Subject: new Hopf listings From: Mark Hovey Date: 08 Dec 1999 13:54:15 -0500 Six new papers this time. I expect you all to be holed up at home writing papers while the world falls apart around us Jan. 1! Mark Hovey New papers uploaded to hopf between 10/25/99 and 12/8/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Christensen/derived Derived categories and projective classes Dan Christensen jdc@math.jhu.edu Abstract: An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show how more general forms of homological algebra also fit into Quillen's framework. Specifically, any set of objects in a complete and cocomplete abelian category A generates a projective class on A, which is exactly the information needed to do homological algebra in A. The main result is that if the generating objects are "small" in an appropriate sense, then the category of chain complexes of objects of A has a model category structure which reflects the homological algebra of the projective class. The motivation for the work is the construction of the "pure derived category" of a ring R. Pure homological algebra has applications to phantom maps in the stable homotopy category and the (usual) derived category of a ring, and these connections will be described. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Christensen-Keller-Neeman/puri ty Failure of Brown Representability in Derived Categories Dan Christensen, Bernhard Keller and Amnon Neeman jdc@math.jhu.edu, keller@math.jussieu.fr, neeman@wintermute.anu.edu.au Abstract: Let T be a triangulated category with coproducts, C the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [Adams71]: All contravariant homological functors C --> Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [Neeman97], it was proved that Adams' theorem remains true as long as C is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T? In a recent paper, Beligiannis made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T = D(R) of rings, and contravariant homological functors C --> Ab which are not restrictions of representables. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/GreenDJ-Minh/ogawa Title: Almost all extraspecial p-groups are Swan groups Authors: David John Green Pham Anh Minh E-mail: green@math.uni-wuppertal.de paminh@bdvn.vnmail.vnd.net Abstract: Let P be an extraspecial p-group which is neither dihedral of order 8, nor of odd order p^3 and exponent p. Let G be a finite group having P as a Sylow p-subgroup. Then the mod-p cohomology ring of G coincides with that of the normalizer N_G(P). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ha-Strom/gray5 THE GRAY FILTRATION ON PHANTOM MAPS L^E MINH HA AND JEFFREY STROM ha@math.wayne.edu jeffrey.strom@dartmouth.edu Abstract This paper is a study of the Gray index of phantom maps. We give a new, tower theoretic, definition of the Gray index, which allows us to study the naturality properties of the Gray index in some detail. McGibbon and Roitberg have shown that if f* is surjective on rational cohomology, then the induced map on phantom sets is also surjective. We show that if f* is surjective just in dimension k, then f induces a surjection on a certain subquotient of the phantom set. If the condition holds for all k, we recover McGibbon and Roitberg's theorem. There is a dual result, and a theorem on phantom maps into spheres which holds one dimension at a time as well. Finally, we examine the set of phantom maps whose Gray in- dex is infinite. The main theorem is a partial verification of our conjecture that if X and Y are nilpotent and of finite type, then every phantom map f : X -! Y must have finite index. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/louis-hs Author: Louis F. McAuley Title: A Proof of the Hilbert-Smith Conjecture E-mail: louis@math.binghamton.edu The Hilbert-Smith conjecture is that if G is a locally compact group which acts effectively on a compact connected n-manifold M as a topological transformation group, then G is a Lie group. If G is not a Lie group, then G contains a group isomorphic to a p-adic group A_p which acts effectively on M. It is shown in this paper that A_p can not act effectively on M and, consequently, the Hilbert-Smith Conjecture is true. (This is a revised version of a previously announced paper). 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strom/hiorder HIGHER ORDER PHANTOM MAPS JEFFREY STROM jeffrey.strom@dartmouth.edu Abstract For each ordinal number ff, we define phantom maps of order ff. We construct universal phantom maps out of X with order ff, and show that under easily verifiable condi- tions, every one of these universal phantom maps is essential. ---------------------Instructions----------------------------- To subscribe or unsubscribe to this list, send a message to Don Davis at dmd1@lehigh.edu with your e-mail address and name. Please make sure he is using the correct e-mail address for you. To see past issues of this mailing list, point your WWW browser to http://www.math.wesleyan.edu/~mhovey/archive/ If this doesn't work or is missing a few issues, try http://www.lehigh.edu/~dmd1/algtop.html which also has the other messages sent to Don's list. To get the papers listed above, point your WWW client (Mosaic, Netscape) to the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There are links to conference announcements, Purdue seminars, and other math related things on this page as well. The largest archive of math preprints is at http://xxx.lanl.gov There is an algebraic topology section in this archive. The most useful way to browse it or submit papers to it is via the front end developed by Greg Kuperberg: http://front.math.ucdavis.edu To get the announcements of new papers in the algebraic topology section at xxx, send e-mail to math@xxx.lanl.gov with subject line "subscribe" (without quotes), and with the body of the message "add AT" (without quotes). You can also access Hopf through ftp. Ftp to hopf.math.purdue.edu, and login as ftp. Then cd to pub. Files are organized by author name, so papers by me are in pub/Hovey. If you want to download a file using ftp, you must type binary before you type get . To put a paper of yours on the archive, cd to /pub/incoming. Transfer the dvi file using binary, by first typing binary then put You should also transfer an abstract as well. Clarence has explicit instructions for the form of this abstract: see http://hopf.math.purdue.edu/pub/new-html/submissions.html In particular, your abstract is meant to be read by humans, so should be as readable as possible. I reserve the right to edit unreadable abstracts. You should then e-mail Clarence at wilker@math.purdue.edu telling him what you have uploaded. I am solely responsible for these messages---don't send complaints about them to Clarence. Thanks to Clarence for creating and maintaining the archive.