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131
Stable model categories are categories of modules
 TOPOLOGY
, 2003
"... A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for ..."
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Cited by 128 (24 self)
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A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R.
On Gorenstein projective, injective and flat dimensions  a functorial description with applications
, 2004
"... For a large class of rings, including all those encountered in algebraic geometry, we establish the conjectured Moritalike equivalence between the full subcategory of complexes of finite Gorenstein flat dimension and that of complexes of finite Gorenstein injective dimension. This functorial descr ..."
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Cited by 98 (19 self)
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For a large class of rings, including all those encountered in algebraic geometry, we establish the conjectured Moritalike equivalence between the full subcategory of complexes of finite Gorenstein flat dimension and that of complexes of finite Gorenstein injective dimension. This functorial description meets the expectations and delivers a series of new results, which allows us to establish a wellrounded theory for Gorenstein dimensions. For any pair of adjoint functors, C
KTheory and Derived Equivalences
, 2003
"... We show that if two rings have equivalent derived categories then they have the same algebraic Ktheory. Similar results are given for Gtheory, and for a large class of abelian categories. ..."
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Cited by 45 (7 self)
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We show that if two rings have equivalent derived categories then they have the same algebraic Ktheory. Similar results are given for Gtheory, and for a large class of abelian categories.
Moduli of objects in dgcategories
, 2006
"... To any dgcategory T (over some base ring k), we define a D −stack MT in the sense of [HAGII], classifying certain T opdgmodules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness ..."
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Cited by 36 (2 self)
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To any dgcategory T (over some base ring k), we define a D −stack MT in the sense of [HAGII], classifying certain T opdgmodules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness conditions on T (e.g. if it is saturated) the D −stack MT is locally geometric (i.e. union of open and geometric substacks). As a consequence we prove the algebraicity of the group of autoequivalences of saturated dgcategories. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as
Compact corigid objects in triangulated categories and cotstructures
"... Abstract. In the work of Hoshino, Kato and Miyachi, [11], the authors look at tstructures induced by a compact object, C, of a triangulated category, T, which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a nondegenerate tstructure on ..."
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Cited by 33 (8 self)
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Abstract. In the work of Hoshino, Kato and Miyachi, [11], the authors look at tstructures induced by a compact object, C, of a triangulated category, T, which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a nondegenerate tstructure on T whose heart is equivalent to Mod(End(C) op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, S, of a triangulated category, T, induces a structure similar to a tstructure which we shall call a cotstructure. We also show that the coheart of this nondegenerate cotstructure is equivalent to Mod(End(S) op), and hence an abelian subcategory of T. Suppose T is a triangulated category with set indexed coproducts and let
Morita theory in abelian, derived and stable model categories, Structured ring spectra
 London Math. Soc. Lecture Note Ser
, 2004
"... These notes are based on lectures given at the Workshop on Structured ring spectra and ..."
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Cited by 27 (0 self)
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These notes are based on lectures given at the Workshop on Structured ring spectra and
From triangulated categories to Lie algebras: A theorem of Peng and Xiao, Trends in representation theory of algebras and related
, 2006
"... In his seminal article [13], Ringel showed how to associate to any finitary ring Λ an associative unital algebra H(Λ), with structure constants encoding information about extensions between finite modules. This generalised the Hall algebra [3, 17], which deals with the ring of padic integers Zp and ..."
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Cited by 26 (1 self)
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In his seminal article [13], Ringel showed how to associate to any finitary ring Λ an associative unital algebra H(Λ), with structure constants encoding information about extensions between finite modules. This generalised the Hall algebra [3, 17], which deals with the ring of padic integers Zp and finite pgroups. In the subsequent article [14] it is shown that if Λ is a representationdirected algebra over a finite field k, then the structure constants are given by evaluating integer polynomials. Using these Hall polynomials as structure constants, one may therefore form the generic RingelHall algebra over Z[T]. Let n(Λ) be the subgroup of H(Λ) generated by the indecomposable modules. If we specialise T ↦ → 1, then Z ⊗ Z[T]n(Λ) becomes a Lie subalgebra of Z ⊗ Z[T]H(Λ). In fact, over the rational numbers, Q ⊗ Z[T] H(Λ) is isomorphic to the universal enveloping algebra of Q ⊗ Z[T] n(Λ). In particular, let Λ be a representationfinite hereditary kalgebra and let g = n − ⊕h⊕n+ be the semisimple complex Lie algebra of the same type as Λ. Then Z ⊗ Z[T]n(Λ) can be identified with the Chevalley Zform of n+, and Z ⊗ Z[T]H(Λ) becomes the Kostant Zform of the universal enveloping algebra U(n+) [15]. For a general finite dimensional hereditary kalgebra Λ one considers the composition algebra, the subalgebra generated by the simple modules. This also has a generic version, constructed as a subalgebra of a direct product over infinitely many finite fields of composition algebras [16]. Green showed in [2] that the generic composition algebra (after twisting the multiplication via the Euler form of the category modΛ) is isomorphic to the quantum group of the same type as Λ. Therefore, we can realise the quantum group of any symmetrisable KacMoody Lie algebra via the module categories of finite dimensional hereditary kalgebras. A natural question is whether it is possible to obtain the full (quantised) enveloping algebra, or at least the full Lie algebra. The latter question was answered by Peng and Xiao in [9] for the affine Lie algebras of type Ã, and in [10] for the
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.
Derived Azumaya algebras and generators for twisted derived categories
, 2011
"... We introduce a notion of derived Azumaya algebras over ring and schemes generalizing the notion of Azumaya algebras of [Gr]. We prove that any such algebra B on a scheme X provides a class φ(B) in H 1 et(X, Z) × H 2 et(X, Gm). We prove that for X a quasicompact and quasiseparated scheme φ defines ..."
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We introduce a notion of derived Azumaya algebras over ring and schemes generalizing the notion of Azumaya algebras of [Gr]. We prove that any such algebra B on a scheme X provides a class φ(B) in H 1 et(X, Z) × H 2 et(X, Gm). We prove that for X a quasicompact and quasiseparated scheme φ defines a bijective correspondence, and in particular that any class in H 2 et(X, Gm), torsion or not, can be represented by a derived Azumaya algebra on X. Our result is a consequence of a more general theorem about the existence of compact generators in twisted derived categories, with coefficients in any local system of reasonable dgcategories, generalizing the well known existence of compact generators in derived categories of quasicoherent sheaves of [BoVa]. A huge part of this paper concerns the treatment of twisted derived categories, as well as the proof that the existence of compact generator locally for the fppf topology implies the existence of a global compact generator. We present explicit examples of derived Azumaya algebras that are not represented by classical Azumaya algebras, as well as applications of our main result to the localization for twisted algebraic Ktheory and to the stability of saturated dgcategories by direct pushforwards along smooth and proper maps. Contents 1 The local theory 5