce its G × ?₂-graded polynomial identities starting from the G-graded polynomial identities of R. As a consequence, we describe a basis for the ? _n  × ?₂-graded identities of the algebras M _n (E). Moreover we give the graded cocharacter sequence of M ₂(E), and show that M ₂(E) is PI-equivalent to M _1,1(E) ? E. This fact is a particular case of a more general result obtained by Kemer.     

The Semigroup Generated by the Similarity Class of a Singular Matrix

Let A be a singular matrix of M _n (𝕂), where 𝕂 is an arbitrary field. Using canonical forms, we give a new proof that the sub-semigroup of ( _n (𝕂), ×) generated by the similarity class of A is the set of matrices of M _n (𝕂) with a rank lesser than or equal to that of A.     

Commutative Rings R Whose C(𝔸𝔾(R)) Consist of Only Triangles

Let R be a commutative ring with identity. The set 𝕀(R) of all ideals of R is a bounded semiring with respect to ordinary addition, multiplication and inclusion of ideals. The zero-divisor graph of 𝕀(R) is called the annihilating-ideal graph of R, denoted by 𝔸𝔾(R). We write 𝒢 for the set of graphs whose cores consist of only triangles. In this paper, the types of the graphs in 𝒢 that can be realized as either the zero-divisor graphs of bounded semirings or the annihilating-ideal graphs of commutative rings are determined. A necessary and sufficient condition for a ring R such that 𝔸𝔾(R) ∈ 𝒢 is given. Finally, a complete characterization in terms of quotients of polynomial rings is established for finite rings R with 𝔸𝔾(R) ∈ 𝒢. Also, a connection between finite rings and their corresponding graphs is realized.