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101
Morava Ktheories and localisation
 Mem. Amer. Math. Soc
, 1999
"... Abstract. We study the structure of the categories of K(n)local and E(n)local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)nilpoten ..."
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Cited by 63 (18 self)
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Abstract. We study the structure of the categories of K(n)local and E(n)local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)nilpotent spectra. We give a number of useful extensions to the theory of vn self maps of finite spectra, and to the theory of Landweber exactness. We show that certain rings of cohomology operations are left Noetherian, and deduce some powerful finiteness results. We study the Picard group of invertible K(n)local spectra, and the problem of grading homotopy groups over it. We prove (as announced by Hopkins and Gross) that the BrownComenetz dual of MnS lies in the Picard group. We give a detailed analysis of some examples when n =1 or 2, and a list of open problems.
Complete modules and torsion modules
 Amer. J. Math
"... Abstract. Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded Rmodules there are naturally defined subcategories of Atorsion objects and of Acomplete objects. Under a finiteness condition on A, we develop a Morita theory for these subc ..."
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Cited by 34 (5 self)
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Abstract. Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded Rmodules there are naturally defined subcategories of Atorsion objects and of Acomplete objects. Under a finiteness condition on A, we develop a Morita theory for these subcategories, find conceptual interpretations for some associated algebraic functors, and, in appropriate commutative situations, identify the associated functors as local homology or local cohomology. Some of the results are suprising even in the case R = Z and A = Z/p. 1.
The stable derived category of a Noetherian scheme
 Compos. Math
"... Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday. Abstract. For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the s ..."
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Cited by 34 (5 self)
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Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday. Abstract. For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an analogue of maximal CohenMacaulay approximations, a construction of Tate cohomology, and an extension of the classical Grothendieck duality. In addition, the relevance of the stable derived category in modular representation theory is indicated.
The additivity of traces in triangulated categories
 Adv. Math
"... Abstract. We explain a fundamental additivity theorem for Euler characteristics and generalized trace maps in triangulated categories. The proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation. The refinement consists of several new axioms relatin ..."
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Cited by 34 (7 self)
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Abstract. We explain a fundamental additivity theorem for Euler characteristics and generalized trace maps in triangulated categories. The proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation. The refinement consists of several new axioms relating products and distinguished triangles. The axioms hold in the examples and shed light on generalized homology and cohomology theories. Contents 1. Generalized trace maps 2 2. Triangulated categories 6 3. Weak pushouts and weak pullbacks 9 4. The compatibility axioms 11
Model category structures on chain complexes of sheaves
 Trans. Amer. Math. Soc
"... of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently ..."
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Cited by 27 (0 self)
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of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently
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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Cited by 25 (1 self)
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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.
Products on MUmodules
 Trans. Amer. Math. Soc
, 1999
"... modules over highly structured ring spectra to give new constructions of MUmodules such as BP, K(n) and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over MU [ 1] ∗ that are concentrated ..."
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Cited by 23 (5 self)
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modules over highly structured ring spectra to give new constructions of MUmodules such as BP, K(n) and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over MU [ 1] ∗ that are concentrated in 2 degrees divisible by 4; this guarantees that various obstruction groups are trivial. We extend these results to the cases where 2 = 0 or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in BaasSullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising 2local MU∗modules as MUmodules. 1.