-theory of one-dimensional rings via pro-excision

Morrow, Matthew

doi:10.1017/s1474748013000194

Cited by 4 publications

(4 citation statements)

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“…The aim of this note is to explore certain analogues of this injectivity conjecture in the case when A is singular, by taking into account nilpotent thickenings of its singular locus. In the case that A is one-dimensional, this problem was first considered by the first author when n = 2 [6], and later by the second author in general [10]. In this note, we extend some of our earlier results to higher dimensions.…”

Section: Introductionsupporting

confidence: 60%

Analogues of Gersten’s conjecture for singular schemes

Krishna¹,

Morrow²

2016

Sel. Math. New Ser.

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Abstract. We formulate analogues, for Noetherian local Q-algebras which are not necessarily regular, of the injectivity part of Gersten's conjecture in algebraic Ktheory, and prove them in various cases. Our results suggest that the algebraic K-theory of such a ring should be detected by combining the algebraic K-theory of both its regular locus and the infinitesimal thickenings of its singular locus.

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Section: Introductionsupporting

confidence: 60%

Analogues of Gersten’s conjecture for singular schemes

Krishna¹,

Morrow²

2016

Sel. Math. New Ser.

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“…Remark M. Morrow has recently demonstrated that the groups

{\underset{lim}{true}}_{i} K_{n} (boldZ / p^{i} boldZ)

play an interesting role in a corrected Gersten conjecture for singular schemes. Although not in this paper, this suggests that they might play a role in future idèle theories , .…”

Section: Adèles With Quasi‐coherent Coefficientsmentioning

confidence: 69%

Geometric two‐dimensional idèles with cycle module coefficients

Braunling

2014

Mathematische Nachrichten

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We give a theory of idèles with coefficients for smooth surfaces over a field. It is an analogue of Beilinson/Huber's theory of higher adèles, but handling cycle module sheaves instead of quasi-coherent ones. We prove that they give a flasque resolution of the cycle module sheaves in the Zariski topology. As a technical ingredient we show the Gersten property for cycle modules on equicharacteristic complete regular local rings, which might be of independent interest.

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“…This provides a strengthening and purely algebraic proof of a result of Geisser and Hesselholt [14,Prop. 3.6], as promised in the author's earlier work [26,Lemma 1.12].…”

Section: The Equivalence Of Pro Unitality Conditionsmentioning

confidence: 85%

“…This has been previously proved by much longer arguments in the following two special cases: A is finitely generated over a characteristic zero field and A is assumed to be smooth, by Krishna [22, Thm. 1.1]; A is one-dimensional, by the author [26].…”

Section: Example 25 (Conductor Ideals) Letmentioning

confidence: 99%

Pro unitality and pro excision in algebraic K-theory and cyclic homology

Morrow

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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The purpose of this paper is to study pro excision in algebraic K-theory and cyclic homology, after Suslin-Wodzicki, Cuntz-Quillen, Cortiñas, and Geisser-Hesselholt, as well as continuity properties of André-Quillen and Hochschild homology. A key tool is first to establish the equivalence of various pro Tor vanishing conditions which appear in the literature.This allows us to prove that all ideals of commutative, Noetherian rings are pro unital in a suitable sense. We show moreover that such pro unital ideals satisfy pro excision in derived Hochschild and cyclic homology. It follows hence, and from the Suslin-Wodzicki criterion, that ideals of commutative, Noetherian rings satisfy pro excision in derived Hochschild and cyclic homology, and in algebraic K-theory.In addition, our techniques yield a strong form of the pro Hochschild-Kostant-Rosenberg theorem; an extension to general base rings of the Cuntz-Quillen excision theorem in periodic cyclic homology; a generalisation of the Feȋgin-Tsygan theorem; a short proof of pro excision in topological Hochschild and cyclic homology; and new Artin-Rees and continuity statements in André-Quillen and Hochschild homology.MSC: 19D55 (primary), 16E40 13D03 (secondary). Pro unitality and pro excisiontheir topological counterparts, and André-Quillen homology. When k is a field, condition (iii) is equivalent to the existing notion of H-unitality of the pro k-algebra I ∞ (see Eg. 1.6), which is central in Wodzicki's original approach to excision [39], as well as in J. Cuntz and D. Quillen's approach to excision in periodic cyclic homology [10]. The importance of (iv) is thus not only its relevance to pro excision in K-theory, but also that it reveals that conditions (i)-(iii) are intrinsic properties of the non-unital ring I, depending neither on the ring A nor on the algebra structure from the base ring k.The first concrete application of Theorem 0.2 is to commutative rings: if I is an ideal of a commutative, Noetherian ring A, then M. André [2] noted that condition (i) is always true (see Lem. 2.1), and so we obtain: Theorem 0.3 (See Thm. 2.3). Let I be an ideal of a commutative, Noetherian ring. Then I is pro Tor-unital. Applying Geisser-Hesselholt's aforementioned pro version of the Suslin-Wodzicki criterion, we have the following consequence of Theorem 0.3 which completely solves the pro excision problem in K-theory for commutative, Noetherian rings: Corollary 0.4 (See Corol. 2.4). Ideals of commutative, Noetherian rings satisfy pro excision in algebraic K-theory. In particular, if A → B is a homomorphism of commutative, Noetherian rings, and I is an ideal of A mapped isomorphically to an ideal of B, then the map of pro abelian groups {K n (A, I r )} r −→ {K n (B, I r )} r is an isomorphism for all n ∈ Z.Now we turn to Hochschild and cyclic homology. Just as for K-theory in the first paragraph of the Introduction, these homology theories do not in general satisfy excision. To further justify the usefulness of pro Tor-unitality, we prove that it is sufficient to ensure that...

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-theory of one-dimensional rings via pro-excision

Cited by 4 publications

References 58 publications

Analogues of Gersten’s conjecture for singular schemes

Analogues of Gersten’s conjecture for singular schemes

Geometric two‐dimensional idèles with cycle module coefficients

Pro unitality and pro excision in algebraic K-theory and cyclic homology

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