Lattice-ordered Abelian groups and Schauder bases of unimodular fans, II

Marra, Vincenzo

doi:10.1090/S0002-9947-2013-05706-6

Lattice-ordered Abelian groups and Schauder bases of unimodular fans, II
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Trans. Amer. Math. Soc. 365 (2013), 2545-2568 Request permission

Abstract:

Unimodular fans are central to toric algebraic geometry, where they correspond to non-singular toric varieties. The Schauder bases mentioned in the title may be described as the standard bases of the free $\mathbb {Z}$-module of support functions (=invariant Cartier divisors) of a unimodular (a.k.a. regular) fan. An abstract, purely algebraic version of Schauder bases was investigated in the first part of the present paper, with motivations coming from the theory of lattice-ordered Abelian groups. The main result obtained there is that such abstract Schauder bases can be characterised in terms of the maximal spectral space of lattice-ordered Abelian groups, with no reference to polyhedral geometry. The results in the present paper will show that abstract Schauder bases can in fact be characterised in the language of lattice-ordered groups by means of an elementary algebraic notion that we call regularity, with no reference to either maximal spectral spaces or to polyhedral geometry. We prove that finitely generated projective lattice-ordered Abelian groups are precisely the lattice-ordered Abelian groups that have a finite, regular set of positive generators. This theorem complements Beynon’s well-known 1977 result that the finitely generated projective lattice-ordered Abelian groups are precisely the finitely presented ones; and the core of the proof consists in showing that finite, regular sets of positive generators are the same thing as abstract Schauder bases. We give three applications of the main result. First, we establish a necessary and sufficient criterion for the lattice-group isomorphism of two lattice-ordered Abelian groups with finite, regular sets of positive generators. Next, we classify in elementary terms (i.e. without reference to spectral spaces) all finitely generated projective lattice-ordered Abelian groups whose maximal spectrum is a closed topological surface. Finally, we show how to explicitly construct $\mathbb {Z}$-module bases of any finitely generated projective lattice-ordered Abelian group.

References

P. S. Alexandrov, Combinatorial topology. Vol. 1, 2 and 3, Dover Publications, Inc., Mineola, NY, 1998. Translated from the Russian; Reprint of the 1956, 1957 and 1960 translations. MR 1643155
Reinhold Baer, Abelian groups without elements of finite order, Duke Math. J. 3 (1937), no. 1, 68–122. MR 1545974, DOI 10.1215/S0012-7094-37-00308-9
Kirby A. Baker, Free vector lattices, Canadian J. Math. 20 (1968), 58–66. MR 224524, DOI 10.4153/CJM-1968-008-x
W. M. Beynon, Combinatorial aspects of piecewise linear functions, J. London Math. Soc. (2) 7 (1974), 719–727. MR 332617, DOI 10.1112/jlms/s2-7.4.719
W. M. Beynon, Duality theorems for finitely generated vector lattices, Proc. London Math. Soc. (3) 31 (197), 114–128. MR 376480, DOI 10.1112/plms/s3-31.1.114
W. M. Beynon, Applications of duality in the theory of finitely generated lattice-ordered abelian groups, Canadian J. Math. 29 (1977), no. 2, 243–254. MR 437420, DOI 10.4153/CJM-1977-026-4
Alain Bigard, Klaus Keimel, and Samuel Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 (French). MR 0552653
Roger D. Bleier, Archimedean vector lattices generated by two elements, Proc. Amer. Math. Soc. 39 (1973), 1–9. MR 329997, DOI 10.1090/S0002-9939-1973-0329997-9
Francis Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994. Basic category theory. MR 1291599
Francis Borceux, Handbook of categorical algebra. 2, Encyclopedia of Mathematics and its Applications, vol. 51, Cambridge University Press, Cambridge, 1994. Categories and structures. MR 1313497
Michael R. Darnel, Theory of lattice-ordered groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 187, Marcel Dekker, Inc., New York, 1995. MR 1304052
C. De Concini and C. Procesi, Complete symmetric varieties. II. Intersection theory, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 481–513. MR 803344, DOI 10.2969/aspm/00610481
Günter Ewald, Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, vol. 168, Springer-Verlag, New York, 1996. MR 1418400, DOI 10.1007/978-1-4612-4044-0
A. M. W. Glass, Partially ordered groups, Series in Algebra, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1791008, DOI 10.1142/3811
A. M. W. Glass and James J. Madden, The word problem versus the isomorphism problem, J. London Math. Soc. (2) 30 (1984), no. 1, 53–61. MR 760872, DOI 10.1112/jlms/s2-30.1.53
A. M. W. Glass and Vincenzo Marra, The underlying group of any finitely generated abelian lattice-ordered group is free, Algebra Universalis 56 (2007), no. 3-4, 467–468. MR 2318248, DOI 10.1007/s00012-007-2015-3
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
John M. Lee, Introduction to topological manifolds, Graduate Texts in Mathematics, vol. 202, Springer-Verlag, New York, 2000. MR 1759845
James Joseph Madden, TWO METHODS IN THE STUDY OF K-VECTOR LATTICES, ProQuest LLC, Ann Arbor, MI, 1984. Thesis (Ph.D.)–Wesleyan University. MR 2633496
Corrado Manara, Vincenzo Marra, and Daniele Mundici, Lattice-ordered abelian groups and Schauder bases of unimodular fans, Trans. Amer. Math. Soc. 359 (2007), no. 4, 1593–1604. MR 2272142, DOI 10.1090/S0002-9947-06-03935-3
Vincenzo Marra, Every abelian $l$-group is ultrasimplicial, J. Algebra 225 (2000), no. 2, 872–884. MR 1741567, DOI 10.1006/jabr.1999.8163
Vincenzo Marra, Weinberg’s theorem, Elliott’s ultrasimplicial property, and a characterisation of free lattice-ordered abelian groups, Forum Math. 20 (2008), no. 3, 505–513. MR 2418203, DOI 10.1515/FORUM.2008.025
Edwin E. Moise, Geometric topology in dimensions $2$ and $3$, Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977. MR 0488059
Daniele Mundici, Farey stellar subdivisions, ultrasimplicial groups, and $K_0$ of AF $C^*$-algebras, Adv. in Math. 68 (1988), no. 1, 23–39. MR 931170, DOI 10.1016/0001-8708(88)90006-0
Daniele Mundici, Free generating sets of lattice-ordered abelian groups, J. Pure Appl. Algebra 211 (2007), no. 2, 400–403. MR 2340457, DOI 10.1016/j.jpaa.2007.03.002
Daniele Mundici, Revisiting the free 2-generator abelian $l$-group, J. Pure Appl. Algebra 208 (2007), no. 2, 549–554. MR 2277694, DOI 10.1016/j.jpaa.2006.01.015
Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR 665919
Zbigniew Semadeni, Schauder bases in Banach spaces of continuous functions, Lecture Notes in Mathematics, vol. 918, Springer-Verlag, Berlin-New York, 1982. MR 653986