A splitting result for the algebraic \(K\)-theory of projective toric schemes. (English) Zbl 1281.18003
Summary: Suppose \(X\) is a projective toric scheme defined over a ring \(R\) and equipped with an ample line bundle \({\mathcal{L}}\). We prove that its \(K\)-theory has a direct summand of the form \(K(R)^{(k+1)}\) where \(k \geq 0\) is minimal such that \(\mathcal{L}^{\otimes (-k-1)}\) is not acyclic. Using a combinatorial description of quasi-coherent sheaves we interpret and prove this result for a ring \(R\) which is either commutative, or else left Noetherian.
MSC:
| 18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
| 18G35 | Chain complexes (category-theoretic aspects), dg categories |
| 19E08 | \(K\)-theory of schemes |
References:
| [1] | Barvinok, A.: Integer points in polyhedra, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008) · Zbl 1154.52009 |
| [2] | Fulton, W.: Introduction to toric varieties, Annals of Mathematics Studies, vol. 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton (1993) · Zbl 0813.14039 |
| [3] | Hartshorne R.: Algebraic geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977) · Zbl 0367.14001 |
| [4] | Hüttemann T.: Finiteness of total cofibres. K-Theory 31, 101-123 (2004) · Zbl 1068.55015 · doi:10.1023/B:KTHE.0000022924.43719.5d |
| [5] | Hüttemann T.: K-theory of non-linear projective toric varieties. Forum Math. 21(1), 67-100 (2009) · Zbl 1165.19002 · doi:10.1515/FORUM.2009.004 |
| [6] | Hüttemann T.: On the derived category of a regular toric scheme. Geom. Dedic. 148, 175-203 (2010) · Zbl 1275.14013 · doi:10.1007/s10711-009-9389-7 |
| [7] | Miller E., Sturmfels B.: Combinatorial commutative algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2005) · Zbl 1090.13001 |
| [8] | Quillen, D.: Higher algebraic K-theory. I, Algebr. K-Theory I. In: Proc. conf. Battelle Inst. 1972, Lecture Notes in Mathematics, vol. 341, pp. 85-147 (1973) · Zbl 0292.18004 |
| [9] | Rosenberg J.: Algebraic K-theory and its applications. Graduate Texts in Mathematics, vol. 147. Springer, New York (1994) · Zbl 0801.19001 |
| [10] | Stanley R.P.: Two poset polytopes. Discret. Comput. Geom. 1(1), 9-23 (1986) · Zbl 0595.52008 · doi:10.1007/BF02187680 |
| [11] | Thomason, R.W., Trobaugh, T.: Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift. A collection of articles written in honor of the 60th birthday of Alexander Grothendieck, Birkhäuser, Boston, pp. 247-435 (1990) · Zbl 0717.00009 |
| [12] | Waldhausen, F.: Algebraic K-theory of spaces. Algebraic and geometric topology. In: Proc. conf., New Brunswick, Lecture Notes in Mathemetics, vol. 1126, pp. 318-419 (1985) · Zbl 0579.18006 |
| [13] | Weibel C.A.: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995) · Zbl 0834.18001 |
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.
