Skip to main content Accessibility help
×
×
Home

Topos points of quasi-coherent sheaves over monoid schemes

  • ILIA PIRASHVILI (a1)
Abstract

Let X be a monoid scheme. We will show that the stalk at any point of X defines a point of the topos of quasi-coherent sheaves over X. As it turns out, every topos point of is of this form if X satisfies some finiteness conditions. In particular, it suffices for M/M× to be finitely generated when X is affine, where M× is the group of invertible elements.

This allows us to prove that two quasi-projective monoid schemes X and Y are isomorphic if and only if and are equivalent.

The finiteness conditions are essential, as one can already conclude by the work of A. Connes and C. Consani [3]. We will study the topos points of free commutative monoids and show that already for ℕ, there are ‘hidden’ points. That is to say, there are topos points which are not coming from prime ideals. This observation reveals that there might be a more interesting ‘geometry of monoids’.

Copyright
References
Hide All
[1] Abramovich, D., Chen, Q., Gillam, D., Huang, Y., Olsson, M., Satriano, M., and Sun, S.. Logarithmic geometry and moduli. arXiv:1006.5870. (2010).
[2] Chu, C., Lorscheid, O. and Santhanam, R.. Sheaves and K-theory for -schemes. Adv. Math. 229, (2012), p. 22392286.
[3] Connes, A.. and Consani, C.. Geometry of the arithmetic site. Adv. Math. 291, (2016), p. 274329.
[4] Connes, A.. and Consani, C.. Schemes over and zeta functions. Compositio Mathematica 146.6, (2010), p. 13831415.
[5] Cortinas, G., Haesemeyer, C., Walker, M. E. and Weibel, C.. Toric variaties, monoid schemes and cdh descent. J. für die Reine u Angewan. Math. 698 (2015), p. 154.
[6] Deitmar, A.. Schemes over . In: Number fields and function field. Two parallel worlds. Ed. by van der Geer, G., Moonen, B., Schoof, R.. Progr. in Math. (2005), 236.
[7] Gabriel, P.. Des catégories abéliennes. Bull. Soc. Math. France 90 (1962), p. 323448.
[8] Garkusha, G. and Prest, M.. Reconstructing projective schemes from Serre subcategories. J. Algebra 319 no. 3 (2008), p. 11321153.
[9] Gilmer, R.. Commutative Semigroup Rings. (The University of Chicago Press 1984).
[10] Kato, K.. Toric singularities. Ameri. J. Math. 11 6. (1994), p. 10731099.
[11] Kato, K.. Logarithmic structures of Fontaine-Illusie, in Algebraic Analysis, Geometry, and Number Theory. Proceedings of the JAMI Inaugural Conference. Supplement to Amer. J. Math., (1989), p. 191224.
[12] Mac Lane, S. and Moerdijk, I.. Sheaves in Geometry and Logic. A First Introduction to Topos Theory. Corrected reprint of the 1992 edition. Universitext. (Springer-Verlag, New York, 1994).
[13] Moerdijk, I.. Classifying Spaces and Classifying Topoi. Lecture Notes in Mathe. vol. 1616, Springer-Verlag, Berlin, (1995).
[14] Pin, J.E.. Tropical semirings. Idempotency (Bristol, 1994) Publ. Newton Inst., 11, (Cambridge University Press, Cambridge, 1998), p. 5059.
[15] Pirashvili, I.. On the spectrum of monoids and semilattices. J. Pure Appl. Algebra 217 (2013), p. 901906.
[16] Pirashvili, I.. On cohomology and vector bundles over monoid schemes. J. Algebra 435 (2015), p. 3351.
[17] Rosenberg, A. L.. The spectrum of abelian categories and reconstruction of schemes. Rings, Hopf algebras and Brauer groups (Antwerp/Brussels 1996) Lecture Notes in Pure and Appl. Math. 197 (1998), p. 257274.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed