Algebraic K-theory and descent for blow-ups

Kerz, Moritz; Strunk, Florian; Tamme, Georg

doi:10.1007/s00222-017-0752-2

Cited by 52 publications

(67 citation statements)

References 42 publications

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“…On the geometry of the blow-up of a complex manifold with a smooth center, the de Rham blow-up formula shows the variant of de Rham cohomology under the blow-up transformations. In literatures, there are many different versions of blow-up formulas for various (co)homology theories; for instance, the cyclic homology [11], the algebraic K-theory [24], and the topological Hochschild homology [7].…”

Section: Introductionmentioning

confidence: 99%

Dolbeault cohomologies of blowing up complex manifolds

Rao

Song

Yang

2019

Journal de Mathématiques Pures et Appliquées

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We prove a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. As corollaries, we present a uniform proof for bimeromorphic invariance of (•, 0)-and (0, •)-Hodge numbers on a compact complex manifold, and obtain the equality for the numbers of the blow-ups and blow-downs in the weak factorization of the bimeromorphic map between two compact complex manifolds with equal (1, 1)-Hodge number or equivalently second Betti number. Many examples of the latter one are listed. Inspired by these, we obtain the bimeromorphic stability for degeneracy of the Frölicher spectral sequences at E1 on compact complex threefolds and fourfolds.

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Section: Introductionmentioning

confidence: 99%

Dolbeault cohomologies of blowing up complex manifolds

Rao

Song

Yang

2019

Journal de Mathématiques Pures et Appliquées

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“…Proof. In view of the main results of [22], the theorem is equivalent to proving that C M K i (X) = 0 for i ≤ −d. Lemma 2.3 allows us to assume that X is reduced.…”

Section: The Main Resultmentioning

confidence: 99%

“…An affirmative answer to this conjecture was obtained recently by Kerz, Strunk and Tamme [22]. For Noetherian rings containing Q, this was earlier solved by Cortiñas, Haesemeyer, Schlichting and Weibel [3] (see also [10], [23] and [38] for older results in positive characteristics).The main technical tool that goes into the proof of Weibel's conjecture in [22] is a pro-cdhdescent theorem for algebraic K-theory. However, the final step in the proof of the conjecture 2010 Mathematics Subject Classification.…”

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confidence: 95%

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Negative 𝐾-theory and Chow group of monoid algebras

Krishna¹,

Sarwar²

2020

𝐾-Theory in Algebra, Analysis And Topology

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We show, for a finitely generated partially cancellative torsion-free commutative monoid M , that K i (R) ∼ = K i (R[M ]) whenever i ≤ −d and R is a quasi-excellent Q-algebra of Krull dimension d ≥ 1. In particular, K i (R[M ]) = 0 for i < −d. This is a generalization of Weibel's K-dimension conjecture to monoid algebras. We show that this generalization fails for X[M ] if X is not an affine scheme. We also show that the Levine-Weibel Chow group of 0-cycles CH LW 0 (k[M ]) vanishes for any finitely generated commutative partially cancellative monoid M if k is an algebraically closed field. 0.1. Weibel's conjecture for monoid algebras. Recall that a famous conjecture of Weibel [36] asserts that if R is a commutative Noetherian ring of Krull dimension d, then K −d (R) ≃ K −d (R[t 1 , · · · , t n ]) and K i (R[t 1 , · · · , t n ]) = 0 for i < −d and n ≥ 0. An affirmative answer to this conjecture was obtained recently by Kerz, Strunk and Tamme [22]. For Noetherian rings containing Q, this was earlier solved by Cortiñas, Haesemeyer, Schlichting and Weibel [3] (see also [10], [23] and [38] for older results in positive characteristics).The main technical tool that goes into the proof of Weibel's conjecture in [22] is a pro-cdhdescent theorem for algebraic K-theory. However, the final step in the proof of the conjecture 2010 Mathematics Subject Classification. Primary 19D50; Secondary 13F15, 14F35.

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“…The strong Bloch-Srinivas conjecture is proven in § 6 using Theorem 1.1, the recent prodescent theorem of Kerz, Strunk and Tamme [23] and some results on the K-theory in positive characteristic from [26]. A question of Kerz-Saito is answered in a special case as an application of our proof of the strong version of the Bloch-Srinivas conjecture.…”

Section: 2mentioning

confidence: 94%