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16
Cyclic homology, cdhcohomology and negative Ktheory
, 2005
"... We prove a blowup formula for cyclic homology which we use to show that infinitesimal Ktheory satisfies cdhdescent. Combining that result with some computations of the cdhcohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic Ktheor ..."
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Cited by 19 (7 self)
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We prove a blowup formula for cyclic homology which we use to show that infinitesimal Ktheory satisfies cdhdescent. Combining that result with some computations of the cdhcohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic Ktheory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.
The nilpotence conjecture in Ktheory of toric varieties
 Inventiones Math
"... Abstract. It is shown that all nontrivial elements in higher Kgroups of toric varieties over a class of regular rings are annihilated by iterations of the natural Frobenius type endomorphisms. This is a higher analog of the triviality of vector bundles on affine toric varieties. 1. Statement of the ..."
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Cited by 9 (4 self)
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Abstract. It is shown that all nontrivial elements in higher Kgroups of toric varieties over a class of regular rings are annihilated by iterations of the natural Frobenius type endomorphisms. This is a higher analog of the triviality of vector bundles on affine toric varieties. 1. Statement of the main result The nilpotence conjecture in Ktheory of toric varieties, treated in our previous works, asserts the following: Conjecture 1.1. Let R be a (commutative) regular ring, M be arbitrary commutative, cancellative, torsion free monoid without nontrivial units, and i be a nonnegative integral number. Then for every sequence c = (c1, c2,...) of natural numbers ≥ 2 and every element x ∈ Ki(R[M]) there exists an index jx ∈ N such that (c1 · · ·cj)∗(x) ∈ Ki(R) for all j> jx. Here R[M] is the monoid Ralgebra of M and for a natural number c the endomorphism of Ki(R[M]), induced by the Ralgebra endomorphism R[M] → R[M], m ↦ → m c, m ∈ M, is denoted by c ∗ (writing the monoid operation multiplicatively). Speaking loosely, this conjecture says that the multiplicative monoid of natural numbers acts nilpotently on Ki(R[M]). The following is a reformulation of Conjecture 1.1 in a typical case: Conjecture. Let R and i be as above and c be a natural number ≥ 2. Assume C ⊂ R n (n ∈ N) is a convex cone, containing no affine lines. Then Ki(R) = Ki(R[C ∩ (c −1 Z) n]). (Here c −1 Z is the additive group of the localization of Z at c.) Although the conjecture is stated for affine (not necessarily normal) toric varieties it yields a similar result for all quasiprojective toric varieties. In fact, the motivation behind the nilpotence conjecture can be described by the diagram of relationships:
Comparison between algebraic and topological Ktheory of locally convex algebras
, 2006
"... This paper is concerned with the algebraic Ktheory of locally convex Calgebras stabilized by operator ideals, and its comparison with topological Ktheory. We show that if L is locally convex and J a Fréchet operator ideal, then all the different variants of topological Ktheory agree on the compl ..."
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Cited by 3 (0 self)
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This paper is concerned with the algebraic Ktheory of locally convex Calgebras stabilized by operator ideals, and its comparison with topological Ktheory. We show that if L is locally convex and J a Fréchet operator ideal, then all the different variants of topological Ktheory agree on the completed projective tensor product Lˆ⊗J, and that the obstruction for the comparison map K(Lˆ⊗J) → Ktop (Lˆ⊗J) to be an isomorphism is (absolute) algebraic cyclic homology. We prove the existence of an exact sequence (Theorem 6.2.1) K top
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.
The Steinberg group of a monoid ring, nilpotence, and algorithms
 J. Algebra
"... Abstract. For a regular ring R and an affine monoid M the homotheties of M act nilpotently on the Milnor unstable groups of R[M]. This strengthens the K2 part of the main result of [G5] in two ways: the coefficient field of characteristic 0 is extended to any regular ring and the stable K2group is ..."
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Abstract. For a regular ring R and an affine monoid M the homotheties of M act nilpotently on the Milnor unstable groups of R[M]. This strengthens the K2 part of the main result of [G5] in two ways: the coefficient field of characteristic 0 is extended to any regular ring and the stable K2group is substituted by the unstable ones. The proof is based on a polyhedral/combinatorial technique, computations in Steinberg groups, and a substantially corrected version of an old result on elementary matrices by Mushkudiani [Mu]. A similar stronger nilpotence result for K1 and algorithmic consequences for factorization of high Frobenius powers of invertible matrices are also derived. 1.
ON RELATIVE AND BIRELATIVE ALGEBRAIC KTHEORY OF RINGS OF FINITE CHARACTERISTIC
, 810
"... Throughout, we fix a prime number p and consider unital associative rings in which p is nilpotent. It was proved by Weibel [26, Cor. 5.3, Cor. 5.4] long ago that, for such rings, the relative Kgroups associated with a nilpotent extension and the birelative Kgroups associated with a Milnor square ..."
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Throughout, we fix a prime number p and consider unital associative rings in which p is nilpotent. It was proved by Weibel [26, Cor. 5.3, Cor. 5.4] long ago that, for such rings, the relative Kgroups associated with a nilpotent extension and the birelative Kgroups associated with a Milnor square are pprimary torsion groups.
On
, 810
"... relative and birelative algebraic Ktheory of rings of finite characteristic ..."
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relative and birelative algebraic Ktheory of rings of finite characteristic