# Extension categories and their homotopy

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## Abstract

Extension categories and their homotopy COMPOSITIO MATHEMATICA AMNONNEEMAN VLADIMIRRETAKH Extension categories and their homotopy Compositio Mathematica, tome 102, no 2 (1996), p. 203-242. <http://www.numdam.org/item?id=CM_1996__102_2_203_0> © Foundation Compositio Mathematica, 1996, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 203 Extension categories and their homotopy AMNON NEEMAN1 * and VLADIMIR RETAKH2**~ Compositio Mathematica 102 : 203-242, 1996. © 1996 Kluwer Academic Publishers. Printed in the Netherlands. 1 Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA. 2Department af Mathematics, Pennsylvania State University, University Park, PA 16802, USA. Received 22 December 1993; accepted in final form 27 April 1995 0. Introduction In [11], Robinson proved that, given a ring Rand two .R-modules A and B, there is a spectrum whose homotopy groups are the torsion groups Tor.R(A B). More precisely, he proves THEOREM 1 of [11] Let R be a ring. Let A be a right R-module, let B be a left R-module. Consider the category TorR(A, B) whose objects are pairs s: P ~ B and 71: P* ~ A, where P is a projective left R-module, and P* = Hom(P, R) is its dual. Then if we geometrically realise the category TorR(A, B) we obtain a space with the natural structure of an infinite loop space, and its homotopy is given by the formula Inspired by Robinson’s work, the second author generalised this to obtain a spec- trum whose - ith homotopy group is Ext2 ( A, B). More precisely,

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