arXiv subjects
Publications and source records attributed to Thomas Huettemann.
Following ideas of Quillen it is shown that the graded K-theory of a Z^n-graded ring with support contained in a pointed cone is entirely determined by the K-theory of the subring of degree-0 elements.
Let X be a quasi-compact scheme, equipped with an open covering by affine schemes. A quasi-coherent sheaf on X gives rise, by taking sections over the covering sets, to a diagram of modules over the various coordinate rings. The resulting "twisted" diagram of modules satisfies a certain gluing condition, stating that the data is compatible with restriction to smaller open sets. In case X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category D(X) of quasi-coherent sheaves on X can be obtained from a category of twisted diagrams which do not necessarily satisfy any gluing condition by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we given an explicit construction of a finite set of weak generators for the derived category. For example, if X is projective n-space then D(X) is generated by n+1 successive twists of the structure sheaf; the present paper gives a new homotopy-theoretic proof of this classical result. The approach taken uses the language of model categories, and the machinery of Bousfield-Hirschhorn colocalisation. The first step is to characterise colocal objects; these turn out to be homotopy sheaves in the sense that chain complexes over different open sets agree on intersections up to quasi-isomorphism only. In a second step it is shown that the homotopy category of homotopy sheaves is the derived category of X.
Twisted diagrams are "diagrams" with components in different categories. Structure maps are defined using auxiliary data which consists of functors relating the various categories to each other. Prime examples of the construction are spectra (in the sense of homotopy theory) and quasi-coherent sheaves on schemes. We develop the basic theory of twisted diagrams, and establish various model structures (which are well-known in special cases). We also introduce a notion of homotopy sheaves, a collection of local data which is compatible up to weak equivalence, and study basic properties of such objects. These objects occur in nature; for example, the notion of an Omega-spectrum fits into this framework. The main purpose of the paper is to provide a convenient reference for model structures on twisted diagrams, and for the language of sheaves and homotopy sheaves as defined here.
By analogy with algebraic geometry, we define a category of non-linear sheaves (quasi-coherent homotopy-sheaves of topological spaces) on projective toric varieties and prove a splitting result for its algebraic K-theory, generalising earlier results for projective spaces. The splitting is expressed in terms of the number of interior lattice points of dilations of a polytope associated to the variety. The proof uses combinatorial and geometrical results on polytopal complexes. The same methods also give an elementary explicit calculation of the cohomology groups of a projective toric variety over any commutative ring.
A subset K of R^n gives rise to a formal Laurent series with monomials corresponding to lattice points in K. Under suitable hypotheses, this series represents a rational function R(K). Michel Brion has discovered a surprising formula relating the rational function R(P) of a lattice polytope P to the sum of rational functions corresponding to the supporting cones subtended at the vertices of P. The result is re-phrased and generalised in the language of cohomology of line bundles on complete toric varieties. Brion's formula is the special case of an ample line bundle on a projective toric variety. - The paper also contains some general remarks on the cohomology of torus-equivariant line bundles on complete toric varieties, valid over noetherian ground rings.
We give an elementary geometric re-proof of a formula discovered by Michel Brion as well as two variants thereof. A subset of R^n gives rise to a formal Laurent series with monomials corresponding to lattice points in the set. Under suitable hypotheses, these series represent rational functions. We will prove formulae relating the rational function of a lattice polytope P to the sum of rational functions corresponding to the supporting cones subtended at the vertices of P. The exposition should be suitable for everyone with a little background in topology.
Let X be a pointed connected simplicial set with loop group G. The linearisation map in K-theory as defined by Waldhausen uses G-equivariant spaces. This paper gives an alternative description using presheaves of sets and abelian groups on the simplex category of X. In other words, the linearisation map is defined in terms of X only, avoiding the use of the less geometric loop group. The paper also includes a comparison of categorical finiteness with the more geometric notion of finite CW objects in cofibrantly generated model categories. The application to the linearisation map employs a model structure on the category of abelian group objects of retractive spaces over X.
If Y is a diagram of spectra indexed by an arbitrary poset C together with a specified sub-poset D, we define the total cofibre Γ(Y) of Y as the strict cofibre of the map from hocolim_D (Y) to hocolim_C (Y). We construct a comparison map from the homotopy limit of Y to a looping of a fibrant replacement of Gamma (Y), and characterise those poset pairs (C,D) for which this comparison map is a stable equivalence. The characterisation is given in terms of stable cohomotopy of spaces related to C and D. For example, if C is a finite polytopal complex with underlying space an m-ball with boundary sphere D, then holim_C (Y) and Γ(Y) agree up to m-fold looping and up to stable equivalence. As an application of the general result we give a spectral sequence for the homotopy groups of Γ(Y) with E_2-term involving higher derived inverse limits of π_* (Y), generalising earlier constructions for space-valued diagrams indexed by the face lattice of a polytope.