Commutative monoids and their corresponding affine $$\Bbbk $$ k -schemes
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Commutative monoids and their corresponding affine k-schemes Alberto Navarro1 · José Navarro2 · Ignacio Ojeda2 Received: 1 August 2019 / Accepted: 30 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this expository note, we give a self-contained presentation of the equivalence between the opposite category of commutative monoids and that of commutative, monoid k-schemes that are diagonalizable, for any field k. Keywords Commutative monoids · Binomial ideals · Congruences
Introduction One of the highlights of toric geometry is that an affine toric variety over a field k is determined by purely combinatorial data; namely, it can be reconstructed by its set of characters, that is a finitely generated, free of torsion and commutative monoid. This point of view has been fruitfully exploited for a long time, allowing for a great development both in the study of commutative monoids and in toric geometry [5,7,8,10,11]. More recently, the analogous relation between more general binomial ideals -not necessarily toric- and their corresponding monoids of characters is also being used in a variety of topics (see [12] and references therein). In fact, an affine toric variety is a
Communicated by Jan Okninski. The second author is partially supported by Junta de Extremadura and FEDER funds, IB18087. The third author is partially supported by Junta de Extremadura, MTM2015-65764-C3-1-P and MTM2017-84890-P.
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José Navarro [email protected] Alberto Navarro [email protected] Ignacio Ojeda [email protected]
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Institut für Mathematik, Universität Zürich, Zurich, Switzerland
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Departamento de Matemáticas, Universidad de Extremadura, Badajoz, Spain
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particular example of a commutative, algebraic k-monoid, and these correspondences are particular cases of a duality that is valid, more generally, for monoid k-schemes that are commutative and diagonalizable (Chapter II, §1, Sections 2.10 and 2.11 of [9]). Nevertheless, this duality seems to be little known among those who approach these objects from a combinatorial point of view, and the aim of this expository note is to provide a self-contained presentation of this correspondence. To be more precise, fix an arbitrary field k. Observe that any commutative monoid S produces an affine k-scheme X S := Spec k[S], which is a commutative monoid k-scheme, i.e., a monoid object in the category of affine k-schemes. Conversely, the set of characters of a commutative monoid k-scheme has the structure of a commutative monoid. The main theorem we prove in this note (Theorem 19) assures that these assignments establish an equivalence between the opposite category of commutative monoids and the category of commutative monoid k-schemes whose linear representations are direct sums of one dimensional representations. Then, it easily follows, as particular cases of this equivalence, the aforementioned relations between affine toric varieties, or more general binomial ideals, with their corresponding monoids
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