K-Theory of non-linear projective toric varieties

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.Comment: v2: Final version, to appear in "Forum Mathematicum". Minor changes only, added more cross-referencing and references for toric geometr

A note on the graded K-theory of certain graded rings

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.Comment: 4 page

Finite domination and Novikov rings. Iterative approach

Suppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,1/x]. Then C is R-finitely dominated, ie, homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules, if and only if the two chain complexes C((x)) and C((1/x)) are acyclic, as has been proved by Ranicki. Here C((x)) is the tensor product over L of C with the Novikov ring R((x)) = R[[x]][1/x] (also known as the ring of formal Laurent series in x); similarly, C((1/x)) is the tensor product over L of C with the Novikov ring R((1/x)) = R[[1/x]][x]. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.Comment: 15 pages; diagrams typeset with Paul Taylor's "diagrams" macro package. Version 2: clarified proof of main theorem, fixed minor typos; Version 3: expanded introduction, now 16 pages; Version 4: corrected mistake on functoriality of mapping tor

EL ajalooline kujunemine ja euroopastumise teooria

BeSt programmi toetusel loodud e-kursuse "EL ajalooline kujunemine ja euroopastumise teooria" õppematerjalid

On the derived category of a regular toric scheme

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.Comment: 35 pages; diagrams need post script viewer or PDF v2: removed "completeness" assumption, changed titl

The "fundamental theorem" for the algebraic K-theory of spaces. II: The canonical involution

Hüttemann T, Klein JR, Vogell W, Waldhausen F, Williams B. The "fundamental theorem" for the algebraic K-theory of spaces. II: The canonical involution. Journal of Pure and Applied Algebra. 2002;167(1):53-82.Let X --> A(X) denote the algebraic K-theory of spaces functor. In the first paper of this series, we showed A(X x S-1) decomposes into a product of a copy of A(X), a delooped copy of A(X) and two homeomorphic nil terms. The primary goal of this paper is to determine how the "canonical involution" acts on this splitting. A consequence of the main result is that the involution acts so as to transpose the nil terms. From a technical point of view, however, our purpose will be to give another description of the involution on A(X) which arises as a (suitably modified) P.-construction. The main result is proved using this alternative discription. (C) 2002 Elsevier Science B.V. All rights reserved

The "fundamental theorem" for the algebraic K-theory of spaces. I

Hüttemann T, Klein JR, Vogell W, Waldhausen F, Williams B. The "fundamental theorem" for the algebraic K-theory of spaces. I. Journal of Pure and Applied Algebra. 2001;160(1):21-52.Let X H A(X) denote the algebraic K-theory of spaces functor. The main objective of this paper is to show that A(X x S-1) admits a functorial splitting. The splitting has four factors: a copy of A(X), a delooped copy of A(X) and two homeomorphic nil terms. One should view the decomposition as the algebraic K-theory of spaces version of the Bass-Heller-Swan theorem. In deducing this splitting, we introduce a new tool: a "non-linear" analogue of the projective line. (C) 2001 Elsevier Science B.V. All rights reserved

The space of Anosov diffeomorphisms

We consider the space \X of Anosov diffeomorphisms homotopic to a fixed automorphism $L$ of an infranilmanifold $M$. We show that if $M$ is the 2-torus $\mathbb T^2$ then \X is homotopy equivalent to $\mathbb T^2$. In contrast, if dimension of $M$ is large enough, we show that \X is rich in homotopy and has infinitely many connected components.Comment: Version 2: referee suggestions result in a better expositio