Subject: new Hopf listings From: Mark Hovey Date: 02 Jan 2000 07:33:38 -0500 Now that we seem to have survived Y2K, the show must go on. 4 new papers this time. Mark Hovey New papers uploaded to hopf between 12/14/99 and 1/2/00. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/CohenR-Lima-Filho/charact Title: An algebraic geometric realization of the Chern character Authors: Ralph L. Cohen and Paulo Lima-Filho Email addresses: ralph@math.stanford.edu and plfilho@math.tamu.edu Text of abstract Using symmetrized Grassmannians we give an algebraic geometric presentation, in the level of classifying spaces, of the Chern character and its relation to Chern classes. This allows one to define, for any projective variety $X$, a Chern character map $ch : K^{-i}_{hol}(X) \to \prod_* L^*H^{2*-i}(X)\otimes Q$ from the "holomorphic $K$-theory of $X$ to its morphic cohomology (introduced by Friedlander and Lawson). The holomorphic $K$-theory of $X$, introduced by Lawson, Lima-Filho and Michelsohn and also by Friedlander and Walker, is defined in terms a group-completion of the space of algebraic morphisms from $X$ into $BU$. It has been further studied by the authors in a companion paper. Holomorphic $K$-theory sits between algebraic $K$-theory and topological $K$-theory in the same way that morphic cohomology sits between motivic cohomology and ordinary cohomology. Our constructions provide a bridge between these two worlds. We also realize Chern classes in the case where $X$ is smooth, and establish a universal relation between the Chern character and the Chern classes. For this we use classical constructions with algebraic cycles and infinite symmetric products of projective spaces. The latter can be seen as the classifying space for motivic cohomology, and under this perspective our constructions are essentially motivic. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/CohenR-Lima-Filho/holo-k-th Title: Holomorphic $K$-theory, algebraic co-cycles, and loop groups Authors: Ralph L. Cohen and Paulo Lima-Filho Email addresses: ralph@math.stanford.edu and plfilho@math.tamu.edu Text of abstract In this paper we study the ``holomorphic $K$-theory" of a projective variety. This theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory was introduced by Lawson, Lima-Filho and Michelsohn, and also by Friedlander and Walker, and a related theory was considered by Karoubi. Using the Chern character studied by the authors in a companion paper, we show that there is a rational isomorphism between holomorphic $K$-theory and the appropriate "morphic cohomology", defined by Lawson and Friedlander. In doing so, we describe a geometric model for rational morphic cohomology groups in terms of algebraic maps from the variety to the ``symmetrized loop group" $\om U(n)/\Sigma_n$ where the symmetric group $\Sigma_n$ acts on $U(n)$ via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians $BU(k)$ by a similar symmetric group action. We then prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic $K$-theory by inverting the Bott class, then it is rationally isomorphic to topological $K$-theory. Finally we produce explicit obstructions to periodicity in holomorphic $K$ - theory, and show that these obstructions vanish for generalized flag manifolds. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/ab Classifying subcategories of modules Mark Hovey mhovey@wesleyan.edu In this paper, we classify certain subcategories of modules over a ring R. A wide subcategory of R-modules is an Abelian subcategory of R-Mod that is closed under extensions. We claim that these wide subcategories are analogous to thick subcategories of the derived category D(R). Indeed, let C_0 denote the wide subcategory generated by R; C_0 is the collection of all finitely presented modules precisely when R is coherent. When R is a quotient of a regular commutative coherent ring by a finitely generated ideal, we classify wide subcategories of C_0. In fact, they are on 1-1 correspondence with thick subcategories of small objects of D(R). The proof relies heavily on Thomason's thick subcategory theorem for D(R). We also classify wide subcategories closed under arbitrary coproducts; these are analogous to localizing subcategories of D(R). In this case, we must assume that R is Noetherian, where we use Neeman's classification of localizing subcategories of D(R). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lawson-Lima-Filho-Michelsohn/a lg-cycles1 Title: Algebraic Cycles and the Classical Groups - Part I, Real Cycles Authors : H. Blaine Lawson, Jr. and Paulo Lima-Filho and Marie-Louise Michelsohn Email addresses: blaine@math.sunysb.edu, plfilho@math.tamu.edu, mmichelsohn@math.sunysb.edu The groups of algebraic cycles on complex projective space P(V) are known to have beautiful and surprising properties. Therefore, when V carries a real structure, it is natural to ask for the properties of the groups of real algebraic cycles on P(V). Similarly, if V carries a quaternionic structure, one can define quaternionic algebraic cycles and ask the same question. In this paper and its sequel the homotopy structure of these cycle groups is completely determined. It turns out to be quite simple and to bear a direct relationship to characteristic classes for the classical groups. It is shown, moreover, that certain functors in K-theory extend directly to these groups. It is also shown that, after taking colimits over dimension and codimension, the groups of real and quaternionic cycles carry E_{\infty}-ring structures, and that the maps extending the K-theory functors are E_{\infty}-ring maps. In fact this stabilized space is a product of (Z/2Z)-equivariant Eilenberg-MacLane spaces indexed at the representations R^{n,n} for n \geq 0. This gives a wide generalization of the results in [BLLMM] on the Segal question. The ring structure on the homotopy groups of these stabilized spaces is explicitly computed. In the real case it is a simple quotient of a polynomial algebra on two generators corresponding to the first Pontrjagin and first Stiefel-Whitney classes. These calculations yield an interesting total characteristic class for real bundles. It is a mixture of integral and mod 2 classes and has nice multiplicative properties. The class is shown to be the (Z/2Z)-equivariant Chern class on Atiyah's KR-theory. ---------------------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.