Higher order derived functors and the Adams spectral sequence

Classical homological algebra studies chain complexes, resolutions, and derived functors in additive categories. In this paper we define higher order chain complexes, resolutions, and derived functors in the context of a new type of algebraic structure, called an algebra of left cubical balls  . We show that higher order resolutions exist in these algebras, and that they determine higher order Ext-groups. In particular, the _Em$E_m$-term of the Adams spectral sequence (m>2)$(m>2)$ is such a higher Ext-group, providing a new way of constructing its differentials.     

Stable functors of derived equivalences and Gorenstein projective modules

From certain triangle functors, called nonnegative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of nonnegative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. In particular, we prove that the stable functor of a derived equivalence between two arbitrary rings provides an explicit triangle equivalence between the stable categories of Gorenstein projective modules. This generalizes a result of Y. Kato. Our results can also be applied to provide shorter proofs of some known results on homological conjectures.     

Homology of generalized partition posets

We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen-Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.     

Weighted locally gentle quivers and Cartan matrices

We introduce and study the class of weighted locally gentle quivers. This naturally extends the class of gentle quivers and gentle algebras, which have been intensively studied in the representation theory of finite-dimensional algebras, to a wider class of potentially infinite-dimensional algebras. Weights on the arrows of these quivers lead to gradings on the corresponding algebras. For natural grading by path lengths, any locally gentle algebra is Koszul. The class of locally gentle algebras consists of the gentle algebras together with their Koszul duals.Our main result is a general combinatorial formula for the determinant of the weighted Cartan matrix of a weighted locally gentle quiver. We show that this weighted Cartan determinant is a rational function which is completely determined by the combinatorics of the quiver-more precisely by the number and the weight of certain oriented cycles.     

Exact categories

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3$3\times 3$-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel–Quillen embedding theorem in an appendix.     

Module structures and the derived functors of iterated loop functors on unstable modules over the Steenrod algebra

The calculation of the iterated loop functors and their left derived functors on the category of unstable modules over the Steenrod algebra is a non-trivial problem; Singer constructed an explicit and functorial chain complex to calculate these functors. The results of Singer are analysed to give information on the behaviour of these functors with respect to the nilpotent filtration of the category of unstable modules.We show that, if an unstable module M supports an action of an unstable algebra K, then the derived functors of the iterated loop functors applied to M support actions of iterated doubles of K. This allows the finiteness results of Henn on unstable modules which support actions of unstable algebras to be applied to deduce structural results on the derived functors of iterated loops on such modules.     

The 1-type of a Waldhausen K-theory spectrum

We give a small functorial algebraic model for the 2-stage Postnikov section of the K-theory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors.     

Calculus of functors and model categories

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large categories, preprint, 2005]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the n-homogeneous model structure, the nth derivative is a Quillen functor to the category of spectra with Σn-action. After taking into account only finitary functors—which may be done in two different ways—the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T.G. Goodwillie [T.G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645-711 (electronic)].     

On exact categories and applications to triangulated adjoints and model structures

We show that Quillen?s small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures. In particular, the interplay of different exact structures on the category of complexes of quasi-coherent sheaves leads to a streamlined and generalized version of recent results obtained by Estrada, Gillespie, Guil Asensio, Hovey, Jørgensen, Neeman, Murfet, Prest, Trlifaj and possibly others.     

Structure sheaves of definable additive categories

We prove that the 2-category of small abelian categories with exact functors is anti-equivalent to the 2-category of definable additive categories. We define and compare sheaves of localisations associated to the objects of these categories. We investigate the natural image of the free abelian category over a ring in the module category over that ring and use this to describe a basis for the Ziegler topology on injectives; the last can be viewed model-theoretically as an elimination of imaginaries result.     

Expansions of abelian categories

Expansions of abelian categories are introduced. These are certain functors between abelian categories and provide a tool for induction/reduction arguments. Expansions arise naturally in the study of coherent sheaves on weighted projective lines; this is illustrated by various applications.     

Mixed algebras and quivers related to cell complexes

In this paper we study finite dimensional algebras arising from categories of perverse sheaves on finite regular cell complexes (cellular perverse algebras). We prove that such algebras are quasi-hereditary and have finite global dimension. We discuss some restrictions, under which cellular perverse algebras are Koszul. We also study the relationship between Koszul duality functors in the derived categories of categories of graded and non-graded modules over an algebra and its quadratic dual.     

Structures de dérivabilité

We introduce a very general framework in which Quillen's theorems of existence, composition and adjunction for derived functors can be proved. We thus generalize and unify previous results by Dwyer, Hirschhorn, Kan and Smith, obtained in their formalism of “homotopical categories,” and by Radulescu-Banu in the context of Cisinski's “derivable categories.”     

Modules in monoidal model categories

This paper studies the existence of and compatibility between derived change of ring, balanced product, and function module derived functors on module categories in monoidal model categories.     

Curved Koszul duality of algebras over unital versions of binary operads

We develop a curved Koszul duality theory for algebras presented by quadratic-linear-constant relations over unital versions of binary quadratic operads. As an application, we study Poisson n-algebras given by polynomial functions on a standard shifted symplectic space. We compute explicit resolutions of these algebras using curved Koszul duality. We use these resolutions to compute derived enveloping algebras and factorization homology on parallelized simply connected closed manifolds with coefficients in these Poisson n-algebras.     

Secondary derived functors and the Adams spectral sequence

Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of “additive groupoid enriched categories”, in which a secondary analog of homological algebra can be performed. We introduce secondary chain complexes and secondary resolutions leading to the concept of secondary derived functors. As a main result we show that the E₃-term of the Adams spectral sequence can be expressed as a secondary derived functor. This result can be used to compute the E₃-term explicitly by an algorithm.     

The Koszul Complex and Duality

This paper is basically about local, Cohen-Macaulay rings although many of the theorems are more general in nature. The major tool that we shall use is the Koszul complex and its self duality. The Koszul complex is closely linked with the functors Γ and λ which we shall define presently. Our main results will be the computation of the derived functors of Γ and λ.     

Abelian categories of modules over a (lax) monoidal functor

Crane and Yetter (Deformations of (bi)tensor categories, Cahier de Topologie et Géometrie Differentielle Catégorique, 1998) introduced a deformation theory for monoidal categories. The related deformation theory for monoidal functors introduced by Yetter (in: E. Getzler, M. Kapranov (Eds.), Higher Category Theory, American Mathematical Society Contemporary Mathematics, Vol. 230, American Mathematical Society, Providence, RI, 1998, pp. 117-134.) is a proper generalization of Gerstenhaber's deformation theory for associative algebras (Ann. Math. 78(2) (1963) 267; 79(1) (1964) 59; in: M. Hazewinkel, M. Gerstenhaber (Eds.), Deformation Theory of Algebras and Structure and Applications, Kluwer, Dordrecht, 1988, pp. 11-264). In the present paper we solidify the analogy between lax monoidal functors and associative algebras by showing that under suitable conditions, categories of functors with an action of a lax monoidal functor are abelian categories. The deformation complex of a monoidal functor is generalized to an analogue of the Hochschild complex with coefficients in a bimodule, and the deformation complex of a monoidal natural transformation is shown to be a special case. It is shown further that the cohomology of a monoidal functor F with coefficients in an F,F-bimodule is given by right derived functors.     

Division theorems for the Koszul complex

We establish a twisted version of Skoda’s estimate for the Koszul complex from which we get global division theorems for the Koszul complex. This generalizes Skoda’s division theorem. We also show how to use Skoda triples to produce division theorems for the Koszul complex.     

Gysin maps in Balmer-Witt theory

We introduce twisted Thom operators and the deformation to the normal cone isomorphisms in the derived Witt theory. Using these ingredients, we define push-forward maps of Witt groups along closed embeddings of smooth varieties and prove their standard properties, the most delicate of which turns out to be the functoriality of such maps.