Sheaves and \(K\)-theory for \(\mathbb F_1\)-schemes. (English) Zbl 1288.19004
In this paper a theory of sheaves and \(K\)-theory of \({\mathbb F}_1\) s in the sense of A. Connes and C. Consani [Compos. Math. 146, No. 6, 1383–1415 (2010; Zbl 1201.14001)] is developed. This extends the corresponding notions for monoid schemes as given in A. Deitmar [Proc. Japan Acad., Ser. A 82, No. 8, 141–146 (2006; Zbl 1173.14004)]. Special attention is paid to normal morphisms and locally projective sheaves, which occur when Quillen’s Q-construction is adopted to a definition of \(G\)-theory and \(K\)-theory of \({\mathbb F}_1\)-schemes. A comparison with Waldhausen’s \(S_\bullet\)-construction yields the ring structure of \(K\)-theory. The paper finishes with some explicit computations of the groups \(K_0\) and \(G_0\).
Reviewer: Anton Deitmar (Tübingen)
MSC:
| 19E08 | \(K\)-theory of schemes |
| 14A15 | Schemes and morphisms |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 55P43 | Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) |
References:
| [1] | Atiyah, M. F.; MacDonald, I. G., Introduction to Commutative Algebra (1969), Addison-Wesley · Zbl 0175.03601 |
| [2] | Barratt, M., A free group functor for stable homotopy, Proc. Sympos. Pure Math., 22, 31-35 (1971) · Zbl 0242.55015 |
| [3] | Borceux, F., Handbook of Categorical Algebra 2, Encyclopedia Math. Appl., vol. 51 (1994) · Zbl 0843.18001 |
| [4] | Gunnar Carlsson, Christopher L. Douglas, Bjorn Ian Dundas, Higher topological cyclic homology and Segal conjecture for tori, preprint, arXiv:0803.2745 [math.AT]; Gunnar Carlsson, Christopher L. Douglas, Bjorn Ian Dundas, Higher topological cyclic homology and Segal conjecture for tori, preprint, arXiv:0803.2745 [math.AT] · Zbl 1223.55004 |
| [5] | Connes, A.; Consani, C., Schemes over \(F_1\) and zeta functions, Compos. Math., 146, 6, 1383-1415 (2010) · Zbl 1201.14001 |
| [6] | Deitmar, A., Schemes over \(F_1\), (Number Fields and Function Fields—Two Parallel Worlds. Number Fields and Function Fields—Two Parallel Worlds, Progr. Math., vol. 239 (2005)), 87-100 · Zbl 1098.14003 |
| [7] | Deitmar, A., Remarks on zeta functions and \(K\)-theory over \(F_1\), Proc. Japan Acad. Ser. A Math. Sci., 82, 141-146 (2006) · Zbl 1173.14004 |
| [8] | Deitmar, A., \(F_1\)-schemes and toric varieties, Contrib. Algebra Geom., 49, 2, 517-525 (2008) · Zbl 1152.14001 |
| [9] | Dwyer, W. G.; Friedlander, E. M.; Mitchell, S. A., The generalized Burnside ring and the \(K\)-theory of ring with roots of unity, \(K\)-Theory, 6, 285-300 (1992) · Zbl 0780.19001 |
| [10] | Elmendorf, A. D.; Mandell, M. A., Rings, modules, and algebras in infinite loop space theory, Adv. Math., 205, 163-228 (2006) · Zbl 1117.19001 |
| [11] | Geisser, Thomas, Topological cyclic homology of schemes, (Algebraic \(K\)-Theory. Algebraic \(K\)-Theory, Seattle, WA, 1997. Algebraic \(K\)-Theory. Algebraic \(K\)-Theory, Seattle, WA, 1997, Proc. Sympos. Pure Math. (1999)) · Zbl 0953.19001 |
| [12] | Gilmer, R., Commutative Semigroup Rings (1984), Univ. of Chicago Press: Univ. of Chicago Press Chicago · Zbl 0566.20050 |
| [13] | Haran, M. J. Shai, Non-additive geometry, Compos. Math., 143, 618-688 (2007) · Zbl 1213.11132 |
| [14] | Hartshorne, R., Algebraic Geometry, Grad. Texts in Math., vol. 52 (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0367.14001 |
| [15] | Hovey, Mark; Shiplely, Brooke; Smith, Jeff, Symmetric spectra, J. Amer. Math. Soc., 13, 149-208 (2000) · Zbl 0931.55006 |
| [16] | Hüttemann, T., Algebraic \(K\)-theory of non-linear projective spaces, J. Pure Appl. Algebra, 170, 185-242 (2002) · Zbl 0993.19001 |
| [17] | Hüttemann, T.; Klein, J.; Vogell, W.; Waldhausen, F.; Williams, B., The fundamental theorem for the algebraic \(K\)-theory of spaces I, J. Pure Appl. Algebra, 160, 21-52 (2001) · Zbl 0982.19001 |
| [18] | M. Kapranov, A. Smirnov, Cohomology determinants and reciprocity laws: number field case, unpublished preprint.; M. Kapranov, A. Smirnov, Cohomology determinants and reciprocity laws: number field case, unpublished preprint. |
| [19] | Kato, K., Toric singularities, Amer. J. Math., 116, 5, 1073-1099 (1994) · Zbl 0832.14002 |
| [20] | Kilp, M.; Knauer, U.; Mikhalev, A., Monoids, Acts and Categories: With Applications to Wreath Products and Graphs, de Gruyter Exp. Math., vol. 29 (2000), Walter de Gruyter: Walter de Gruyter Berlin · Zbl 0945.20036 |
| [21] | Knauer, U., Projectivity of acts and Morita equivalence of monoids, Semigroup Forum, 3, 359-370 (1972) · Zbl 0231.18013 |
| [22] | Lewis, L. G.; May, J. P.; Steinberger, M.; McClure, J. E., Equivariant Stable Homotopy Theory, Lecture Notes in Math., vol. 1213 (1996), Springer-Verlag |
| [23] | López Peña, J.; Lorscheid, O., Torified varieties and their geometries over \(F_1\), Math. Z., 267, 3, 605-643 (2011) · Zbl 1220.14021 |
| [24] | López Peña, J.; Lorscheid, O., Mapping \(F_1\)-land: an overview of geometries over the field with one element, (Consani, Caterina; Connes, Alain, Noncommutative Geometry, Arithmetic, and Related Topics: Proceedings of the Twenty-First Meeting of the Japan-U.S. Mathematics Institute (2012), The Johns Hopkins University Press) · Zbl 1271.14003 |
| [25] | O. Lorscheid, Algebraic groups over the field with one element, Math. Z., doi:10.1007/s00209-011-0855-1arXiv:0907.3824 [math.AG]; O. Lorscheid, Algebraic groups over the field with one element, Math. Z., doi:10.1007/s00209-011-0855-1arXiv:0907.3824 [math.AG] |
| [26] | Manin, Y. I., Lectures on zeta functions and motives (according to Deninger and Kurokawa), Columbia University Number-Theory Seminar (1992). Columbia University Number-Theory Seminar (1992), Astérisque, 228, 4, 121-163 (1995) · Zbl 0840.14001 |
| [27] | Priddy, S., On \(\Omega^\infty \Sigma^\infty\) and the infinite symmetric group, Proc. Sympos. Pure Math., 22, 217-220 (1971) · Zbl 0242.55011 |
| [28] | Quillen, D., Higher algebraic \(K\)-theory I, (Algebraic \(K\)-Theory I: Higher \(K\)-Theories. Algebraic \(K\)-Theory I: Higher \(K\)-Theories, Seattle, 1972. Algebraic \(K\)-Theory I: Higher \(K\)-Theories. Algebraic \(K\)-Theory I: Higher \(K\)-Theories, Seattle, 1972, Lecture Notes in Math., vol. 341 (1973), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0292.18004 |
| [29] | Quillen, D., Letter to J. Milnor, (Lecture Notes in Math., vol. 551 (1976), Springer-Verlag: Springer-Verlag New York), 135-158 · Zbl 0351.55003 |
| [30] | Segal, G. B., Categories and cohomology theories, Topology, 13, 213-221 (1974) |
| [31] | Soulé, C., Les variétés sur le corps à un élément, Mosc. Math. J., 4, 217-244 (2004) · Zbl 1103.14003 |
| [32] | M. Szczesny, On the Hall algebra of coherent sheaves on \(\mathbb{P}^1\mathbb{F}_1\) arXiv:1009.3570v3; M. Szczesny, On the Hall algebra of coherent sheaves on \(\mathbb{P}^1\mathbb{F}_1\) arXiv:1009.3570v3 |
| [33] | J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, in: Colloque dʼalgèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956 (1957), pp. 261-289.; J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, in: Colloque dʼalgèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956 (1957), pp. 261-289. · Zbl 0084.15902 |
| [34] | tom Dieck, Tammo, Orbittypen und äquivariante Homologie. II, Arch. Math. (Basel), 26, 650-662 (1975) · Zbl 0334.55004 |
| [35] | Waldhausen, F., Algebraic \(K\)-theory of spaces, (Algebraic and Geometric Topology. Algebraic and Geometric Topology, New Brunswick, NJ, 1983. Algebraic and Geometric Topology. Algebraic and Geometric Topology, New Brunswick, NJ, 1983, Lecture Notes in Math., vol. 1126 (1985), Springer-Verlag: Springer-Verlag Berlin), 318-419 · Zbl 0579.18006 |
| [36] | Weibel, C., A survey of products in algebraic \(K\)-theory, (Algebraic \(K\)-Theory. Algebraic \(K\)-Theory, Evanston, 1980. Algebraic \(K\)-Theory. Algebraic \(K\)-Theory, Evanston, 1980, Lecture Notes in Math., vol. 854 (1981), Springer-Verlag), 494-517 · Zbl 0482.18004 |
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