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Ideals in triangulated categories: Phantoms, ghosts and skeleta
 Adv. in Math
, 1998
"... ABSTRACT. We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral seq ..."
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Cited by 41 (5 self)
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ABSTRACT. We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik–Schnirelmann category of a space, and we calculate this stable analogue for lowdimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case of A ∞ modules over an A ∞ ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps from X to Y can always be described as a lim1 ←− group. The last two sections focus
Absolute, Relative, And Tate Cohomology Of Modules Of Finite Gorenstein Dimension
, 2000
"... this paper: ..."
TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
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MONADS AND COMONADS ON MODULE CATEGORIES
"... known in module theory that any Abimodule B is an Aring if and only if the functor − ⊗A B: MA → MA is a monad (or triple). Similarly, an Abimodule C is an Acoring provided the functor − ⊗A C: MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of − ⊗A B and comodu ..."
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Cited by 12 (10 self)
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known in module theory that any Abimodule B is an Aring if and only if the functor − ⊗A B: MA → MA is a monad (or triple). Similarly, an Abimodule C is an Acoring provided the functor − ⊗A C: MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of − ⊗A B and comodules (or coalgebras) of − ⊗A C are well studied in the literature. On the other hand, the right adjoint endofunctors HomA(B, −) and HomA(C, −) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find
Two kinds of derived categories, Koszul duality, and comodulecontramodule correspondence
, 2009
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Comparison of relative cohomology theories with respect to semidualizing modules, preprint
, 2007
"... Abstract. We compare and contrast various relative cohomology theories that arise from resolutions involving semidualizing modules. We prove a general balance result for relative cohomology over a CohenMacaulay ring with a dualizing module, and we demonstrate the failure of the naive version of bal ..."
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Cited by 8 (5 self)
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Abstract. We compare and contrast various relative cohomology theories that arise from resolutions involving semidualizing modules. We prove a general balance result for relative cohomology over a CohenMacaulay ring with a dualizing module, and we demonstrate the failure of the naive version of balance one might expect for these functors. We prove that the natural comparison morphisms between relative cohomology modules are isomorphisms in several cases, and we provide a Yonedatype description of the first relative Ext functor. Finally, we show by example that each distinct relative cohomology construction does in fact result in a different functor.
Homological algebra in bivariant Ktheory and other triangulated categories
"... Abstract. Bivariant (equivariant) Ktheory is the standard setting for noncommutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from homological algebra: phantom maps, exact chain complexes, pro ..."
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Cited by 6 (2 self)
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Abstract. Bivariant (equivariant) Ktheory is the standard setting for noncommutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from homological algebra: phantom maps, exact chain complexes, projective resolutions, and derived functors. We introduce these notions and apply them to examples from bivariant Ktheory. An important observation of Beligiannis is that we can approximate our
C ∗ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED KTHEORY
, 810
"... Abstract. We define the filtrated Ktheory of a C ∗algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated Ktheory. For finite spaces with totally ordered lattice of open subsets, this spect ..."
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Cited by 2 (0 self)
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Abstract. We define the filtrated Ktheory of a C ∗algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated Ktheory. For finite spaces with totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated Ktheory is not yet a complete invariant. We describe two C ∗algebras over a space X with four points that have isomorphic filtrated Ktheory without being KK(X)equivalent. For this space X, we enrich filtrated Ktheory by another Ktheory functor to a complete invariant up to KK(X)equivalence that satisfies a Universal Coefficient Theorem. 1.