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39
On presentations of commutative monoids
 INT. J. ALGEBRA COMPUT
, 1999
"... In this paper, we introduce the concept of a strongly reduced monoid and we characterize the minimal presentations for such monoids. As a consequence, we give a method to obtain a presentation for any commutative monoid. ..."
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In this paper, we introduce the concept of a strongly reduced monoid and we characterize the minimal presentations for such monoids. As a consequence, we give a method to obtain a presentation for any commutative monoid.
Polytopal linear retractions
 Trans. Amer. Math. Soc
"... Abstract. We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension 1 retractions factor ..."
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Abstract. We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension 1 retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general. This paper is a part of the project started in [BG1, BG2], where we have investigated the graded automorphism groups of polytopal algebras. Part of the motivation comes from the observation that there is a reasonable ‘polytopal ’ generalization of linear algebra (and, subsequently, that of algebraic Ktheory). 1.
Combinatorics of binomial primary decomposition
, 2008
"... An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristicfree combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely ..."
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An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristicfree combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup. These results are intimately connected to hypergeometric differential equations in several variables.
Consani Characteristic 1, entropy and the absolute point
, 1997
"... We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point Spec F1. After introducing the notion of “perfect” semiring of characteristi ..."
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We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point Spec F1. After introducing the notion of “perfect” semiring of characteristic one, we explain how to adapt the construction of the Witt ring in characteristic p> 1 to the limit case of characteristic one. This construction also unveils an interesting connection with entropy and thermodynamics, while shedding a new light on the classical Witt construction itself. We simplify our earlier construction of the geometric realization of an F1scheme and extend our earlier computations of the zeta function to cover the case of F1schemes with torsion. Then, we show that the study of the additive structures on monoids provides a natural map M ↦ → A(M) from monoids to sets which comes close to fulfill the requirements for the hypothetical curve Spec Z over the absolute point Spec F1. Finally, we test the computation of the zeta function on elliptic
ON THE ARITHMETIC OF KRULL MONOIDS WITH INFINITE CYCLIC CLASS GROUP
, 2009
"... Let H be a Krull monoid with infinite cyclic class group G and let GP ⊂ G denote the set of classes containing prime divisors. We study under which conditions on GP some of the main finiteness properties of factorization theory—such as local tameness, the finiteness and rationality of the elasticit ..."
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Let H be a Krull monoid with infinite cyclic class group G and let GP ⊂ G denote the set of classes containing prime divisors. We study under which conditions on GP some of the main finiteness properties of factorization theory—such as local tameness, the finiteness and rationality of the elasticity, the structure theorem for sets of lengths, the finiteness of the catenary degree, and the existence of monotone and of near monotone chains of factorizations—hold in H. In many cases, we derive explicit characterizations.
An algorithm for the QuillenSuslin theorem for monoid rings
 J. PURE APPL. ALGEBRA 117
, 1997
"... Let k be a field, and let M be a commutative, seminormal, finitely generated monoid, which is torsionfree, cancellative, and has no nontrivial units. J. Gubeladze proved that finitely generated projective modules over kM are free. This paper contains an algorithm for finding a free basis for a finit ..."
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Cited by 5 (1 self)
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Let k be a field, and let M be a commutative, seminormal, finitely generated monoid, which is torsionfree, cancellative, and has no nontrivial units. J. Gubeladze proved that finitely generated projective modules over kM are free. This paper contains an algorithm for finding a free basis for a finitely generated projective module over kM. As applications one obtains alternative algorithms for the QuillenSuslin Theorem for polynomial rings and Laurent polynomial rings, based on Quillen’s proof.
Homological aspects of semigroup gradings on rings and algebras
 MR 1701322, Zbl 0934.16038
, 1999
"... Abstract. This article studies algebras R over a simple artinian ring A, presented by a quiver and relations and graded by a semigroup . Suitable semigroups often arise from a presentation of R. Throughout, the algebras need not be finite dimensional. The graded K0, along with the graded Cartan end ..."
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Abstract. This article studies algebras R over a simple artinian ring A, presented by a quiver and relations and graded by a semigroup . Suitable semigroups often arise from a presentation of R. Throughout, the algebras need not be finite dimensional. The graded K0, along with the graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties. A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert series in the associated path incidence ring. The rationality of theEuler characteristic, the Hilbertseries and the PoincaréBettiseries is studied when is torsionfree commutative and A is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.
Monoids of IGtype and maximal orders
 J. ALGEBRA
, 2006
"... Let G be a finite group that acts on an abelian monoid A. If φ: A → G is a map so that φ(aφ(a)(b)) = φ(a)φ(b), for all a, b ∈ A, then the submonoid S = {(a, φ(a))  a ∈ A} of the associated semidirect product A⋊G is said to be a monoid of IGtype. If A is a finitely generated free abelian monoid o ..."
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Let G be a finite group that acts on an abelian monoid A. If φ: A → G is a map so that φ(aφ(a)(b)) = φ(a)φ(b), for all a, b ∈ A, then the submonoid S = {(a, φ(a))  a ∈ A} of the associated semidirect product A⋊G is said to be a monoid of IGtype. If A is a finitely generated free abelian monoid of rank n and G is a subgroup of the symmetric group Sym n of degree n, then these monoids first appeared in the work of GatevaIvanova and Van den Bergh (they are called monoids of Itype) and later in the work of Jespers and Okniński. It turns out that their associated semigroup algebras share many properties with polynomial algebras in finitely many commuting variables. In this paper we first note that finitely generated monoids S of IGtype are epimorphic images of monoids of Itype and their algebras K[S] are Noetherian and satisfy a polynomial identity. In case the group of fractions SS −1 of S also is torsionfree then it is characterized when K[S] also is a maximal order. It turns out that they often are, and hence these algebras again share arithmetical properties with natural classes of commutative algebras. The characterization is in terms of prime ideals of S, in particular Gorbits of minimal prime ideals in A play a crucial role. Hence, we first describe the prime ideals of S. It also is described when the group SS −1 is torsionfree.
OVERWEIGHT DEFORMATIONS OF AFFINE TORIC VARIETIES AND LOCAL UNIFORMIZATION
"... Abstract. Given an equicharacteristic complete noetherian local ring R with algebraically closed residue field k, we first present a combinatorial proof of embedded local uniformization for zerodimensional valuations of R whose associated graded ring grνR with respect to the filtration defined by t ..."
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Abstract. Given an equicharacteristic complete noetherian local ring R with algebraically closed residue field k, we first present a combinatorial proof of embedded local uniformization for zerodimensional valuations of R whose associated graded ring grνR with respect to the filtration defined by the valuation is a finitely generated kalgebra. The main idea here is that some of the birational toric maps which provide embedded pseudoresolutions for the affine toric variety corresponding to grνR also provide local uniformizations for ν on R. These valuations are necessarily Abhyankar (for zerodimensional valuations this means that the value group is Zr with r = dimR). In a second part we show that conversely, given an excellent noetherian equicharacteristic local domain R with algebraically closed residue field, if the zerodimensional valuation ν of R is Abhyankar, there are local domains R ′ which are essentially of finite type over R and dominated by the valuation ring Rν (νmodifications of R) such that the semigroup of values of ν on R ′ is finitely generated, and therefore so is the kalgebra grνR ′. Combining the two results and using the fact that Abhyankar valuations behave well under completion gives a proof of local uniformization for rational Abhyankar valuations and, by a specialization argument, for all Abhyankar valuations. As a byproduct we obtain a description of the valuation ring of a rational Abhyankar valuation as an inductive limit indexed by N of birational toric maps of regular local rings. One of our main tools, the valuative Cohen theorem, is then used to study the extensions of rational monomial Abhyankar valuations of the ring k[[x1,..., xr]] to monogenous integral extensions and the nature of their key polynomials. In the conclusion we place the results in the perspective of local embedded resolution of singularities by a single toric modification after an appropriate reembedding. 2000 Mathematics Subject Classification. 14M25, 14E15, 14B05. Key words and phrases. Toric geometry, Valuations, Key polynomials. 1 ha l0