共查询到20条相似文献,搜索用时 46 毫秒
1.
Thomas Kahle Johannes Rauh 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2014,84(2):187-201
The toric fiber product is an operation that combines two ideals that are homogeneous with respect to a grading by an affine monoid. The Segre product is a related construction that combines two multigraded rings. The quotient ring by a toric fiber product of two ideals is a subring of the Segre product, but in general this inclusion is strict. We contrast the two constructions and show that any Segre product can be presented as a toric fiber product without changing the involved quotient rings. This allows to apply previous results about toric fiber products to the study of Segre products. We give criteria for the Segre product of two affine toric varieties to be dense in their toric fiber product, and for the map from the Segre product to the toric fiber product to be finite. We give an example that shows that the quotient ring of a toric fiber product of normal ideals need not be normal. In rings with Veronese type gradings, we find examples of toric fiber products that are always Segre products, and we show that iterated toric fiber products of Veronese ideals over Veronese rings are normal. 相似文献
2.
We study the graphs G for which their toric ideals I G are complete intersections. In particular, we prove that for a connected graph G such that I G is a complete intersection all of its blocks are bipartite except for at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. In this case, the generators of the toric ideal correspond to even cycles of G except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of these graphs satisfy the odd cycle condition. Finally, we characterize all complete intersection toric ideals of graphs which are normal. 相似文献
3.
Daniel Carando Silvia Lassalle 《Journal of Mathematical Analysis and Applications》2008,347(1):243-254
We study the existence of atomic decompositions for tensor products of Banach spaces and spaces of homogeneous polynomials. If a Banach space X admits an atomic decomposition of a certain kind, we show that the symmetrized tensor product of the elements of the atomic decomposition provides an atomic decomposition for the symmetric tensor product , for any symmetric tensor norm μ. In addition, the reciprocal statement is investigated and analogous consequences for the full tensor product are obtained. Finally we apply the previous results to establish the existence of monomial atomic decompositions for certain ideals of polynomials on X. 相似文献
4.
We study global primary decompositions in the category of sheaves on a scheme which are equivariant under the action of an algebraic group. We show that equivariant primary decompositions exist if the group is connected. As main application we consider the case of varieties which are quotients of a quasi-affine variety by the action of a diagonalizable group and thus admit a homogeneous coordinate ring, such as toric varieties. Comparing these decompositions with primary decompositions of graded modules over the homogeneous coordinate ring, we show that these are equivalent if the action of the diagonalizable group is free. We give some specific examples for the case of toric varieties. 相似文献
5.
Nathan Owen Ilten 《manuscripta mathematica》2011,134(1-2):123-137
For a complete, smooth toric variety Y, we describe the graded vector space ${T_Y^1}$ . Furthermore, we show that smooth toric surfaces are unobstructed and that a smooth toric surface is rigid if and only if it is Fano. For a given toric surface we then construct homogeneous deformations by means of Minkowski decompositions of polyhedral subdivisions, compute their images under the Kodaira-Spencer map, and show that they span ${T_Y^1}$ . 相似文献
6.
In combinatorial commutative algebra and algebraic statistics many toric ideals are constructed from graphs. Keeping the categorical structure of graphs in mind we give previous results a more functorial context and generalize them by introducing the ideals of graph homomorphisms. For this new class of ideals we investigate how the topology of the graphs influences the algebraic properties. We describe explicit Gröbner bases for several classes, generalizing results by Hibi, Sturmfels, and Sullivant. One of our main tools is the toric fiber product, and we employ results by Engström, Kahle, and Sullivant. The lattice polytopes defined by our ideals include important classes in optimization theory, as the stable set polytopes. 相似文献
7.
Using a generalized notion of matching in a simplicial complex and circuits of vector configurations, we compute lower bounds for the minimum number of generators, the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Prime lattice ideals are toric ideals, i.e. the defining ideals of toric varieties. 相似文献
8.
Markus Perling 《Geometriae Dedicata》2007,127(1):121-129
We describe the construction of a class of toric varieties as spectra of homogeneous prime ideals.
相似文献
9.
P. Jipsen 《Annals of Pure and Applied Logic》2009,161(2):228-234
It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FLw-algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras are represented as Esakia products of simple n-potent MV-algebras. 相似文献
10.
Shunsuke Takagi Kei-ichi Watanabe 《Transactions of the American Mathematical Society》2004,356(10):3951-3961
Demailly, Ein and Lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals. We prove that, in the two-dimensional case, the subadditivity theorem holds on log terminal singularities. However, in the higher dimensional case, we have several counterexamples. We consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three-dimensional toric ring.
11.
Thomas Kahle 《Annals of the Institute of Statistical Mathematics》2010,62(4):727-745
We present Binomials, a package for the computer algebra system Macaulay 2, which specializes well-known algorithms to binomial ideals. These
come up frequently in algebraic statistics and commutative algebra, and it is shown that significant speedup of computations
like primary decomposition is possible. While central parts of the implemented algorithms go back to a paper of Eisenbud and
Sturmfels, we also discuss a new algorithm for computing the minimal primes of a binomial ideal. All decompositions make significant
use of combinatorial structure found in binomial ideals, and to demonstrate the power of this approach we show how Binomials was used to compute primary decompositions of commuting birth and death ideals of Evans et al., yielding a counterexample
for their conjectures. 相似文献
12.
Nik Stopar 《Semigroup Forum》2012,85(2):322-336
In this paper we investigate the ascending chain conditions on principal left and right ideals for semidirect products of semigroups and show how this is connected to the corresponding problem for rings of skew generalized power series. Let S be a left cancellative semigroup with a unique idempotent e, T a right cancellative semigroup with an idempotent f and $\omega: T \to \operatorname {End}(S)$ a semigroup homomorphism such that ??(f)=id S . We show that in this case the semidirect product S? ?? T satisfies the ascending chain condition for principal left ideals (resp. right ideals) if and only if S and T satisfy the ascending chain condition for principal left ideals (resp. right ideals and $\operatorname {Im}\omega(t)$ is closed for complete inverses for all t??T). We also give several examples to show that for more general semigroups these implications may not hold. 相似文献
13.
There are two seemingly unrelated ideals associated with a simplicial complex Δ: one is the Stanley–Reisner ideal I Δ , the monomial ideal generated by minimal non-faces of Δ, well-known in combinatorial commutative algebra; the other is the toric ideal I M(Δ) of the facet subring of Δ, whose generators give a Markov basis for the hierarchical model defined by Δ, playing a prominent role in algebraic statistics. In this note we show that the complexity of the generators of I M(Δ) is determined by the Betti numbers of I Δ . The unexpected connection between the syzygies of the Stanley–Reisner ideal and degrees of minimal generators of the toric ideal provide a framework for further exploration of the connection between the model and its many relatives in algebra and combinatorics. 相似文献
14.
V. V. Bavula 《Ukrainian Mathematical Journal》1993,45(3):329-334
For generalized Weyl algebras containing the universal enveloping algebra Usl (2,K) of the Lie algebra sl (2) over a field with characteristic zero, bilateral ideals are classified. We show that a product of ideals is commutative and any proper ideal can be uniquely decomposed into a product of primary ideals. 相似文献
15.
We give explicit formulas for the Bhattacharya function of 𝔪-primary complete monomial ideals in two variables in terms of the vertices of the Newton polyhedra or in terms of the decompositions of the ideals as products of simple ideals. 相似文献
16.
17.
We introduce the concept of Stanley decompositions in the localized polynomial ring S f where f is a product of variables, and we show that the Stanley depth does not decrease upon localization. Furthermore it is shown that for monomial ideals ${J \subset I \subset S_f}$ the number of maximal Stanley spaces in a Stanley decomposition of I/J is an invariant of I/J. For the proof of this result we introduce Hilbert series for ${\mathbb{Z}^n}$ -graded K-vector spaces. 相似文献
18.
Nathan Owen Ilten 《Journal of Pure and Applied Algebra》2009,213(6):1086-1096
In the case of two-dimensional cyclic quotient singularities, we classify all one-parameter toric deformations in terms of certain Minkowski decompositions introduced by Altmann [Minkowski sums and homogeneous deformations of toric varieties, Tohoku Math. J. (2) 47 (2) (1995) 151-184.]. In particular, we show how to induce each deformation from a versal family, describe exactly to which reduced versal base space components each such deformation maps, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions. 相似文献
19.
Laura Felicia Matusevich Christopher ONeill 《Journal of Pure and Applied Algebra》2019,223(1):380-394
Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing “mesoprimary decompositions” determined by their underlying monoid congruences. Monoid congruences (and therefore, binomial ideals) can present many subtle behaviors that must be carefully accounted for in order to produce general results, and this makes the theory complicated. In this paper, we examine their results in the presence of a positive A-grading, where certain pathologies are avoided and the theory becomes more accessible. Our approach is algebraic: while key notions for mesoprimary decomposition are developed first from a combinatorial point of view, here we state definitions and results in algebraic terms, which are moreover significantly simplified due to our (slightly) restricted setting. In the case of toral components (which are well-behaved with respect to the A-grading), we are able to obtain further simplifications under additional assumptions. We also provide counterexamples to two open questions, identifying (i) a binomial ideal whose hull is not binomial, answering a question of Eisenbud and Sturmfels, and (ii) a binomial ideal I for which is not binomial, answering a question of Dickenstein, Miller and the first author. 相似文献
20.
A. A. Tuganbaev 《Mathematical Notes》1996,60(2):186-203
A module is called distributive (is said to be a chain module) if the lattice of all its submodules is distributive (is a chain). Let a ringA be a finitely generated module over its unitary central subringR. We prove the equivalence of the following conditions:
- A is a right or left distributive semiprime ring;
- for any maximal idealM of a subringR central inA, the ring of quotientsA M is a finite direct product of semihereditary Bézout domains whose quotient rings by the Jacobson radicals are finite direct products of skew fields;
- all right ideals and all left ideals of the ringA are flat (right and left) modules over the ringA, andA is a distributive ring, without nonzero nilpotent elements, all of whose quotient rings by prime ideals are semihereditary orders in skew fields.