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Abstract. We define the spectrum of a tensor triangulated category K as the set of so-called prime ideals, endowed with a suitable topology. In this very generality, the spectrum is the universal space in which one can define supports for objects of K. This construction is functorial with respect to all tensor triangulated functors. Several elementary properties of schemes hold for such spaces, e.g. the existence of generic points and some quasi-compactness. Locally trivial morphisms are proved to be nilpotent. We establish in complete generality a classification of thick ⊗-ideal subcategories in terms of arbitrary unions of closed subsets with quasi-compact complements (Thomason’s theorem for schemes, mutatis mutandis). Finally, we compute examples and show that our spectrum unifies both the underlying spaces of schemes in algebraic geometry and of support varieties in modular representation theory.

We discuss the Picard group, the Grothendieck ring, and the Burnside ring of a symmetric monoidal category, and we consider examples from algebra, homological algebra, topology, and algebraic geometry.

From the point of view taken in these lectures, motivic cohomology with coefficients in an abelian group A is a family of contravariant functors H p,q (−, A) : Sm/k → Ab from smooth schemes over a given field k to abelian groups, indexed by

There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a shift of the left adjoint specified by an appropriate "dualizing object". Typically the left adjoint is well understood while the right adjoint is more mysterious, and the result identifies the right adjoint in familiar terms. We give a categorical discussion of such results. One essential point is to di#erentiate between the classical framework that arises in algebraic geometry and a deceptively similar, but genuinely di#erent, framework that arises in algebraic topology.

Given a good homology theory E and a topological space X, E∗X is not just an E∗-module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A, Γ). That is, we construct

The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point index that give a converse to the Lefschetz fixed point theorem. An important part of this theorem is the identification of these different invariants. We define a generalization of the trace in symmetric monoidal categories to a trace in bicategories with shadows. We show the invariants used in the converse of the Lefschetz fixed point theorem are examples of this trace and that the functoriality of the trace provides some of the necessary identifications. The methods used here do not use simplicial techniques and so generalize readily to other contexts. iii Contents

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In an important recent paper [12], May gave an axiomatic description of the properties of triangulated categories with a symmetric tensor product. The main point of the current article is that there are two other results in the literature which can be used to shed considerable light on May’s work. The first is a construction of Verdier’s, which appeared in Beilinson, Bernstein and Deligne’s [4, Prop. 1.1.11, pp. 24-25]. The second and more important is the beautiful work of Happel, in [9], which can be used to better organise May’s axioms.

Theorem. The following are equivalent for j ≥ 3. (1) The element h 2 j in the Adams spectral sequence for the sphere survives to a stable homotopy class of order 2. (2) There is a 3-cell complex with a mod 2 Moore space at the bottom, the relative attaching map of the top cell is detected by h 2 j, and Sq 2j+1 is nontrivial from bottom to top. (3) The element hj+1e0 in the Adams spectral sequence for the mod 2 Moore space survives to a stable homotopy class (where e0 is the bottom homology class in the Moore space). The fact which ties these together is the Adams differential d2hj+1 = h0h 2 j, j ≥ 3 Proof. (2) clearly implies each of the other claims. We start by proving that (3) implies (2). That is to say, the relative attaching map of the top cell of the mapping cone of an element of the homotopy of a Moore space represented by hj+1e0 is represented by h 2 j.