------------------------------------------------------------ Hmm, I seem to be losing momentum on these Hopf announcements. 9 new papers this time, from Behrens-Davis, Gillespie-Hovey, Gonzalez-Landweber, Hovey (2), Hovey-Lockridge, Kashiwabara, Kuhn, and Monico-Neusel. Mark Hovey New papers appearing on hopf between 5/15/08 and 9/6/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens-Davis/funcspec15 Title of Paper: The homotopy fixed point spectra of profinite Galois extensions Authors: Mark Behrens, Daniel G. Davis AMS Classification numbers: 55N20, 55P43 ArXiv ID: math.AT/0808.1092 Abstract: Let E be a k-local profinite G-Galois extension of an E_\infty-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite setting. We show the function spectrum F_A((E^hH)_k, (E^hK)_k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]])^hK)_k where H and K are closed subgroups of G. Applications to Morava E-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Gillespie-Hovey/gorenstein Gorenstein model structures and generalized derived categories James Gillespie and Mark Hovey In a previous paper, the second author introduced the Gorenstein projective and Gorenstein injective model structures on $R$-Mod, the category of $R$-modules, where $R$ is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce; the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable homotopy category of $R$. Here we show that if such a ring $R$ has finite global dimension, the graded ring $R[x]/(x^2)$ is Gorenstein and the three associated Gorenstein model structures on $R[x]/(x^2)$-Mod, the category of graded $R[x]/(x^2)$-modules, are nothing more than the usual projective, injective and flat model structures on Ch($R$), the category of chain complexes of $R$-modules. Although these correspondences only recover these model structures on Ch($R$) when $R$ has finite global dimension, we can set $R = \Z$ and use general techniques from model category theory to lift the projective model structure from Ch($\Z$) to Ch($R$) for an arbitrary ring $R$. This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when $\Z[x]/(x^2)$ is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring $R$ as well as the derived category of $R$ and we give some examples of such generalizations. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gonzalez-Landweber/symmotion Title: Symmetric topological complexity of projective and lens spaces Authors: Jesus Gonzalez and Peter Landweber Adresses: Departamento de Matematicas, CINVESTAV-IPN, Mexico City 07000, MEXICO Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA Abstract: For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps, and (c) the topological complexity are known to be three facets of the same problem. This paper describes the corresponding relationship between the symmetrized versions of (b) and (c) to the Euclidean embedding dimension of projective spaces. Extensions to the case of lens spaces and complex projective spaces are discussed. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/bpbp1 The homotopy of MString and MU<6> at large primes Mark Hovey We use Hopf rings to compute the homotopy rings $\pi_{*}\MO{8}$ and $\pi_{*}\MU{6}$ at primes $>3$. In this case, the additive structure is well-known, but the ring structure is not polynomial. Instead, these rings are quotients of polynomial rings by infinite regular sequences. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/intersection-homology Intersection homological algebra Mark Hovey We investigate the abelian category which is the target of intersection homology. Recall that, given a stratified space $X$, we get intersection homology groups $I^{\perversity{p}}H_{n}X$ depending on the choice of an $n$-perversity $\perversity{p}$. The $n$-perversities form a lattice, and we can think of $IH_{n}X$ as a functor from this lattice to abelian groups, or more generally $R$-modules. Such perverse $R$-modules form a closed symmetric monoidal abelian category. We study this category and its associated homological algebra. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/ssrs Semisimple ring spectra Mark Hovey and Keir Lockridge Abstract. We define global dimension and weak dimension for the structured ring spectra that arise in algebraic topology. We provide a partial classification of ring spectra of global dimension 0, the semisimple ring spectra of the title. These ring spectra are closely related to classical rings whose projective modules admit the structure of a triangulated category. Applications to two analogues of the generating hypothesis in algebraic topology are given. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Kashiwabara/hqs2008 The Hopf ring for Bockstein-nil homology of QSn Takuji Kashiwabara Institut Fourier UMR au CNRS 5582 BP 74 38402, St-Martin-d'H`eres CEDEX FRANCE In this paper we give a generator-relation description of mod $p$ Bockstein-nil homology of $QS^n$ for odd prime $p$. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/loopSS Title: Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology Author: Nicholas J. Kuhn Address: University of Virginia, Charlottesville, VA 22904 AMS classification number: 55S10 arXiv:0806.3281 Abstract: We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg--Moore spectral sequence by a single use of the spectral sequence converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie tower for the suspension spectrum of Omega^nX. Much of the paper develops basic properties of this spectral sequence. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Monico-Neusel/counting TITLE: Counting Special Monomials AUTHORS: Chris Monico and Mara D. Neusel Department of Mathematics and Statistics, MS 1042, Texas Tech University, Lubbock, Texas 79409 ABSTRACT: In this paper we study the number of orbits of special monomials of G acting by permutations on the polynomials in n variables. We give formulae for several crucial families of groups, for direct sums of representations, as well as for vector invariants. In addition we give two algorithms for arbitrary permutation groups, one relying on the geometry of G acting on the underlying vector space, the other relying on the representation theory of the symmetric groups. -----------------