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The spectrum of prime ideals in tensor triangulated categories
 J. Reine Angew. Math
"... Abstract. We define the spectrum of a tensor triangulated category K as the set of socalled 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 ..."
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Abstract. We define the spectrum of a tensor triangulated category K as the set of socalled 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 quasicompactness. 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 quasicompact 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.
Lecture notes on motivic cohomology
 of Clay Mathematics Monographs. American Mathematical Society
, 2006
"... 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 ..."
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Cited by 21 (2 self)
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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
Picard groups, Grothendieck rings, and Burnside rings of categories
, 2000
"... 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. ..."
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Cited by 18 (0 self)
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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.
Isomorphisms Between Left And Right Adjoints
 Theory Appl. Categ
, 2003
"... 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 le ..."
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Cited by 15 (2 self)
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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.
Homotopy theory of comodules over a Hopf algebroid
, 2003
"... 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 wellbehaved Hopf algebroid (A, Γ). That is, we construct ..."
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Cited by 13 (3 self)
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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 wellbehaved Hopf algebroid (A, Γ). That is, we construct
Fixed point theory and trace for bicategories
, 2007
"... 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 inde ..."
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Cited by 4 (1 self)
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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
The Connection Between May’s Axioms for a Triangulated Tensor Product and Happel’s Description of the Derived Category of the Quiver D_4
 DOCUMENTA MATH.
, 2002
"... 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. ..."
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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. 2425]. The second and more important is the beautiful work of Happel, in [9], which can be used to better organise May’s axioms.
The refined transfer, bundle structures and algebraic ktheory
, 2007
"... We give new homotopy theoretic criteria for deciding when a fibration with homotopy finite fibers admits a reduction to a fiber bundle with compact topological manifold fibers. The criteria lead to an unexpected result about homeomorphism groups of manifolds. A tool used in the proof is a surjectiv ..."
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We give new homotopy theoretic criteria for deciding when a fibration with homotopy finite fibers admits a reduction to a fiber bundle with compact topological manifold fibers. The criteria lead to an unexpected result about homeomorphism groups of manifolds. A tool used in the proof is a surjective splitting of the assembly map for Waldhausen’s functor A(X). We also give concrete examples of fibrations having a reduction to a fiber bundle with compact topological manifold fibers but which fail to admit a compact fiber smoothing. The examples are detected by algebraic Ktheory invariants. We consider a refinement of the BeckerGottlieb transfer. We show that a version of the axioms described by Becker and Schultz uniquely determines the refined transfer for the class of fibrations admitting a reduction to a fiber bundle with compact topological manifold fibers. In an