We use homological ideals in triangulated categories to get a sufficient criterion for a pair of subcategories in a triangulated category to be complementary. We apply this criterion to construct the Baum-Connes assembly map for locally compact groups and torsion-free discrete quantum groups. Our methods are related to the abstract version of the Adams spectral sequence by Brinkmann and Christensen.
Bivariant (equivariant) K-theory is the standard setting for non-commutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from homological algebra: phantom maps, exact chain complexes, projective resolutions, and derived functors. We introduce these notions and apply them to examples from bivariant K-theory. An important observation of Beligiannis is that we can approximate our category by an Abelian category in a canonical way, such that our homological concepts reduce to the corresponding o…
Let $\ell$ be a commutative ring with involution $*$ containing an element $λ$ such that $λ+λ^*=1$ and let $\operatorname{Alg}^*_\ell$ be the category of $\ell$-algebras equipped with a semilinear involution and involution preserving homomorphisms. We construct a triangulated category $kk^h$ and a functor $j^h:\operatorname{Alg}^*_\ell\to kk^h$ that is homotopy invariant, matricially and hermitian stable and excisive and is universal initial with these properties. We prove that a version of Karoubi's fundamental theorem holds in $kk^h$. By the universal property of the latter, this implies tha…
We give an elementary introduction to the theory of triangulated categories covering their axioms, homological algebra in triangulated categories, triangulated subcategories, and Verdier localization. We try to use a minimal set of axioms for triangulated categories and derive all other statements from these, including the existence of biproducts. We conclude with a list of examples.
kk$^{\text{ban}}$ is a bivariant K-theory for Banach algebras that has reasonable homological properties, a product and is Morita invariant in a very general sense. We define it here by a universal property and ensure its existence in a rather abstract manner using triangulated categories. The definition ensures that there is a natural transformation from Lafforgue's theory KK$^{\text{ban}}$ into it so that one can take products of elements in KK$^{\text{ban}}$ that lie in kk$^{\text{ban}}$.
Using the machinery of weak fibration categories due to Schlank and the first author, we construct a convenient model structure on the pro-category of separable $C^*$-algebras $\mathrm{Pro}(\mathtt{SC^*})$. The opposite of this model category models the $\infty$-category of pointed noncommutative spaces $\mathtt{N}\mathcal{S_*}$ defined by the third author. Our model structure on $\mathrm{Pro}(\mathtt{SC^*})$ extends the well-known category of fibrant objects structure on $\mathtt{SC^*}$. We show that the pro-category $\mathrm{Pro}(\mathtt{SC^*})$ also contains, as a full coreflective subcateg…
A kind of motivic stable homotopy theory of algebras is developed. Explicit fibrant replacements for the $S^1$-spectrum and $(S^1,\mathbb G)$-bispectrum of an algebra are constructed. As an application, unstable, Morita stable and stable universal bivariant theories are recovered. These are shown to be embedded by means of contravariant equivalences as full triangulated subcategories of compact generators of some compactly generated triangulated categories. Another application is the introduction and study of the symmetric monoidal compactly generated triangulated category of $K$-motives. It i…
This paper is the first part of a project aimed at understanding deformations of triangulated categories, and more precisely their dg and A infinity models, and applying the resulting theory to the models occurring in the Homological Mirror Symmetry setup. In this first paper, we focus on models of derived and related categories, based upon the classical construction of twisted objects over a dg or $A_{\infty}$-algebra. For a Hochschild 2 cocycle on such a model, we describe a corresponding "curvature compensating" deformation which can be entirely understood within the framework of twisted ob…
We introduce and study several homological notions which generalise the discrete derived categories of D. Vossieck. As an application, we show that Vossieck discrete algebras have this property with respect to all bounded t-structures. We give many examples of triangulated categories regarding these notions.
Let T be a triangulated category with coproducts, C the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [Adams71]: All contravariant homological functors C --> Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [Neeman97], it was proved that Adams' theorem remains true as long as C is countable, but can fail in general. The failure exhibited was that…
Using a homological invariant together with an obstruction class in a certain Ext^2-group, we may classify objects in triangulated categories that have projective resolutions of length two. This invariant gives strong classification results for actions of the circle group on C*-algebras, C*-algebras over finite unique path spaces, and graph C*-algebras with finitely many ideals.
We study Grothedieck groups of triangulated categories using weight structures, weight complexes, and the corresponding pure (co)homological functors. We prove some general statements on $K_0$ of weighted categories and apply it to Voevodsky motives endowed with so-called Chow weight structures. We obtain certain "motivic substitutes" for smooth compactifications of smooth varieties over arbitrary perfect fields; this enables us to make certain unramified cohomology and Euler characteristic calculations that are closely related to results of T. Ekedahl and B. Kahn.
Extension conjecture states that if a simple module over an artin algebra has nonzero first self-extension group then it has nonzero i-th self-extension group for infinitely many positive integers i. It is shown by recollement of triangulated categories and differential graded homological algebra approach that extension conjecture is true for finite-dimensional elementary algebras over a field, particularly, for finite-dimensional algebras over an algebraically closed field. Moreover, bimodule approach is introduced to strong no loop conjecture, which provides two new proofs of Igusa-Liu-Paque…
We study (not necessarily connected) Z-graded A-infinity-algebras and their A-infinity-modules. Using the cobar and the bar construction and Quillen's homotopical algebra, we describe the localisation of the category of A-infinity-algebras with respect to A-infintity-quasi-isomorphisms. We then adapt these methods to describe the derived category of an augmented A-infinity-algebra A. The case where A is not endowed with an augmentation is treated differently. Nevertheless, when A is strictly unital, its derived category can be described in the same way as in the augmented case. Next, we compar…
