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.
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