Abstract
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 ≥ 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.
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Barvinok, A.: Integer points in polyhedra, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008)
Fulton, W.: Introduction to toric varieties, Annals of Mathematics Studies, vol. 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton (1993)
Hartshorne R.: Algebraic geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)
Hüttemann T.: Finiteness of total cofibres. K-Theory 31, 101–123 (2004)
Hüttemann T.: K-theory of non-linear projective toric varieties. Forum Math. 21(1), 67–100 (2009)
Hüttemann T.: On the derived category of a regular toric scheme. Geom. Dedic. 148, 175–203 (2010)
Miller E., Sturmfels B.: Combinatorial commutative algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)
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)
Rosenberg J.: Algebraic K-theory and its applications. Graduate Texts in Mathematics, vol. 147. Springer, New York (1994)
Stanley R.P.: Two poset polytopes. Discret. Comput. Geom. 1(1), 9–23 (1986)
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)
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)
Weibel C.A.: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995)
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Communicated by Michael Weiss.
This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/H018743/1].
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Hüttemann, T. A splitting result for the algebraic K-theory of projective toric schemes. J. Homotopy Relat. Struct. 7, 1–30 (2012). https://doi.org/10.1007/s40062-012-0003-6
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DOI: https://doi.org/10.1007/s40062-012-0003-6