Presenting higher stacks as simplicial schemes

We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zero-slice then a relative version of Voevodsky’s conjecture on the slices of MGL holds true. We outline the picture for K-theory and rational slices.

https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/preprints/Preprints2008/36-2008.pdf

Relations between slices and quotients of the algebraic cobordism spectrum

We prove existence results a la Jeff Smith for left-induced model category structures, of which the injective model structure on a diagram category is an important example. We further develop the notions of fibrant generation and Postnikov presentation from [9], which are dual to a weak form of cofibrant generation and cellular presentation. As examples, for k a field and H a differential graded Hopf algebra over k, we produce a left-induced model structure on augmented H-comodule algebras and show that the category of bounded below chain complexes of finite-dimensional k-vector spaces has a Postnikov presentation. To conclude, we investigate the fibrant generation of (generalized) Reedy categories. In passing, we also consider cofibrant generation, cellular presentation, and the small object argument for Reedy diagrams.

/pdf/left-induced-model-structures-and-diagram-categories-2hb9gr82xo.pdf

Left-induced model structures and diagram categories

Given a diagram of rings, one may consider the category of modules over them. We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories M(s) (as s runs through the diagram), we consider the category of diagrams where the object X(s) at s comes from M(s). We develop model structures on such categories of diagrams, and Quillen adjunctions that relate categories based on different diagram shapes. Under certain conditions, cellularizations (or right Bousfield localizations) of these adjunctions induce Quillen equivalences. As an application we show that a cellularization of a category of modules over a diagram of ring spectra (or differential graded rings) is Quillen equivalent to modules over the associated inverse limit of the rings. Another application of the general machinery here is given in work by the authors on algebraic models of rational equivariant spectra. Some of this material originally appeared in the preprint “An algebraic model for rational torus-equivariant stable homotopy theory”, arXiv:1101.2511, but has been generalized here.

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Homotopy theory of modules over diagrams of rings

We describe derived moduli functors for a range of problems involving schemes and quasi-coherent sheaves, and give cohomological conditions for them to be representable by derived geometric n-stacks. Examples of problems represented by derived geometric 1-stacks are derived moduli of polarised projective varieties, derived moduli of vector bundles, and derived moduli of abelian varieties.

Derived moduli of schemes and sheaves

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

Twisted diagrams and homotopy sheaves

We consider a non-standard totalisation functor to produce a cochain complex from a given double complex D∗,∗: instead of sums or products, totalisation is defined via truncated products of modules. We give an elementary proof of the fact that a double complex with exact rows (resp., columns) yields an acyclic cochain complex under totalisation using right (resp., left) truncated products. As an application we consider the algebraic mapping torus T (h) of a self map h of a cochain complex C. We show that if C consists of finitely presented modules then T (h) has trivial negative Novikov cohomology; if in addition h is a quasi-isomorphism, then T (h) has trivial positive Novikov cohomology as well. As a consequence we obtain a new proof that a finitely dominated cochain complex over a Laurent polynomial ring has trivial Novikov cohomology. Finiteness conditions for chain complexes of modules play an important role in both algebra and topology. For example, given a group G one might ask whether the trivial G-module Z admits a resolution by finitely generated projective Z[G]-modules; existence of such resolutions is relevant for the study of group cohomology of G, and has applications in the theory of duality groups [B75]. For topologists, finite domination of chain complexes is related, among other things, to questions about finiteness of CW complexes, the topology of ends of manifolds, and obstructions for the existence of nonsingular closed 1-forms [Ran95, S06]. A cochain complex C of R[z, z−1]-modules is called finitely dominated if it is homotopy equivalent, as a complex of R-modules, to a bounded complex of finitely generated projective R-modules. Finite domination of C can be characterised in various ways; Brown considered compatibility of the functors M 7→ H∗(C;M) and M 7→ H∗(C;M) with products and direct limits, respectively [B75, Theorem 1], while Ranicki showed that C is finitely dominated if and only if the Novikov cohomology of C is trivial [Ran95, Theorem 2] (see also Definition 2.3 and Corollary 2.7 below). Our approach to Novikov cohomology involves a non-standard totalisation functor. The key fact is that certain double complexes are converted into acyclic cochain complexes (Proposition 1.2) which is proved by an elementary calculation. As an application we obtain a new result for vanishing of Novikov cohomology of algebraic mapping tori (Theorem 2.5), and recover the “only-if” part of Ranicki’s criterion for finite domination over Date: 08.09.2011. 2000 Mathematics Subject Classification. Primary 18G35; Secondary 55U15. This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/H018743/1]. 1

Double complexes and vanishing of Novikov cohomology

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.

/pdf/finite-domination-and-novikov-rings-iterative-approach-u26of1n30g.pdf

Finite domination and Novikov rings. Iterative approach

Let L be a unital Z-graded ring, and let C be a bounded chain complex of finitely generated L-modules. We give a homological characterisation of when C is homotopy equivalent to a bounded complex of finitely generated projective L
 0-modules, generalising known results for twisted Laurent polynomial rings. The crucial hypothesis is that L is a strongly graded ring.

/pdf/finite-domination-and-novikov-homology-over-strongly-z-5nhnwlyapg.pdf

Finite domination and Novikov homology over strongly Z-graded rings

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.

Thomas Huettemann

Papers

Twisted diagrams and homotopy sheaves

Double complexes and vanishing of Novikov cohomology

Finite domination and Novikov rings. Iterative approach

Finite domination and Novikov homology over strongly Z-graded rings

On the derived category of a regular toric scheme