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5 new papers this month, from Bisson-Tsemo, ChornyB, and Neusel(3). 

                  Mark Hovey

New papers appearing on hopf between 6/8/07 and 8/13/07

1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Bisson-Tsemo/AlgGeomOps

Title: Extended powers and Steenrod operations in algebraic geometry 
(Preliminary Draft, July 2007 version)

Authors: Terrence Bisson 
     and Aristide Tsemo

Abstract: Steenrod operations have been defined by
Voedvodsky in motivic cohomology in order to show the Milnor
and Bloch-Kato conjectures. These operations have also been
constructed by Brosnan for Chow rings. The purpose of this
paper is to provide a setting for the construction of the
Steenrod operations in algebraic geometry, for generalized
cohomology theories whose formal group law has order two. We
adapt the methods used by Bisson-Joyal in studying Steenrod
and Dyer-Lashof operations in unoriented cobordism and mod 2
cohomology.

2.
http://hopf.math.purdue.edu/cgi-bin/generate?/ChornyB/BrownRep

Title: Brown representability for space-valued functors

Author(s): Boris Chorny

Abstract: In this paper we prove two theorems which resemble the
classical cohomological and homological Brown representability
theorems. The main difference is that our results classify small
contravariant functors from spaces to spaces up to weak equivalence of
functors.

In more detail, we show that every small contravariant functor from
spaces to spaces which takes coproducts to products up to homotopy and
takes homotopy pushouts to homotopy pullbacks is naturally weakly
equivalent to a representable functor.

The second representability theorem states: every contravariant
continuous functor from the category of finite simplicial sets to
simplicial sets taking homotopy pushouts to homotopy pullbacks is
equivalent to the restriction of a representable functor. This theorem
may be considered as a contravariant analog of Goodwillie's
classification of linear functors.

3.
http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/hanoi

Inseparable Extensions of Algebras over the Steenrod Algebra with
Applications to Modular Invariant Theory of Finite Groups II

author: Mara D. Neusel

abstract:
We continue our study of the homological properties of the purely
inseparable extensions of integrally closed unstable Noetherian integral
domains over the Steenrod algebra. It turns out that the projective
dimension of an algebra is a lower bound for the projective dimension of
its inseparable closure. Furthermore, its depth is an upper bound for
the depth of its inseparable closure.  Moreover, both algebras have the
same global dimension. We apply these results to invariant theory.

4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/schmid

Degree bounds and the regular representation

author: Mara D. Neusel

abstract:

Let rho : G --> GL(n , F) be a faithful representation of a finite group
G. Denote by beta(F[V]^G) the maximal degree of an F-algebra generator
of the ring of polynomial invariants F[V]^G in a minimal generating set.
We prove the old conjecture that in the nonmodular case

beta(F[V]^G)<= beta(F[FG]^G),

where FG is the regular representation. Along the way we show that rings
of permutation invariants that are Cohen-Macaulay always satisfy
Noether's bound. Furthermore, we show that rings of invariants of sums
of permutation representations that are Cohen-Macaulay are generated by
polarizations.

5.
http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/unstable

The Unstable Parts Functor and Injective Objects

author: Mara D. Neusel

abstract:
The unstable part functor Un assigns to an arbitrary module over the
Steenrod algebra the largest unstable submodule. We start by showing
some general properties of this functor.  Then we study the functor Un
S^{-1} obtained from Un by precomposition with a localization.  We show
that Un S^{-1} is an exact functor from the category of unstable
noetherian modules over some unstable noetherian algebra to
itself. Along the lines we describe the injective objects in this
category.

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