On the derived category of a regular toric scheme

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U _σ = Spec A ^σ . A quasi-coherent sheaf on X gives rise, by taking sections over the U _σ , to a diagram of modules over the coordinate rings A ^σ , indexed by the intersection poset Σ of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent sheaves on X can be obtained from a category of Σ^op-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we show that there is a finite set of weak generators, one for each cone in the fan Σ. The approach taken uses the machinery of Bousfield–Hirschhorn colocalisation of model categories. 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 U _σ agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves is equivalent to the derived category of X.     

On a category of module bundles

It is shown, that the category ℰ _A (X) ofA-bundles over a topological spaceX is an additive pseudo-Abelian category for each Waelbroeck algebraA. This research was supported by Greek State Scholarship Foundation and Estonian Science Foundation grant 6205.     

Coefficient Quantization in Banach Spaces

Let (e _i ) be a dictionary for a separable infinite-dimensional Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a ‘finite alphabet’. We investigate several approximation properties of this type and connect them to the Banach space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c ₀. We also show that, for every ε>0, the unit ball of every separable infinite-dimensional Banach space X contains a dictionary (x _i ) such that the additive group generated by (x _i ) is (3+ε)⁻¹-separated and 1/3-dense in X.      

Generalized Mukai conjecture for special Fano varieties

Let X be a Fano variety of dimension n, pseudoindex i _X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i _X −1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i _X ≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρ_X> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.     

Triangulated functors,homological functors,tilts, and lifts

 Let T be a triangulated category and let X be an object of T. This paper studies the questions: Does there exist a triangulated functor G : D(ℤ)  T with G(ℤ)≌X? Does there exist a triangulated functor H : T  D(ℤ) with h⁰ ⊚ H ⋍ Hom_T (X, −)? To what extent are G and H unique? One spin off is a proof that the homotopy category of spectra is not the stable category of any Frobenius category with set indexed coproducts. Received: 8 March 2002 / Revised version: 18 October 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 18E30, 55U35     

An example of L-fuzzy join space

On a generalized deMorgan lattice (X, ≤, ∨, ∧,′) we introduce a family of join hyperoperations * _p , parametrized by a parameterp εX. As a result we obtain a family of join spaces (X, * _p ). We show that: for everya,b εX the family {a*_pb} _pεX can be considered as thep-cuts of aL-fuzzy seta*b; in this manner we synthesize aL-fuzzy hyperoperation * which takes pairs fromX toL-fuzzy subsets ofX. We then show that (X, * _p ) is aL-fuzzy hypergroup (in the sense of Corsini) and can be considered as aL-fuzzy join space. Furthermore,a*b is aL-fuzzy interval for alla,b εX.     

On genera of polyhedra

We consider the stable homotopy category S of polyhedra (finite cell complexes). We say that two polyhedra X,Y are in the same genus and write X ∼ Y if X _p ≅ Y _p for all prime p, where X _p denotes the image of Xin the localized category S _p . We prove that it is equivalent to the stable isomorphism X∨B ₀ ≅Y∨B ₀, where B ₀ is the wedge of all spheres S ⁿ such that π _n ^S (X) is infinite. We also prove that a stable isomorphism X ∨ X ≅ Y ∨ X implies a stable isomorphism X ≅ Y.     

The Lyusternik–Shnirel'man category,¶vector bundles, and immersions of manifolds

One of our two main results exhibits for a vector bundle over a compact Hausdorff space X an interplay between its span, its possible splittings, and the Lyusternik–Shnirel'man category of X. The other main result, also on vector bundles and the Lyusternik–Shnirel'man category, enables us to derive certain inequalities connecting the immersion codimension, the stable span, and the Lyusternik–Shnirel'man category of a smooth closed manifold which is not stably parallelizable. Our results are applicable in various situations of general interest. Received: 4 September 1997     

Sobolev embeddings into BMO, VMO, andL ∞

LetX be a rearrangement-invariant Banach function space onR ⁿ and letV ¹ X be the Sobolev space of functions whose gradient belongs toX. We give necessary and sufficient conditions onX under whichV ¹ X is continuously embedded into BMO or intoL _∞. In particular, we show thatL _{n, ∞} is the largest rearrangement-invariant spaceX such thatV ¹ X is continuously embedded into BMO and, similarly,L _{n, 1} is the largest rearrangement-invariant spaceX such thatV ¹ X is continuously embedded intoL _∞. We further show thatV ¹ X is a subset of VMO if and only if every function fromX has an absolutely continuous norm inL _{n, ∞} . A compact inclusion ofV ¹ X intoC ⁰ is characterized as well.     

Microlocal branes are constructible sheaves

Let X be a compact real analytic manifold, and let T* X be its cotangent bundle. In a recent paper with Zaslow (J Am Math Soc 22:233–286, 2009), we showed that the dg category Sh _c (X) of constructible sheaves on X quasi-embeds into the triangulated envelope F(T* X) of the Fukaya category of T* X. We prove here that the quasi-embedding is in fact a quasi-equivalence. When X is a complex manifold, one may interpret this as a topological analogue of the identification of Lagrangian branes in T* X and regular holonomic D_X{{\mathcal D}_X} -modules developed by Kapustin (A-branes and noncommutative geometry, arXiv:hep-th/0502212) and Kapustin and Witten (Commun Number Theory Phys 1(1):1–236, 2007) from a physical perspective. As a concrete application, we show that compact connected exact Lagrangians in T* X (with some modest homological assumptions) are equivalent in the Fukaya category to the zero section. In particular, this determines their (complex) cohomology ring and homology class in T* X, and provides a homological bound on their number of intersection points. An independent characterization of compact branes in T* X has recently been obtained by Fukaya et al. (Invent Math 172(1):1–27, 2008).     

Remote points and the like in pointfree topology

A point p of β, where X is a Tychonoff space, is called a remote point if for any nowhere dense D ⊆, X, p ∉ cl_βx D. A subset S of X is called round if whenever the closure in βX of a zero-set of X contains S, then it is a neighborhood of S. The purpose of this paper is to study these notions in the pointfree context. In the process, we introduce N-homomorphisms and show how the Stone extension of an N-homomorphism transfers remote points back and forth.      

Hyperspaces of nowhere topologically complete spaces

It is proved that ifX is a connected locally continuumwise connected coanalytic nowhere topologically complete space, then the hyperspace 2 ^X of all nonempty compact subsets ofX is strongly universal in the class of all coanalytic spaces. Moreover, 2 ^X is homeomorphic to Π₂ ifX is a Baire space, and toQ∖Π₁ ifX contains a dense absoluteG _δ-setG ⊂X such that the intersectionG ∩U is connected for any open connectedU ⊂X. (Here Π₁, Π₁⊂X are the standard subsets of the Hilbert cubeQ absorbing for the classes of analytic and coanalytic spaces, respectively.) Similar results are obtained for higher projective classes. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 35–51, July, 1997. Translated by O. V. Sipacheva     

The Lyusternik-Shnirel’man category, vector bundles, and immersions of manifolds

One of our two main results exhibits for a vector bundle over a compact Hausdorff spaceX an interplay between its span, its possible splittings, and the Lyusternik-Shnirel’man category ofX. The other main result, also on vector bundles and the Lyusternik-Shnirel’man category, enables us to derive certain inequalities connecting the immersion codimension, the stable span, and the Lyusternik-Shnirel’man category of a smooth closed manifold which is not stably parallelizable. Our results are applicable in various situations of general interest.     

Polish group actions,nice topologies,and admissible sets

Let G be a closed subgroup of S_∞ and X be a Polish G ‐space. To every x ∈ X we associate an admissible set A _x and show how questions about X which involve Baire category can be formalized in A _x . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The plus construction, Postnikov towers and universal central module extensions

Given a connected spaceX, we consider the effect of Quillen’s plus construction on the homotopy groups ofX in terms of its Postnikov decomposition. Specifically, using universal properties of the fibration sequenceAX→X→X ⁺, we explain the contribution of π _n X to π _n X ⁺, π _n+1 X ⁺ and π _n AX, π _n+1 AX explicitly in terms of the low dimensional homology of π _n X regarded as a module over π₁ X. Key ingredients developed here for this purpose are universal II-central fibrations and a theory of universal central extensions of modules, analogous to universal central extensions of perfect groups. Research partially supported by NSERC of Canada.     

A note on approximation of semigroups of contractions on Hilbert spaces

We show that every contractive C ₀-semigroup on a separable, infinite-dimensional Hilbert space X can be approximated by unitary C ₀-groups in the weak operator topology uniformly on compact subsets of ℝ₊. As a consequence we get a new characterization of a bounded H ^∞-calculus for the negatives of generators of bounded holomorphic semigroups. Applications of our results to the study of a topological structure of the set of (almost) weakly stable contractive C ₀-semigroups on X are also discussed. The author was partially supported by the Marie Curie “Transfer of Knowledge” programme, project “TODEQ”, and by a MNiSzW grant Nr. N201384834.     

Product of lattice-valued measures on topological spaces

X ₁ and X ₂ are completely regular Hausdorff spaces, E ₁, E ₂ and F are Dedekind complete Banach lattices, 〈·,·〉: E ₁ × E ₂ → F is a bilinear mapping, and μ ₁ and μ ₂ are, respectively, E ₁ and E ₂ valued positive, countably additive Baire or Borel measures (countable additivity relative to order convergence) on X ₁ and X ₂. Under certain conditions the existence and uniqueness of the F-valued, positive, product measure is proved.      

Hyperspaces of Banach Spaces with the Attouch—Wets Topology

Let X be an infinite-dimensional Banach space with weight τ. By Cld _AW (X), we denote the hyperspace of nonempty closed sets in X with the Attouch—Wets topology. By Fin _AW (X), Comp _AW (X) and Bdd _AW (X), we denote the subspaces of Cld _AW (X) consisting of finite sets, compact sets and bounded closed sets, respectively. In this paper, it is proved that Fin _AW (X)≈Comp _AW (X)≈ℓ₂(τ)×ℓ₂ ^f ℓandℓBdd _AW (X)≈ℓ₂(2^τ)×ℓ₂ ^f where ≈ means ‘is homeomorphic to’, ℓ₂(τ) is the Hilbert space with weight τ (ℓ₂(ℵ₀)=ℓ₂ the separable Hilbert space) and ℓ₂ ^f ={(x _i ) _iεN εℓ₂∣x _i =0 except for finitely many iεN}.     

On the classes of hereditarily ℓp Banach spaces

Let X denote a specific space of the class of X _α,p Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily ℓ_p Banach spaces. We show that for p > 1 the Banach space X contains asymptotically isometric copies of ℓ_p. It is known that any member of the class is a dual space. We show that the predual of X contains isometric copies of ℓ_p where 1/p + 1/q = 1. For p = 1 it is known that the predual of the Banach space X contains asymptotically isometric copies of c ₀. Here we give a direct proof of the known result that X contains asymptotically isometric copies of ℓ₁.