Subject: new Hopf listings From: Mark Hovey Date: 16 Jun 1999 10:37:59 -0400 I missed a few papers this morning, so here are four more. By the way, in the latest issue of Notices of the AMS, there is an interview with outgoing DMS director (i.e. the math division of the NSF) D.J. Lewis. In it he says "And one that should be under the same strain is Geometry/Topology. But quite frankly, over the last three years that program made too many very small grants, and so the strain is hidden." He also says later "We're under terrific pressure to increase the size of our grants. If we did what the [National Science] Board wants us to do, we would fund 800 people instead of 1,400." I completely disagree with the tenor of these remarks! Perhaps I am wrong about this, but it seems to me that doubling the size of each grant while giving half as many total grants would have a completely negative effect on mathematics! I don't understand why the NSF is so stupid as to want to do this--it seems like very poor management and not very cost-effective. Perhaps somebody can explain this to me, or perhaps together we can complain about it. It follows from my opinion that I think what Prof. Lewis says should be taken as an indication of the excellent job Ralph Krause has been doing! I am very sorry he is leaving. Even with the best will in the world, his successor will probably not be in a position to resist the pressure to cut the numbers of grants as effectively as Ralph was able to do. One more completely annoying statement that appears in the interview: "We fund proposals, not individuals". I think that in practice, how good a job the NSF is doing can be measured by how much the section heads manage to get around this stated policy. Following this policy would mean Andrew Wiles would not have been funded while he was working on Fermat. It is the exact opposite of how I would run things. Mark Hovey New papers uploaded to hopf between 5/17/99 and 6/16/99, part 2. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/KrauseH/brown Title: Brown representability and flat covers Author: Henning Krause Status: Submitted Address: University of Bielefeld, Germany E-mail: henning@mathematik.uni-bielefeld.de Abstract: We exhibit a surprising connection between the following two concepts: Brown representability which arises in stable homotopy theory, and flat covers which arise in module theory. It is shown that Brown representability holds for a compactly generated triangulated category if and only if for every additive functor from the category of compact objects into the category of abelian groups a flat cover can be constructed in a canonical way. The proof also shows that Brown representability for objects and morphisms is a consequence of Brown representability for objects and isomorphisms. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/KrauseH/idempotent Title: Decomposing thick subcategories of the stable module category Author: Henning Krause Status: Math. Ann. 313 (1999), 95-108 Address: University of Bielefeld, Germany E-mail: henning@mathematik.uni-bielefeld.de Abstract: Let stmod kG be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a Krull-Remak-Schmidt theorem for thick subcategories of stmod kG. It is shown that every thick tensor-ideal C of stmod kG (i.e. a thick subcategory which is a tensor ideal) has a (usually infinite) unique decomposition C=\coprod_{i\in I}C_i into indecomposable thick tensor-ideals. This decomposition follows from a decomposition of the corresponding idempotent kG-module E_C into indecomposable modules. If C=C_W is the thick tensor-ideal corresponding to a closed homogeneous subvariety W of the maximal ideal spectrum of the cohomology ring H^*(G,k), then the decomposition of C reflects the decomposition W=\bigcup_{i=1}^nW_i of W into connected components. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/KrauseH-Reichenbach/endofinite ness Title: Endofiniteness in stable homotopy theory Author: Henning Krause and Ulrike Reichenbach Status: Submitted Address: University of Bielefeld, Germany E-mail: henning@mathematik.uni-bielefeld.de reichenb@mathematik.uni-bielefeld.de Abstract: We study endofinite objects in a compactly generated triangulated category in terms of ideals in the category of compact objects. Our results apply in particular to the stable homotopy category. This leads, for example, to a new interpretation of stable splittings for classifying spaces of finite groups. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Strickland/strickland_chapx Chern approximations for generalised group cohomology Neil P. Strickland University of Sheffield N.P.Strickland@sheffield.ac.uk Let G be a finite group, and let E be a generalised cohomology theory, subject to certain technical conditions. We study a certain ring C(E,G) that is the best possible approximation to E^0BG that can be built using only knowledge of the complex representations of G. There is a natural map C(E,G) -> E^0BG, whose image is the subring of E^0BG generated over E^0 by all Chern classes of such representations. There is ample precedent for considering this subring in the parallel case of ordinary cohomology. However, although the generators of this subring come from representation theory, the same cannot be said for the relations; one purpose of our construction is to remedy this. We also also develop a kind of generalised character theory which gives good information about the rationalisation of C(E,G). In the few cases that we have been able to analyse completely, either C(E,G) is rationally different from E^0BG for easy character-theoretic reasons, or we have C(E,G)=E^0BG. Rather than working directly with rings, we will study the formal schemes X(G)=spf(E^0BG) and XCh(G)=spf(C(E,G)). Suitably interpreted, our main definition is that XCh(G) is the scheme of homomorphisms from the Lambda-semiring R^+(G) of complex representations of G to the Lambda-semiring scheme of divisors on the formal group associated to E. ---------------------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/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.