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22
On differential graded categories
 INTERNATIONAL CONGRESS OF MATHEMATICIANS. VOL. II
, 2006
"... Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié. ..."
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Cited by 66 (3 self)
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Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié.
Localization theorems in topological Hochschild homology and topological cyclic homology
, 2008
"... We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of ..."
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Cited by 16 (2 self)
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We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of ThomasonTrobaugh in Ktheory. We also deduce versions of Thomason’s blowup formula and the projective bundle formula for THH and TC.
Kregularity, cdhfibrant Hochschild homology, and a conjecture of Vorst
 J. Amer. Math. Soc
, 2006
"... Abstract. In this paper we prove that for an affine scheme essentially of finite type over a field F and of dimension d, Kd+1regularity implies regularity, assuming that the characteristic of F is zero. This verifies a conjecture of Vorst. ..."
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Abstract. In this paper we prove that for an affine scheme essentially of finite type over a field F and of dimension d, Kd+1regularity implies regularity, assuming that the characteristic of F is zero. This verifies a conjecture of Vorst.
spectral sequences on singular schemes and failure of generalized Gersten conjecture
 Proceedings of the American Mathematical Society 137, no 1 (2009
"... Abstract. We construct a new localglobal spectral sequence for Thomason’s nonconnective Ktheory, generalizing the Quillen spectral sequence to possibly nonregular schemes. Our spectral sequence starts at the E1page where it displays Gerstentype complexes. It agrees with Thomason’s hypercohomo ..."
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Abstract. We construct a new localglobal spectral sequence for Thomason’s nonconnective Ktheory, generalizing the Quillen spectral sequence to possibly nonregular schemes. Our spectral sequence starts at the E1page where it displays Gerstentype complexes. It agrees with Thomason’s hypercohomology spectral sequence exactly when these Gerstentype complexes are locally exact, a condition which fails for general singular schemes, as we indicate. Our main result is the following application of abstract triangular geometry [2]. Theorem 1. Let X be a (topologically) noetherian scheme of finite Krull dimension. Then there exists a spectral sequence whose first page is
Geometric description of the connecting homomorphism for Witt groups, preprint
, 2008
"... Abstract. We give a geometric setup in which the connecting homomorphism in the localization long exact sequence for Witt groups decomposes as the pullback to the exceptional fiber of a suitable blowup followed by a pushforward. 1. ..."
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Abstract. We give a geometric setup in which the connecting homomorphism in the localization long exact sequence for Witt groups decomposes as the pullback to the exceptional fiber of a suitable blowup followed by a pushforward. 1.
THE KTHEORY OF TORIC VARIETIES
"... Abstract. Recent advances in computational techniques for Ktheory allow us to describe the Ktheory of toric varieties in terms of the Ktheory of fields and simple cohomological data. 1. ..."
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Abstract. Recent advances in computational techniques for Ktheory allow us to describe the Ktheory of toric varieties in terms of the Ktheory of fields and simple cohomological data. 1.
STABILIZATION OF THE WITT GROUP
"... In this Note, using an idea due to Thomason [8], we define a “homology theory ” on the category of rings which satisfies excision, exactness, homotopy (in the algebraic sense) and periodicity of order 4. For regular noetherian rings, we find Balmers’s higher Witt groups. For more general rings, this ..."
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In this Note, using an idea due to Thomason [8], we define a “homology theory ” on the category of rings which satisfies excision, exactness, homotopy (in the algebraic sense) and periodicity of order 4. For regular noetherian rings, we find Balmers’s higher Witt groups. For more general rings, this homology is isomorphic to the KTtheory of Hornbostel [3], inspired by the work of Williams [9]. For real or complex C*algebras, we recover up to 2 torsion topological Ktheory. 1. Let A be a ring with an antiinvolution a € a and let ε be an element of the center of A such that εε = 1. We assume also that 2 is invertible in the ring. There are now well known definitions of the higher hermitian Kgroup (denoted by εLn (A), as in [5]) and the higher Witt group εWn (A) : this is the cokernel of the map induced by the hyperbolic functor Kn (A) zzc εLn (A) where the K n (A) denote the Quillen Kgroup (which is defined for all values of n [ Z). One of the fundamental results of higher Witt theory is the periodicity isomorphism (where Z ’ = Z[1/2], cf.[4]) εWn (A) * Z ’ – εWn2 (A) * Z’ It is induced by the cupproduct with a genuine element u 2 [1 L2 (Z’). By analogy with algebraic topology, we shall call u 2 the Bott element in Witt theory. This element is explicitly described in the following way. We consider the 2 x 2 matrix (with the involution defined by z = z1 and t = t1 and where we put λ = λ = 1/2).