Category Of Complexes Research Articles

$L$-homology on ball complexes and products

We construct homology theories with coefficients in L-spectra on the category of ball complexes and we define products in this setting. We also obtain signatures of geometric situations in these homology groups and prove product formulae which we hope will clarify products used in the theory of the total surgery obstruction.

Equivariant noncommutative motives

Given a finite group G, we develop a theory of G-equivariant noncommutative motives. This theory provides a well-adapted framework for the study of G-schemes, Picard groups of schemes, G-algebras, 2-cocycles, G-equivariant algebraic K-theory, etc. Among other results, we relate our theory with its commutative counterpart as well as with Panin’s theory. As a first application, we extend Panin’s computations, concerning twisted projective homogeneous varieties, to a large class of invariants. As a second application, we prove that whenever the category of perfect complexes of a G-scheme X admits a full exceptional collection of G-invariant (≠G-equivariant) objects, the G-equivariant Chow motive of X is of Lefschetz type. Finally, we construct a G-equivariant motivic measure with values in the Grothendieck ring of G-equivariant noncommutative Chow motives.

Integral transforms for coherent sheaves

The theory of integral, or Fourier–Mukai, transforms between derived categories of sheaves is a well established tool in noncommutative algebraic geometry. General “kernel theorems” represent all reasonable linear functors between categories of perfect complexes (or their “large” version, quasi-coherent complexes) on schemes and stacks over some fixed base as integral kernels in the form of complexes (of the same nature) on the fiber product. However, for many applications in mirror symmetry and geometric representation theory one is interested instead in the bounded derived category of coherent sheaves (or its “large” version, ind-coherent sheaves), which differs from perfect complexes (and quasi-coherent complexes) once the underlying variety is singular. In this paper, we prove general kernel theorems for linear functors between derived categories of coherent sheaves over a base in terms of integral kernels on the fiber product. Namely, we identify coherent kernels with functors taking perfect complexes to coherent complexes (an analogue of the classical Schwartz kernel theorem), and kernels which are coherent relative to the source with functors taking all coherent complexes to coherent complexes. The proofs rely on key aspects of the “functional analysis” of derived categories, namely the distinction between small and large categories and its measurement using t-structures. These are used in particular to correct the failure of integral transforms on ind-coherent complexes to correspond to ind-coherent complexes on a fiber product. The results are applied in a companion paper to the representation theory of the affine Hecke category, identifying affine character sheaves with the spectral geometric Langlands category in genus one.

Neeman's characterization of K(R-Proj) via Bousfield localization

Let R be an associative ring with unit and denote by K(R-Proj) the homotopy category of complexes of projective left R-modules. Neeman proved the theorem that K(R-Proj) is ℵ1-compactly generated, with the category K+(R-proj) of left bounded complexes of finitely generated projective R-modules providing an essentially small class of such generators. Another proof of Neeman's theorem is explained, using recent ideas of Christensen and Holm, and Emmanouil. The strategy of the proof is to show that every complex in K(R-Proj) vanishes in the Bousfield localization K(R-Flat)/〈K+(R-proj)〉.

Thick tensor ideals of right bounded derived categories

Let $R$ be a commutative noetherian ring. Denote by $D^-(R)$ the derived category of cochain complexes $X$ of finitely generated $R$-modules with $H^i(X)=0$ for $i\gg0$. Then $D^-(R)$ has the structure of a tensor triangulated category with tensor product $-\otimes_R^L-$ and unit object $R$. In this paper, we study thick tensor ideals of $D^-(R)$, i.e., thick subcategories closed under the tensor action by each object in $D^-(R)$, and investigate the Balmer spectrum $Spc\,D^-(R)$ of $D^-(R)$, i.e., the set of prime thick tensor ideals of $D^-(R)$. First, we give a complete classification of the thick tensor ideals of $D^-(R)$ generated by bounded complexes, establishing a generalized version of the Hopkins-Neeman smash nilpotence theorem. Then, we define a pair of maps between the Balmer spectrum $Spc\,D^-(R)$ and the Zariski spectrum $Spec\,R$, and study their topological properties. After that, we compare several classes of thick tensor ideals of $D^-(R)$, relating them to specialization-closed subsets of $Spec\,R$ and Thomason subsets of $Spc\,D^-(R)$, and construct a counterexample to a conjecture of Balmer. Finally, we explore thick tensor ideals of $D^-(R)$ in the case where $R$ is a discrete valuation ring.

Gorenstein flat and projective (pre)covers

We consider a right coherent ring R. We prove that the class of Gorenstein flat complexes is covering in the category of complexes of left R-modules Ch(R). When R is also left n-perfect, we prove that the class of Gorenstein projective complexes is special precovering in Ch(R).

Factorization homology and calculus $\textit{à la}$ Kontsevich Soibelman

We use factorization homology over manifolds with boundaries in order to construct operations on Hochschild cohomology and Hochschild homology. These operations are parametrized by a colored operad involving disks on the surface of a cylinder defined by Kontsevich and Soibelman. The formalism of the proof extends without difficulties to a higher dimensional situation. More precisely, we can replace associative algebras by algebras over the little disks operad of any dimensions, Hochschild homology by factorization (also called topological chiral) homology and Hochschild cohomology by higher Hochschild cohomology. Our result works in categories of chain complexes but also in categories of modules over a commutative ring spectrum giving interesting operations on topological Hochschild homology and cohomology.

Tannaka duality revisited

We establish several strengthened versions of Lurie's Tannaka duality theorem for certain classes of spectral algebraic stacks. Our most general version of Tannaka duality identifies maps between stacks with exact symmetric monoidal functors between ∞-categories of quasi-coherent complexes which preserve connective and pseudo-coherent complexes.

Adelic descent theory

A result of André Weil allows one to describe rank$n$vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set$\text{GL}_{n}(\mathbb{A})$of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to$G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles$\mathbb{A}_{X}^{\bullet }$, we have an equivalence$\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$between perfect complexes on$X$and cartesian perfect complexes for$\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal$\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

The family Floer functor is faithful

Family Floer theory is used to construct a functor from the Fukaya category of a symplectic manifold admitting a Lagrangian torus fibration to a (twisted) category of perfect complexes on the mirror rigid analytic space. This functor is shown to be faithful by a degeneration argument involving moduli spaces of annuli.

A Zariski-local notion of F-total acyclicity for complexes of sheaves

We study a notion of total acyclicity for complexes of flat sheaves over a scheme. It is Zariski-local—i.e. it can be verified on any open affine covering of the scheme—and for sheaves over a quasi-compact semi-separated scheme it agrees with the categorical notion. In particular, it agrees, in their setting, with the notion studied by Murfet and Salarian for sheaves over a noetherian semi-separated scheme. As part of the study we recover, and in several cases extend the validity of, recent results on existence of covers and precovers in categories of sheaves. One consequence is the existence of an adjoint to the inclusion of these totally acyclic complexes into the homotopy category of complexes of flat sheaves.

Sullivan minimal models of operad algebras

We prove the existence of Sullivan minimal models of operad algebras for a quite wide family of operads in the category of complexes of vector spacesover a field of characteristic zero. Our construction is an adaptation of Sullivan’s original step by step construction to the setting of operad algebras. The family of operads that we consider includes all operads concentrated in degree 0 as well as their minimal models. In particular, this gives Sullivan minimal models for algebras over Com, Ass, and Lie, as well as over their minimal models Com∞, Ass∞, and Lie∞. Other interesting operads, such as the operad Ger encoding Gerstenhaber algebras, also fit in our study.

GORENSTEIN MODULES AND GORENSTEIN MODEL STRUCTURES

AbstractGiven a complete hereditary cotorsion pair$(\mathcal{X}, \mathcal{Y})$, we introduce the concept of$(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integern, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤nover a left Noetherian ringR. Similarly, ifRis a left coherent ring in which all flat leftR-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤n. These structures have their analogous in the category of chain complexes.

Formula omitted]-motives

Considering a (co)homology theory T on a base category C as a fragment of a first-order logical theory we here construct an abelian category A[T] which is universal with respect to models of T in abelian categories. Under mild conditions on the base category C, e.g. for the category of algebraic schemes, we get a functor from C to Ch(Ind(A[T])) the category of chain complexes of ind-objects of A[T]. This functor lifts Nori's motivic functor for algebraic schemes defined over a subfield of the complex numbers. Furthermore, we construct a triangulated functor from D(Ind(A[T])) to Voevodsky's motivic complexes.

Blow-up in Manifolds with Generalized Corners

We construct a functor from the category of manifolds with generalized corners to the category of complexes of toric monoids, and for every `refinement' of the complex associated to a manifold, we show there is a unique `blow-up', i.e., a new manifold mapping to the original one, which satisfies a universal property and whose complex realizes the refinement. This was inspired in part by the work of Gillam and Molcho, though we work with manifolds with generalized corners, as developed by Joyce, which have embedded boundary faces, for which the appropriate objects (i.e., complexes of monoids) are simpler than they would be otherwise (i.e., monoidal spaces in the sense of Kato).

Arithmetic mirror symmetry for genus 1 curves with n marked points

We establish a \({\mathbb {Z}}[[t_1,\ldots , t_n]]\)-linear derived equivalence between the relative Fukaya category of the 2-torus with n distinct marked points and the derived category of perfect complexes on the n-Tate curve. Specialising to \(t_1= \cdots =t_n=0\) gives a \({\mathbb {Z}}\)-linear derived equivalence between the Fukaya category of the n-punctured torus and the derived category of perfect complexes on the standard (Neron) n-gon. We prove that this equivalence extends to a \({\mathbb {Z}}\)-linear derived equivalence between the wrapped Fukaya category of the n-punctured torus and the derived category of coherent sheaves on the standard n-gon. The corresponding results for \(n=1\) were established in Lekili and Perutz (Arithmetic mirror symmetry for the 2-torus (preprint) arXiv:1211.4632, 2012).

Some remarks on Bridgeland’s Hall algebras

ABSTRACTWe study the functorial properties of Bridgeland’s Hall algebras. Specifically, let 𝒜 and ℬ be two categories satisfying certain conditions for the definitions of Bridgeland’s Hall algebras, and let F:𝒜→ℬ be a fully faithful exact functor, which preserves projectives, then F induces an embedding of algebras from the Bridgeland’s Hall algebra of 𝒜 to the one of ℬ. In addition, let A be a finite-dimensional algebra over a finite field and B some special quotient algebra of A, then the Bridgeland’s Hall algebra of B is the quotient algebra of the one of A. Moreover, we consider the BGP-reflection functors on the category of 2-cyclic complexes and obtain some homomorphisms of algebras among the subalgebras of Bridgeland’s Hall algebras.

Gorenstein AC-projective dimension of unbounded complexes

ABSTRACTRecently, the notion of Gorenstein AC-projective (resp., Gorenstein AC-injective) modules was introduced in [3] by which the so-called “Gorenstein AC-homological algebra” was established. Here, we define and study a notion of Gorenstein AC-projective dimension for complexes (not necessarily bounded) over associative rings, which is inspired by Veliche’s construction of defining Gorenstein projective dimension. In particular, we show that such a dimension can be closely related to the “proper” Gorenstein AC-projective resolutions of complexes induced by a complete and hereditary cotorsion pair in the category of complexes of modules. This enables us to interpret this dimension of a complex in terms of vanishing of the derived functor RHomR(−,−). As applications, some characterizations of the Gorenstein AC-projective dimension of a module are also obtained.

Geometricity for derived categories of algebraic stacks

We prove that the dg category of perfect complexes on a smooth, proper Deligne-Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field.

STABLE ISOTOPE GEOCHEMISTRY OF MASSIVE ICE

The paper summarises stable-isotope research on massive ice in the Russian and North American Arctic, and includes the latest understanding of massive-ice formation. A new classification of massive-ice complexes is proposed, encompassing the range and variability of massive ice. It distinguishes two new categories of massive-ice complexes: homogeneous massive-ice complexes have a similar structure, properties and genesis throughout, whereas heterogeneous massive-ice complexes vary spatially (in their structure and properties) and genetically within a locality and consist of two or more homogeneous massive-ice bodies. Analysis of pollen and spores in massive ice from Subarctic regions and from ice and snow cover of Arctic ice caps assists with interpretation of the origin of massive ice. Radiocarbon ages of massive ice and host sediments are considered together with isotope values of heavy oxygen and deuterium from massive ice plotted at a uniform scale in order to assist interpretation and correlation of the ice.