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 \(\Bbbk \)-schemes that are diagonalizable, for any field \(\Bbbk \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The term monoid \(\Bbbk \)-scheme is also used in the literature for a different, but related, concept (cf. [6]).
If X is not commutative, then \(V^*\) is a linear representation of the opposite monoid \(X^{op}\), or a linear representation on the right of the monoid X.
References
Álvarez, A., Sancho, C., Sancho, P.: Algebra schemes and their representations. J. Algebra 296(1), 110–144 (2006)
Brion, M.: Local structure of algebraic monoids. Mosc. Math. J. 8, 647–666 (2008)
Brion, M.: On algebraic semigroups and monoids. In: Can, M., Li, Z., Steinberg, B., Wang, Q. (eds.) Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics. Fields Institute Communications, vol. 71. Springer, New York (2014)
Brion, M.: On algebraic semigroups and monoids, II. Semigroup Forum 88, 250–272 (2014)
Campillo, A., Pisón, P.: Toric mathematics from semigroup viewpoint in ring theory and algebraic geometry, Ch. 5. In: Granja, A., Hermida, J.A., Verschoren, A. (eds.) Series: Lecture Notes in Pure and Applied Mathematics, vol. 221, pp. 95–112. Marcel Dekker, New York (2001)
Cortiñas, G., Haesemeyer, C., Walker, M.E., Weibel, C.: Toric varieties, monoid schemes and cdh-descent. J. Reine Angew. Math. 698, 1–54 (2015)
Cox, D., Little, B., Schenk, H.: Toric varieties, Graduate Studies in Mathematics, vol. 124. The American Mathematical Society, Providence (2011)
D’Anna, M., García-Sánchez, P.A., Micale, V., Tozzo, L.: Good subsemigroups of \(\mathbb{N}^n\). Int. J. Algebra Comput. 28(2), 179–206 (2018)
Demazure, M., Gabriel, P.: Introduction to Algebraic Geometry and Algebraic Groups. Mathematics Studies 39. North-Holland, Amsterdam (1980)
Eisenbud, D., Sturmfels, B.: Binomial ideals. Duke Math. J. 84(1), 1–45 (1996)
Kahle, T., Miller, E.: Decompositions of commutative monoid congruences and binomial ideals. Algebra Number Theory 8(6), 1297–1364 (2014)
Matusevich, L.F., Ojeda, I.: Binomial ideals and congruences on \(\mathbb{N}^n\). In: Greuel, G.M., et al. (eds.) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics Festschrift for Antonio Campillo on the Occasion of his 65th Birthday, pp. 429–454. Springer, Berlin (2018)
Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Vol. 227 of Graduate Texts in Mathematics. Springer, New York (2005)
Neeb, K.-H.: Toric varieties and algebraic monoids. Sem. Sophus Lie 2, 159–187 (1992)
Putcha, M.: Linear Algebraic Monoids, London Mathematical Society Lecture Note, vol. 133. Cambridge University Press, Cambridge (1988)
Renner, L.: Linear Algebraic Monoids, Encyclopedia of Mathematics, vol. 134. Springer, Berlin (2005)
Rosales, J.C., García-Sánchez, P.A.: Finitely Generated Commutative Monoids. Nova Science Publishers Inc., New York (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jan Okninski.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
Rights and permissions
About this article
Cite this article
Navarro, A., Navarro, J. & Ojeda, I. Commutative monoids and their corresponding affine \(\Bbbk \)-schemes. Semigroup Forum 101, 421–434 (2020). https://doi.org/10.1007/s00233-019-10069-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-019-10069-2