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Charles E. Watts

Bio: Charles E. Watts is an academic researcher. The author has contributed to research in topics: Unit (ring theory) & Functor. The author has an hindex of 1, co-authored 1 publications receiving 160 citations.

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01 Jan 1960
TL;DR: In this article, the intrinsic functorial characterizations of the functors Hom and X were obtained for homological algebra, and they were used to account in part for the distinguished role played by then in homology algebra.
Abstract: Our purpose here is to obtain intrinsic functorial characterizations of the functors Hom and X and thus to account in part for the distinguished role played by then in homological algebra. In all that follows, A, r are rings with unit, Z the ring of integers. The category of all F-A-bimodules with r operating on the left, A on the right, is denoted by r91lA, the category of left A-modules (F-modules) by A91(rM1Z), etc. All functors are assumed additive. We use throughout the terminology of [I].

171 citations


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TL;DR: In this paper, it was shown that the relation between d-jointness nd triples is reversible and that the adoint functor T is faithful (Theorem 2.2).
Abstract: A riple F (F, ,) in ctegory a consists of functor F a nd morphisms la F, F F stisfying some identities (see 2, (T.1)-(T.3)) nlogous to those stisfied in monoid. Cotriples re defined dually. It has been recognized by Huber [4] that whenever one hs pir of adoint functors T a , S a (see 1), then the functor TS (with appropriate morphisms resulting from the adjointness relation) constitutes a triple in nd similarly ST yields cotriple in a. The main objective of this pper is to show that this relation between d-jointness nd triples is in some sense reversible. Given triple Y in a we define new ctegory a nd adoint functors T a a, S a a such that the triple given by TS coincides with. There my be mny adoint pirs which in this wy generate the triple Y, but among those there is a universal one (which therefore is in a sense the \"best possible one\") nd for this one the functor T is faithful (Theorem 2.2). This construction cn best be illustrated by n example. Let a be the ctegory of modules over a commu-tative ring K nd let A be K-lgebm. The functor F A@ together with morphisms nd resulting from the morphisms K A, h @ A A given by the K-algebra structure of A, yield then a triple Y a. The ctegory a is then precisely the ctegory of A-modules. The general construction of a closely resembles this example. As another example, let a be the category of sets nd let F be the functor which to ech set A ssigns the underlying set of the free group generated by A. There results triple Y in a nd a is the category of groups.

340 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present conditions générales d'utilisation (http://www.numdam.org/conditions), i.e., Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Abstract: © Publications mathématiques de l’I.H.É.S., 1992, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

174 citations

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113 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is etale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group.
Abstract: We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is etale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. The proof uses an equivariant version of Artin's algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.

101 citations