On the Buchsbaumness of the Associated Graded Ring of a One-Dimensional Local Ring

Let (R, m) be a Noetherian, one-dimensional, local ring, with |R/m|=∞. We study when its associated graded ring G(m) is Buchsbaum; in particular, we give a theoretical characterization for G(m) to be Buchsbaum not Cohen–Macaulay. Finally, we consider the particular case of R being the semigroup ring associated to a numerical semigroup S: we introduce some invariants of S, and we use them in order to give a necessary and a sufficient condition for G(m) to be Buchsbaum.     

Generalized Cohen-Macaulay modules over rings with approximation property

Let (A, m) be an excellent Henselian ring with isolated singularity and letR be its completion. Then every indecomposable maximal Buchsbaum (resp. generalized Cohen-Macaulay)R-module is isomorphic with the completion of an indecomposable maximal Buchsbaum (resp. generalized Cohen-Macaulay)A-module. Hence one gets examples of non-complete, non-regular rings having finite Buchsbaum representation type.     

Tangent Cone of Numerical Semigroup Rings of Embedding Dimension Three

In this article, we give new characterizations of the Buchsbaum and Cohen–Macaulay properties of the tangent cone gr _𝔪 (R), where (R, 𝔪) is a numerical semigroup ring of embedding dimension 3. In particular, we confirm the conjectures raised by Sapko on the Buchsbaumness of gr _𝔪 (R).     

Buchsbaum Stanley–Reisner Rings and Cohen–Macaulay Covers

First, we give a new criterion for Buchsbaum Stanley–Reisner rings to have linear resolutions. Next, we prove that every (d ? 1)-dimensional complex Δ of initial degree d is contained in the same dimensional Cohen–Macaulay complex whose (d ? 1)th reduced homology is isomorphic to that of Δ. We call such a simplicial complex a Cohen–Macaulay cover of Δ. And we also show that all the intermediate complexes between Δ and its Cohen–Macaulay cover are Buchsbaum provided that Δ is Buchsbaum. As an application, we determine the h-vectors of the 3-dimensional Buchsbaum Stanley–Reisner rings with initial degree 3.     

Enumeration and Generating Functions of Differential Rota–Baxter Words

In this paper, the methods and results in enumeration and generation of Rota–Baxter words in Guo and Sit (Algebraic and Algorithmic Aspects of Differential and Integral Operators (AADIOS), Math. Comp. Sci., vol. 4, Sp. Issue (2,3), 2011) are generalized and applied to a free, non-commutative, non-unitary, ordinary differential Rota–Baxter algebra with one generator. A differential Rota–Baxter algebra is an associative algebra with two operators modeled after the differential and integral operators, which are related by the First Fundamental Theorem of Calculus. Differential Rota–Baxter words are words formed by concatenating differential monomials in the generator with images of words under the Rota–Baxter operator. Their totality is a canonical basis of a free, non-commutative, non-unitary, ordinary differential Rota–Baxter algebra. A free differential Rota–Baxter algebra can be constructed from a free Rota–Baxter algebra on a countably infinite set of generators. The order of the derivation gives another dimension of grading on differential Rota–Baxter words, enabling us to generalize and refine results from Guo and Sit to enumerate the set of differential Rota–Baxter words and outline an algorithm for their generation according to a multi-graded structure. Enumeration of a basis is often a first step to choosing a data representation in implementation of algorithms involving free algebras, and in particular, free differential Rota–Baxter algebras and several related algebraic structures on forests and trees. The generating functions obtained can be used to provide links to other combinatorial structures.     

Homotopies for the Generalized Bar Complex Associated to Certain 3-Rowed Weyl Modules

In Buchsbaum and Rota (1994 Buchsbaum , D. , Rota , G.-C. ( 1994 ). A new construction in homological algebra . Proc. Nat. Acad. Sci. 91 : 4115 – 4119 . [CSA] [CROSSREF] [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), the authors presented a generalized bar complex associated to certain 3-rowed Weyl modules and proved that this complex is in fact a resolution via an induction on the number of overlaps between the second and third rows and a fundamental exact sequence (Akin and Buchsbaum, 1985 Akin , K. , Buchsbaum , D. ( 1985 ). Characteristic-free representation theory of the general linear group . Adv. Math 58 ( 2 ): 149 – 200 . [CSA]  [Google Scholar]). In this article we study the structure of this resolution by constructing a splitting contracting homotopy for the complexes corresponding to certain shapes.     

Buchsbaum Stanley-Reisner rings with minimal multiplicity

In this paper, we study Buchsbaum Stanley-Reisner rings with linear free resolution. We introduce the notion of Buchsbaum Stanley-Reisner rings with minimal multiplicity of initial degree , which extends the notion of Buchsbaum rings with minimal multiplicity defined by Goto. As an application, we give many examples of non-Cohen-Macaulay Buchsbaum Stanley-Reisner rings with linear resolution.

     

Irreducible buchsbaum curves

For every biliaison class C _M of Buchsbaum curves of π ³ we prove that the leftmost shift in which there are smooth and connected curves is the same as for irreducible curves. As a consequence, every irreducible Buchsbaum curve has a flat deformation with cohomology and Hartshorne-Rao module constant which is smooth and connected.     

The homogeneous q-difference operator

We introduce a q-differential operator D_xy on functions in two variables which turns out to be suitable for dealing with the homogeneous form of the q-binomial theorem as studied by Andrews, Goldman, and Rota, Roman, Ihrig, and Ismail, et al. The homogeneous versions of the q-binomial theorem and the Cauchy identity are often useful for their specializations of the two parameters. Using this operator, we derive an equivalent form of the Goldman–Rota binomial identity and show that it is a homogeneous generalization of the q-Vandermonde identity. Moreover, the inverse identity of Goldman and Rota also follows from our unified identity. We also obtain the q-Leibniz formula for this operator. In the last section, we introduce the homogeneous Rogers–Szegö polynomials and derive their generating function by using the homogeneous q-shift operator.     

Socles of Buchsbaum modules, complexes and posets

The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel's conjecture for the maximum value of the Euler characteristic of a 2k-dimensional simplicial manifold on n vertices as well as Kalai's conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension, number of vertices, and the first Betti number.     

Enumeration and Generating Functions of Rota–Baxter Words

In this paper, we prove results on enumerations of sets of Rota–Baxter words ( ${{{\tt RBWs}}}$ ) in a single generator and one unary operator. Examples of operators are integral operators, their generalization to Rota–Baxter operators, and Rota–Baxter type operators. ${{{\tt RBWs}}}$ are certain words formed by concatenating generators and images of words under the operators. Under suitable conditions, they form canonical bases of free Rota–Baxter type algebras which are studied recently in relation to renormalization in quantum field theory, combinatorics, number theory, and operads. Enumeration of a basis is often a first step to choosing a data representation in implementation. We settle the case of one generator and one operator, starting with the idempotent case (more precisely, the exponent 1 case). Some integer sequences related to these sets of ${{{\tt RBWs}}}$ are known and connected to other sequences from combinatorics, such as the Catalan numbers, and others are new. The recurrences satisfied by the generating series of these sequences prompt us to discover an efficient algorithm to enumerate the canonical basis of certain free Rota–Baxter algebras.     

On the Lüroth semigroup of curves lying on a smooth quadric

For a smooth irreducible complete algebraic curveC the “gaps” are the integersn such that every linear series of degreen has at least a base point. The Lüroth semigroup S_C of a curveC is the subsemigroup ofN whose elements are not gaps. In this paper we deal with irreducible smooth curves of type (a, b) on a smooth quadricQ. The main result is an algorithm by which we can say if some integer λ∈N is a gap or is in S_C. In the general case there are integers λ which are undecidable. For curves such as complete intersection, arithmetically Cohen-Macaulay or Buchsbaum, we are able to describe explicitly “intervals” of gaps and “intervals” of integers which belong to S_C. For particular cases we can completely determine S_C, by giving just the type of the curve (in particular the degree and the genus). Work done with financial support of M.U.R.S.T. while the authors were members of G.N.S.A.G.A. of C.N.R.     

Rota–Baxter Hom-Lie–Admissible Algebras

We study Hom-type analogs of Rota–Baxter and dendriform algebras, called Rota–Baxter G-Hom–associative algebras and Hom-dendriform algebras. Several construction results are proved. Free algebras for these objects are explicitly constructed. Various functors between these categories, as well as an adjunction between the categories of Rota–Baxter Hom-associative algebras and of Hom-(tri)dendriform algebras, are constructed.     

Curve aritmeticamente Buchsbaum su superfici di grado 3 e 4 dello spazio proiettivo

Riassunto Si studiano curve aritmeticamente Buchsbaum nello spazio proiettivoP ³, tali che l’ordine minimo di una superficie che le contiene è 3 o 4. Per tali curve si determinano l’ordine, il genere aritmetico, il carattere numerico connesso, il modulo di Hartshorne-Rao e la curva legata di ordine minimo. Nel caso di curve situate su superfici cubiche lisce si determinano anche i multigradi corrispondenti.

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Summary In this paper we study arithmetically Buchsbaum curves in the projective spaceP ³, such that the minimal degree of a surface containing them is 3 or 4. For such curves we determine the degree, the aritmethic genus, the connected numerical character, the Hartshorne-Rao module, and the linked curves having minimal degree. For curves lying on smooth cubic surfaces ofP ³ we determine also the associated multidegrees.

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Lavoro eseguito sotto gli auspici del G.N.S.A.G.A. del C.N.R.     

Some remarks on quasi-complete intersection space curves

Summary This paper is devoted to the study of quasi-complete intersection space curves. First, we give a Castelnuovo bound on the index of regularity fork-Buchsbaum, quasi-complete intersection space curves. Then, we prove that, smooth, arithmetically Buchsbaum, quasi-complete intersection space curves of maximal rank are unobstructed. We conclude by studying some examples and adding some remarks.

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Riassunto Questo articolo è dedicato allo studio delle curve spaziali quasi-complete intersezioni. Dapprima, noi diamo un limite di Castelnuovo per l'indice di regolarità delle curvek-Buchsbaum, quasi-complete intersezioni. Inoltre, dimostriamo che le curve liscie, aritmeticamente di Buchsbaum, di rango massimo quasi complete intersezioni sono non ostruite. Si conclude studiando alcuni esempi e aggiungendo alcune osservazioni.

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To Joan     

Generalized Shuffles Related to Nijenhuis and TD-Algebras

Shuffle type products are well known in mathematics and physics. They are intimately related to Loday's dendriform algebras and were extensively used to give explicit constructions of free Rota–Baxter algebras. In the literature there exist at least two other Rota–Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle, the so-called left-shift shuffle, can be used to give an explicit construction of the free unital commutative TD-algebra. We explore some basic properties of TD-operators. Our construction is related to Loday's unital commutative tridendriform algebra, including the involutive case. The concept of Rota–Baxter, Nijenhuis and TD-bialgebras is introduced at the end, and we show that any commutative bialgebra provides such objects.     

An algorithm to determine the Buchsbaum property and invariant for polynomial ideals

In 1913 F. S. Macaulay introduced the concept of perfectness. We extend Macaulay’s original ideas in order to include homogeous Buchsbaum ideals in a polynomial ring. Our computational methods enable us to decide if a given ideal is or is not Buchsbaum. We also obtain a new property of the Buchsbaum invariant for these ideals in terms of generating sets, which allows us to calculate this invariant.

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Sunto Nel 1913 F.S. Macaulay introdusse il concetto di ideale perfetto. Noi estendiamo l’idea originale di Macaulay per includere ideali omogenei di Buchsbaum in un anello di polinomi. I nostri metodi computazionali ci permettono di decidere se un dato ideale è o no di Buchsbaum. Inoltre otteniamo una nuova proprietà dell’invariante di Buchsbaum per questi ideali in termini degli insiemi generatori che ci permette di calcolarlo.

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This author would like to thank Massey University for financial support and the Department of Mathematics for its friendly atmosphere while finishing this paper.     

On Buchsbaum bundles on quadric hypersurfaces

Let E be an indecomposable rank two vector bundle on the projective space ℙ ⁿ , n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q _n ⊂ ℙ ⁿ⁺¹, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q ₃, k ≥ 2, we prove two boundedness results.     

Zu einem problem von H. Bresinsky über monomiale Buchsbaum-Kurven

By means of a class of examples it is shown that all nonnegative integers are assumed by the difference between the Buchsbaum invariant and the length of the semigroup ideal for monomial Buchsbaum curves. This answers a question of Bresinsky in [1].