\(K\)-theory of toric varieties revisited. (English) Zbl 1312.19002
From the author’s abstract: “After surveying higher \(K\)-theory of toric varieties, we present Totaro’s old (c.1997) unpublished result (see B. Totaro [Forum Math. Sigma 2, Article ID e17, 25 p. (2014; Zbl 1329.14018)]) on expressing the corresponding homotopy theory via singular cohomology. It is a higher analog of the rational Chern character isomorphism for general toric schemes. In the special case of a projective simplicial toric scheme over a regular ring one obtains a rational isomorphism between the homotopy K-theory and the direct sum of \(m\) copies of the \(K\)-theory of the ground ring, \(m\) being the number of maximal cones in the underlying fan.”
Reviewer: Boris Goldfarb (Albany)
MSC:
| 19L47 | Equivariant \(K\)-theory |
| 19D35 | Negative \(K\)-theory, NK and Nil |
| 14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
Keywords:
toric variety; homotopy \(K\)-theory; Adams operations; nilpotence; Mayer-Vietoris spectral sequence; singular cohomologyCitations:
Zbl 1329.14018References:
| [1] | Arapura, P., Bakhtary, D., Wlodarczyk, J.: Weights on cohomology, invariants of singularities, and dual complexes. Preprint. http://arxiv.org/abs/1102.4370 · Zbl 1280.14005 |
| [2] | Bass, H.: Algebraic \[K\]-theory. W. A. Benjamin, Inc., New York/Amsterdam (1968) · Zbl 0174.30302 |
| [3] | Bruns, W., Gubeladze, J.: Polytopes, rings, and \[K\]-theory. Springer Monographs in Mathematics. Springer, Dordrecht (2009) · Zbl 1168.13001 |
| [4] | Cisinski, D.-C.: Descente par éclatements en \[K\]-thórie invariante par homotopie. Ann. Math. (to appear) · Zbl 1264.19003 |
| [5] | Cortiñas, G., Haesemeyer, C., Walker, M.E., Weibel, C.: The \[K\]-theory of toric varieties. Trans. Am. Math. Soc. 361, 3325-3341 (2009) · Zbl 1170.19001 |
| [6] | Cortiñas, G., Haesemeyer, C., Walker, M.E., Weibel, C.: The \[{K}\]-theory of toric varieties in positive characteristic. Preprint. http://arxiv.org/abs/1207.2891 · Zbl 1309.14017 |
| [7] | Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. In: Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011) · Zbl 1223.14001 |
| [8] | Deligne, P: Poids dans la cohomologie des variétés algébriques. In: Proceedings of the International Congress of Mathematicians, vol. 1, pp. 79-85. Vancouver B. C. (1974) · Zbl 0334.14011 |
| [9] | Fulton, W., MacPherson, R., Sottile, F., Sturmfels, B.: Intersection theory on spherical varieties. J. Algebraic Geom. 4, 181-193 (1995) · Zbl 0819.14019 |
| [10] | Fulton, W.: Intersection theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 2. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2nd edn. Springer, Berlin (1998) · Zbl 0885.14002 |
| [11] | Geller, S., Reid, L., Weibel, C.: The cyclic homology and \[K\]-theory of curves. J. Reine Angew. Math. 393, 39-90 (1989) · Zbl 0649.14006 |
| [12] | Gillet, H., Soulé, C.: Filtrations on higher algebraic \[K\]-theory. In: Algebraic \[K\]-theory, Seattle, 1997. Proceedings of the Symposium on Pure Mathematics, vol. 67, pp. 89-148 (1999) · Zbl 0951.19003 |
| [13] | Gubeladz, J.: The Anderson conjecture and a maximal class of monoids over which projective modules are free. Mat. Sb. (N.S.). 135(177), 169-185 (1988) · Zbl 0654.13013 |
| [14] | Gubeladze, J: Classical algebraic \[K\]-theory of monoid algebras. \[In: K\]-Theory and Homological Algebra, Tbilisi, 1987-88. Lecture Notes in Mathematics, vol. 1437, pp. 36-94. Springer, Berlin (1990) · Zbl 0731.19001 |
| [15] | Gubeladze, J.: Nontriviality of \[SK_1(R[M])\]. J. Pure Appl. Algebra. 104, 169-190 (1995) · Zbl 0840.19002 · doi:10.1016/0022-4049(94)00130-1 |
| [16] | Gubeladze, J.: Higher \[K\]-theory of toric varieties. \[K\]-Theory. 28, 285-327 (2003) · Zbl 1039.19001 |
| [17] | Gubeladze, Joseph: Toric varieties with huge Grothendieck group. Adv. Math. 186, 117-124 (2004) · Zbl 1061.14056 · doi:10.1016/j.aim.2003.07.009 |
| [18] | Gubeladze, Joseph: The nilpotence conjecture in \[K\]-theory of toric varieties. Invent. Math. 160, 173-216 (2005) · Zbl 1075.14051 · doi:10.1007/s00222-004-0410-3 |
| [19] | Gubeladze, Joseph: The Steinberg group of a monoid ring, nilpotence, and algorithms. J. Algebra. 307, 461-496 (2007) · Zbl 1125.19002 · doi:10.1016/j.jalgebra.2006.03.006 |
| [20] | Gubeladze, Joseph: Global coefficient ring in the nilpotence conjecture. Proc. Am. Math. Soc. 136, 499-503 (2008) · Zbl 1134.19002 · doi:10.1090/S0002-9939-07-09106-X |
| [21] | Haesemeyer, C.: Descent properties of homotopy \[K\]-theory. Duke Math. J. 125, 589-620 (2004) · Zbl 1079.19001 · doi:10.1215/S0012-7094-04-12534-5 |
| [22] | Hüttemann, T.: A splitting result for the algebraic k-theory of projective toric schemes. J. Homotopy Relat. Struct. 1, 1-30 (2012) · Zbl 1281.18003 · doi:10.1007/s40062-012-0003-6 |
| [23] | Kratzer, Ch.: \[ \lambda \]-structure en \[K\]-théorie algébrique. Comment. Math. Helv. 55:233-254 (1980) · Zbl 0444.18008 |
| [24] | Lecomte, F:. Opérations d’Adams en \[K\]-théorie algébrique. \[K\]-Theory. 13,179-207 (1998) · Zbl 0954.19003 |
| [25] | Massey, A.: The \[{HK}\]-theory of complete simplicial toric varieties and the algebraic k-theory of weighted projective spaces. Preprint. http://arxiv.org/abs/1207.0123 · Zbl 1328.19001 |
| [26] | Quillen, D.: Higher algebraic \[K\]-theory. I. In: Algebraic \[K\]-Theory, I: Higher \[K\]-Theories, Proceedings of the Conference held at the Battelle Memorial Institute, Seattle, Wash., 1972, pp. 85-147. Lecture Notes in Mathematics, vol. 341. Springer, Berlin (1973) · Zbl 0292.18004 |
| [27] | Joël, R.: Algebraic \[K\]-theory, \[{\\\mathbf{A}}^1\]-homotopy and Riemann-Roch theorems. J. Topol. 3, 229-264 (2010) · Zbl 1202.19004 |
| [28] | Roberts, Leslie G., Singh, Balwant: Subintegrality, invertible modules and the Picard group. Compos. Math. 85, 249-279 (1993) · Zbl 0782.13006 |
| [29] | Soulé, C.: Opérations en \[K\]-théorie algébrique. Canad. J. Math. 37, 488-550 (1985) · Zbl 0575.14015 · doi:10.4153/CJM-1985-029-x |
| [30] | Stienstra, J.: Operations in the higher \[K\]-theory of endomorphisms. In: Current Trends in Algebraic Topology, Part 1 (London, Ont., 1981) CMS Conference Proceedings, volume 2, pp. 59-115. Am. Math. Soc. (1982) · Zbl 0545.18004 |
| [31] | Swan, R.G.: Gubeladze’s proof of Anderson’s conjecture. In: Azumaya Algebras, Actions, and Modules, Bloomington, IN, 1990. Contemporary Mathematics, vol. 124, pp. 215-250. Am. Math. Soc. (1992) · Zbl 0742.13005 |
| [32] | Thomason, R.W., Trobaugh, T.: Higher algebraic \[K\]-theory of schemes and of derived categories. In: The Grothendieck Festschrift, vol. III. Programming Mathematics, vol. 88, pp. 247-435. Birkhäuser Boston, Boston (1990) · Zbl 0731.14001 |
| [33] | Totaro, B.: Chow groups, chow cohomology, and linear varieties. J. Alg. Geom (to appear) · Zbl 1329.14018 |
| [34] | Vezzosi, Gabriele, Vistoli, Angelo: Higher algebraic \[K\]-theory for actions of diagonalizable groups. Invent. Math. 153, 1-44 (2003) · Zbl 1032.19001 · doi:10.1007/s00222-002-0275-2 |
| [35] | Waldhausen, F.: Algebraic \[K\]-theory of spaces. In: Algebraic and Geometric Topology, New Brunswick, N.J., 1983. Lecture Notes in Mathematics, vol. 1126, pp. 318-419. Springer, Berlin (1985) · Zbl 0579.18006 |
| [36] | Weibel, C.A.: Homotopy algebraic \[K\]-theory. In: Algebraic \[K\]-Theory and Algebraic Number Theory, Honolulu, HI, 1987. Contemporary Mathematics, vol. 83, pp. 461-488. Am. Math. Soc. · Zbl 0669.18007 |
| [37] | Weibel, Charles A.: Module structures on the \[K\]-theory of graded rings. J. Algebra 105, 465-483 (1987) · Zbl 0611.13012 · doi:10.1016/0021-8693(87)90210-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
