Spectra, spectra, spectra – Tensor triangular spectra versus Zariski spectra of endomorphism rings

Abstract

We construct a natural continuous map from the triangular spectrum of a tensor triangulated category to the algebraic Zariski spectrum of the endomorphism ring of its unit object. We also consider graded and twisted versions of this construction. We prove that these maps are quite often surjective but far from injective in general. For instance, the stable homotopy category of finite spectra has a triangular spectrum much bigger than the Zariski spectrum of Z. We also give a first discussion of the spectrum in two new examples, namely equivariant KK -theory and stable A 1 -homotopy theory.18E30; 14F05, 19K35, 20C20, 55P42, 55U35 IntroductionAlgebraic geometers, stable topologists, finite group representation theorists, motivic theorists, noncommutative geometers and many other mathematicians have triangulated categories in common: The derived category of sheaves of modules over a scheme, the stable homotopy category of topological spectra, the derived category or the stable category of representations of a finite group or finite group scheme, the various motivic derived categories, Morel and Voevodsky's stable A 1 -homotopy category and equivariant KK -theory or E -theory of C -algebras are famous examples. In several cases, a tensor structure is also available and is especially well-behaved on the triangulated subcategory of compact objects. In the above examples, this leads us to focus on perfect complexes, finite topological spectra, finite dimensional representations, geometric motives, etc.This profusion of examples motivates the study of tensor triangulated categories per se. Emphasizing the geometric aspects of this unified theory leads us to a subject called tensor triangular geometry, to which the present paper belongs.To explain these ideas, let us denote by one of our tensor triangulated categories (say, of compact objects), by˝W ! the symmetric tensor x˝y D y˝x and by 1 2 the˝-unit: 1˝x D x . We began our geometric study in [1] with the definition of a topological space Spc./ called the spectrum of (see also Definition 1.3 below). We call it the triangular spectrum here, to avoid confusion with other meanings of the word "spectrum". This fundamental space Spc./ is the canvas on which to draw tensor triangular geometry. For instance, every object x 2 has a support, supp.x/ Spc./, which is a closed subset behaving nicely with respect to exact triangles and tensor product. In all applications though, the crucial anchor point is the computation of the triangular spectrum Spc./ in the first place. Without this knowledge, abstract results of tensor triangular geometry are difficult to translate into concrete terms. It is therefore a major challenge to compute the spectrum Spc./ in as many examples as possible, or at least to provide some information about that space when the full determination of Spc./ lies beyond reach of the community's current forces.Actually, by our paper [1, Theorem 4.10, Theorem 5.2] and Buan, Krause and Solberg [9, Corollary 5.2], we know that the information about contained in the ...

Higher comparison maps for the spectrum of a tensor triangulated category

“…Finally, it is worth mentioning that [DS14] has also defined generalizations of the original comparison maps from [Bal10a]. However, that work focuses on invertible objects in the category and goes in quite a different direction than our theory.…”

“…The basic reference is [Bal05] but the theory has been developed in a series of papers. The survey [Bal10b] provides a good overview and for our purposes the papers [Bal07,Bal10a] are particularly relevant.…”

“…We would like to go the other way around: develop general techniques for computing the spectrum Spc(K) and then obtain a classification of the thick ⊗-ideals in K by invoking Theorem 4.2.3. Balmer took the first step in this direction in [Bal10a] by defining two maps…”

“…For example, a (nice) scheme can be recovered from the tensor triangulated structure of its derived category of 1 perfect complexes, but not from the triangulated structure alone (see [Bal10a, Remark 64], for example). Using the ⊗-structure, Balmer [Bal05] has introduced the spectrum of a tensor triangulated category.…”

“…In any tensor triangulated category, the endomorphism ring of the unit is commutative, and the first step towards saying something about the spectrum of a general tensor triangulated category was taken in [Bal10a] where continuous maps In this dissertation, we will construct new comparison maps which generalize those mentioned above. More specifically, for each object X in a tensor triangulated category K we will define maps ρ X : supp(X ) → Spec(R X ) and ρ…”