Champs algébriques

Laumon, Gérard; Moret-Bailly, Laurent

doi:10.1007/978-3-540-24899-6

Cited by 509 publications

(919 citation statements)

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“…The use of these ideas has expanded to the point where there is now a whole cottage industry on stacks, and they are supposed to be part of our basic tool kit. The standard book reference on this subject is [27] and a concise introduction to the algebraic geometry point of view is in the appendix to [41]. A relative recent introductory paper also exists; see [16].…”

Section: An Extremely Short Introduction To Stacksmentioning

confidence: 99%

(PRE-)Sheaves of Ring Spectra Over the Moduli Stack of Formal Group Laws

Goerss¹

2004

Axiomatic, Enriched and Motivic Homotopy Theory

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In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem.While some of what I say is quite general, the ring spectra I have in mind will arise from the chromatic point of view, which uses the geometry of formal groups to organize stable homotopy theory. Thus, a subsidiary aim here is to reemphasize this connection between algebraic geometry and homotopy theory.

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Section: An Extremely Short Introduction To Stacksmentioning

confidence: 99%

(PRE-)Sheaves of Ring Spectra Over the Moduli Stack of Formal Group Laws

Goerss¹

2004

Axiomatic, Enriched and Motivic Homotopy Theory

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“…5.1.5] asserts that we can, in effect, mod out by the automorphisms of the stack of stable vector bundles to obtain a quotient stack without automorphisms, and having the same "moduli space" (i.e., algebraic space representing the sheafification). But since this quotient is already automorphism-free, by [16,Cor. 8.1.1] it is an algebraic space, and hence is itself the sheafification of the stack of stable bundles.…”

Section: Lemma 43mentioning

confidence: 99%

“…By Lemma 3.4, Frobenius-unstable vector bundles form a substack of the stack of semi-stable vector bundles, and we see by the same lemma that they must lie within the stable locus. We thus need to check that this is a locally closed sub-stack; since both stacks deal only with quasi-coherent sheaves which are (locally) of finite presentation, and maps between such sheaves, they are clearly locally of finite presentation (i.e., on the level of rings they commute with direct limits; see [16,Prop. 4.15]), so it is easy to see that we may restrict to schemes T of finite type over S, and in particular schemes which are Noetherian.…”

Section: Lemma 43mentioning

confidence: 99%

Mochizuki’s Crys-Stable Bundles: A Lexicon and Applications

Osserman

2007

Publ. Res. Inst. Math. Sci.

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Mochizuki's work on torally crys-stable bundles [18] has extensive implications for the theory of logarithmic connections on vector bundles of rank 2 on curves, once the language is translated appropriately. We describe how to carry out this translation, and give two classes of applications: first, one can conclude almost immediately certain results classifying Frobenius-unstable vector bundles on curves; and second, it follows from the results of [22] that one also obtains results on rational functions with prescribed ramification in positive characteristic. §1. IntroductionMochizuki's theory of torally crys-stable bundles and torally indigenous bundles developed in [18] has, after appropriate translation, immediate implications for logarithmic connections on vector bundles of rank 2 on curves. This in turn has immediate implications to a subject which has recently been studied by a number of different people (see, for instance,

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“…(Quasi-separatedness is a running assumption throughout [LMB], along with separatedness of the diagonal.) More specifically, specializing [LMB,5.7.2] to the case of algebraic spaces, one gets: Lemma A.2.10. Let S be a quasi-separated algebraic space.…”

Section: Appendix a Some Foundational Facts For Algebraic Spacesmentioning

confidence: 99%