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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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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
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.
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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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". Typica ..."
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Cited by 26 (3 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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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
THE MUKAI PAIRING, I: THE HOCHSCHILD STRUCTURE
, 2003
"... We study the Hochschild structure of a smooth space or orbifold, emphasizing the importance of a pairing defined on Hochschild homology which generalizes a similar pairing introduced by Mukai on the cohomology of a K3 surface. We discuss those properties of the structure which can be derived witho ..."
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Cited by 23 (2 self)
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We study the Hochschild structure of a smooth space or orbifold, emphasizing the importance of a pairing defined on Hochschild homology which generalizes a similar pairing introduced by Mukai on the cohomology of a K3 surface. We discuss those properties of the structure which can be derived without appealing to the HochschildKostantRosenberg isomorphism and Kontsevich formality, namely: – functoriality of homology, commutation of pushforward with the Chern character, and adjointness with respect to the generalized pairing; – formal HirzebruchRiemannRoch and the Cardy condition from physics; – invariance of the full Hochschild structure under FourierMukai transforms. Connections with homotopy theory and TQFT’s are discussed in an appendix. A separate paper [9] treats consequences of the HKR isomorphism. Applications of these results to the study of a mirror symmetric analogue of ChenRuan’s orbifold product will be presented in a
Kaplansky classes and derived categories
 Math. Z
"... chain complexes of quasicoherent sheaves over a quasicompact and semiseparated scheme X. The approach generalizes and simplifies the method used by the author in [Gil04] and [Gil06] to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of ..."
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chain complexes of quasicoherent sheaves over a quasicompact and semiseparated scheme X. The approach generalizes and simplifies the method used by the author in [Gil04] and [Gil06] to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed, much of the paper is dedicated to showing that in any Grothendieck category G, any nice enough class of objects F induces a model structure on the category Ch(G) of chain complexes. The main technical requirement on F is the existence of a regular cardinal κ such that every object F ∈ F satisfies the following property: Each κgenerated subobject of F is contained in another κgenerated subobject S for which S, F/S ∈ F. Such a class F is called a Kaplansky class. Kaplansky classes first appeared in [ELR02] in the context of modules over a ring R. We study in detail the connection between Kaplansky classes and model categories. We also find simple conditions to put on F which will guarantee that our model structure is monoidal. We will see that in several categories the class of flat objects form such Kaplansky classes, and hence induce monoidal model structures on the associated chain complex categories. We will also see that in any Grothendieck category G, the class of all objects is a Kaplansky class which induces the usual (nonmonoidal) injective model structure on Ch(G). 1.