Happy New Year! 10 new papers this time, from Anton, DavisDaniel, Harper (2) (that is John E. Harper of Notre Dame, not John Harper of Rochester), Hovey-Lockridge (2), Neusel (2), and Yagita (2). Mark Hovey New papers appearing on hopf between 11/29/07 and 1/18/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Anton/homologicalSymbols Title: Homological symbols and the Quillen Conjecture Author(s): Marian F. Anton Abstract: We formulate a "correct" version of the Quillen conjecture on the cohomology of linear groups by defining an unstable form of Milnor K-theory and show that this version can be solved by a finite process. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/fibrantmodel4 Title: Explicit fibrant replacement for discrete G-spectra Author: Daniel G. Davis Abstract: If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a profinite group, the fibrant objects in the model category of discrete G-spectra are also difficult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in C, under certain finiteness conditions. Similarly, in this paper, we show that if G has finite virtual cohomological dimension and X is a discrete G-spectrum, then there is an explicit fibrant model for X. Also, we give several applications of this concrete model related to closed subgroups of G. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Harper/modules-operads-monoidal Title: Homotopy theory of modules over operads and non-Sigma operads in monoidal model categories Author: John E. Harper Author's mailing address: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA Comments: 49 pages, uses xy-pic; we have compiled the .tex file without using the better looking dvips,ps options in xy-pic, so the .dvi file should be device independent, but the diagrams may appear jagged etc. Abstract: This paper studies the existence of model category structures on modules and algebras over operads in monoidal model categories. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Harper/modules-spectra Title: Homotopy theory of modules over operads in symmetric spectra Author: John E. Harper Author's mailing address: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA Comments: 21 pages, uses xy-pic; we have compiled the .tex file without using the better looking dvips,ps options in xy-pic, so the .dvi file should be device independent, but the diagrams may appear jagged etc. Abstract: This paper establishes model category structures on modules and algebras over operads in symmetric spectra, and studies when a morphism of operads induces a Quillen equivalence between corresponding categories of modules (resp. algebras) over operads. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/gen-gen-hyp The ghost dimension of a ring Mark Hovey Wesleyan University Keir Lockridge Wake Forest University We introduce the concept of the ghost dimension of a ring R. This is the longest nontrivial chain of maps in the derived category emanating from a perfect complex such that each map is zero on homology. We show that the ghost dimension of R is less than or equal to the weak dimension of R, with equality if R is coherent or has weak dimension 1. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge/triproj Triangulations of Projective Modules Mark Hovey Wesleyan University Keir Lockridge Wake Forest University We show that the category of projective modules over a graded commutative ring admits a triangulation with respect to module suspension if and only if the ring is a finite product of graded fields and exterior algebras on one generator over a graded field (with a unit in the appropriate degree). We also classify the ungraded commutative rings for which the category of projective modules admits a triangulation with respect to the identity suspension. Applications to two analogues of the generating hypothesis in algebraic topology are given, and we translate our results into the setting of modules over a symmetric ring spectrum or $S$-algebra, where semisimple and von Neumann regular ring spectra are defined and discussed. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/hilbert Title: On the Hilbert Ideal Author: Mara D. Neusel Abstract: We prove the Hilbert number conjecture. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/schmid Title: Degree Bounds and the Regular Representation Author: Mara D. Neusel Abstract: This is a revised version of the paper with the same name posted during last summer. We prove Schmid's inequality in the general case, and Killius' conjecture for permutation representations. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/coniveaufilt Title: Coniveau filtration of cohomology of group Author: Nobuaki Yagita Abstract: We consider natural filtrations of mod p cohomology of a classifying space BG for a compact Lie group G, such that the reduced power operation preserves the filtration but the Bockstein opration descends the filtration degree one. An example of such filtrations is defined by the image from the motivic cohomology. For example, when BG=BO(n), this filtration coincides the coniveau filtration defined by Grothendieck. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/torsorEE Title: Note on Chow rings of nontrivial G-torsors over a field. Author: Nobuaki Yagita Abstract: Let G(k) be a split reductive group over a field k corresponding to a compact Lie group G. Let E be a nontrivial G(k)-torsor over a field k. In this paper we study the Chow ring of nontrivial G(k)-torsors E. For example when (G,p)=(F_4,3), we see that the positive degree of the mod 3 Chow ring of E is zero. ----------------