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4 new papers this month, from Badzioch-Chung-Voronov,
Broto-Castellana-Grodal-Levi-Oliver, Christensen-Isaksen, and KrauseH.

                  Mark Hovey

New papers appearing on hopf between 3/1/04 and 4/5/04

1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Badzioch-Chung-Voronov/bcv

Title: Yet another delooping machine
Authors: Bernard Badzioch, Kuerak Chung, and Alexander A. Voronov
Author's e-mail address: voronov@math.umn.edu
Authors' mailing address: School of Mathematics, University of
  Minnesota, Minneapolis, MN 55455
Included ps or eps files: mor.eps
AMS classification number: 55P48 (Primary); 18C10 (Secondary)
ArXiv submission number: math.AT/0403098

Abstract: We suggest a new delooping machine, which is based on
  recognizing an n-fold loop space by a collection of operations acting
  on it, like the traditional delooping machines of Stasheff, May,
  Boardman-Vogt, Segal, and Bousfield. Unlike in the traditional
  delooping machines, which carefully select a nice space of such
  operations, we consider all natural operations on n-fold loop spaces,
  resulting in the algebraic theory Map (V_. S^n, V_. S^n).  The
  advantage of this new approach is that the delooping machine is
  universal in a certain sense, the proof of the recognition principle
  is more conceptual, works the same way for all values of n, and does
  not need the test space to be connected.

2.
http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Castellana-Grodal-Levi-Oliver/bcglo1

Subgroup families controlling $p$-local finite groups

by C. Broto, N. Castellana, J. Grodal, R. Levi, B. Oliver

A $p$-local finite group consists of a finite $p$-group $S$, together with 
a pair of categories which encode ``conjugacy'' relations among subgroups 
of $S$, and which are modelled on the fusion in a Sylow $p$-subgroup of a 
finite group.  It contains enough information to define a classifying 
space which has many of the same properties as $p$-completed  classifying 
spaces of finite groups.  In this paper, we examine which subgroups 
control this structure.  More precisely, we prove that the question of 
whether an abstract fusion system $F$ over a finite $p$-group $S$ is 
saturated can be determined by just looking at smaller classes of 
subgroups of $S$.  We also prove that the homotopy type of the classifying 
space of a given $p$-local finite group is independent of the family of 
subgroups used to define it, in the sense that it remains unchanged when 
that family ranges from the set of $F$-centric $F$-radical subgroups (at a 
minimum) to the set of $F$-quasicentric subgroups (at a maximum).  
Finally, we look at constrained fusion systems, analogous to 
$p$-constrained finite groups, and prove that they in fact all arise from 
groups.

3.
http://hopf.math.purdue.edu/cgi-bin/generate?/Christensen-Isaksen/duality

                    Duality and Pro-spectra

          J. Daniel Christensen and Daniel C. Isaksen
                 jdc@uwo.ca       isaksen@math.wayne.edu

Keywords: Spectrum, pro-spectrum, Spanier-Whitehead duality, 
          closed model category, colocalization

Arxiv: math.AT/0403451

MSC-class: 55P42 (Primary); 55P25, 18G55, 55U35, 55Q55 (Secondary)

Abstract:

Cofiltered diagrams of spectra, also called pro-spectra, have arisen
in diverse areas, and to date have been treated in an ad hoc manner.
The purpose of this paper is to systematically develop a homotopy
theory of pro-spectra and to study its relation to the usual homotopy
theory of spectra, as a foundation for future applications.  The
surprising result we find is that our homotopy theory of pro-spectra
is Quillen equivalent to the opposite of the homotopy theory of
spectra.  This provides a convenient duality theory for all spectra,
extending the classical notion of Spanier-Whitehead duality which
works well only for finite spectra.  Roughly speaking, the new duality
functor takes a spectrum to the cofiltered diagram of the
Spanier-Whitehead duals of its finite subcomplexes.  In the other
direction, the duality functor takes a cofiltered diagram of spectra
to the filtered colimit of the Spanier-Whitehead duals of the spectra
in the diagram.  We prove the equivalence of homotopy theories by
showing that both are equivalent to the category of ind-spectra
(filtered diagrams of spectra).  

To construct our new homotopy theories, we prove a general existence
theorem for colocalization model structures generalizing known results
for cofibrantly generated model categories.

4.
http://hopf.math.purdue.edu/cgi-bin/generate?/KrauseH/stable

Title: The stable derived category of a noetherian scheme
Author: Henning Krause
E-mail: hkrause@math.upb.de

Abstract: For a noetherian scheme, we introduce its unbounded stable
derived category. This leads to a recollement which reflects the
passage from the bounded derived category of coherent sheaves to the
quotient modulo the subcategory of perfect complexes. Some
applications are included, for instance an analogue of maximal
Cohen-Macaulay approximations, a construction of Tate cohomology, and
an extension of the classical Grothendieck duality.

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