High dimension, low sample size data are emerging in various areas of science. We find a common structure underlying many such data sets by using a non-standard type of asymptotics: the dimension tends to ∞ while the sample size is fixed. Our analysis shows a tendency for the data to lie deterministically at the vertices of a regular simplex. Essentially all the randomness in the data appears only as a random rotation of this simplex. This geometric representation is used to obtain several new statistical insights. Copyright 2005 Royal Statistical Society.

Grothendieck proved that if f : X ⟶ Y f:X\longrightarrow Y is a proper morphism of nice schemes, then R f ∗ Rf_* has a right adjoint, which is given as tensor product with the relative canonical bundle. The original proof was by patching local data. Deligne proved the existence of the adjoint by a global argument, and Verdier showed that this global adjoint may be computed locally. In this article we show that the existence of the adjoint is an immediate consequence of Brown’s representability theorem. 1It follows almost as immediately, by “smashing” arguments, that the adjoint is given by tensor product with a dualising complex. Verdier’s base change theorem is an easy consequence.

In this survey we present the relatively new concept of approximable triangulated categories. We will show that the definition is natural, that it leads to powerful new results, and that it throws new light on old, familiar objects.

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The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel Annales scientifiques de l'É.N.S. 4 e série, tome 25, n o 5 (1992), p. 547-566 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1992, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) 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. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/