Two more postings related to Hovey's question....DMD
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Subject: Re: question about stable homotopy ring
From: "Nicholas J. Kuhn" <njk4x@cms.mail.virginia.edu>
Date: Thu, 14 Jul 2005 11:17:11 -0400
>
> For a specific question, is it known that for any x in pi_* S,
> there is a positive degree element y with xy nonzero?
> Mark Hovey


A positive answer to this question would, of course, imply that 
there are nontrivial products of arbitrary length in pi_*S.  
I think I remember hearing talks in the mid 1980's 
(post Nilpotence Theorem) speculating about such things 
(by Mahowald and/or Hopkins ??), but I do not think they 
were resolved then, and thus, I assume, not resolved now.

At the Adams SS E_2 level, is there perhaps a conjecture 
that if a in Ext_A(Z/2,Z/2) is nonzero, then so is ah_i for i>>0?  
WH Lin and Minami have thought about such things.

Nick Kuhn
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Here is one place where I, your moderator, can step in and
say that I proved that, in JPAA 21 (1981) 145-150.
Thm. If a is nonzero in Ext_A^{s,t}(Z/2,Z/2) for 0 < t-s \le 2^j, 
then ah_i is nonzero for i \ge 2j+1. 

Don Davis
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Subject: Re: three quick responses
From: Dan Christensen <jdc@uwo.ca>
Date: Thu, 14 Jul 2005 13:47:55 -0400

Here are the seven references to T.Y. Lin's work that I could find via
MathSciNet.  I include them here since it took me a few minutes to
isolate them (either T.Y. switched to fuzzy logic after 1977, or MSN
has merged the work of two authors together).

If I recall correctly, the fourth and fifth are the most relevant to
Mark's question.

I should also point out that I wasn't able to understand one or
two of the proofs, so you might want to check for errors before
using these results in further work.  In particular, I couldn't
follow the proof of the result Rudyak quoted, namely that modules
over pi_* are either free or have infinite homological dimension.
Knowing this would have allowed me to make my thesis two words
shorter.   :-) 

Dan

@article {MR0450309,
    AUTHOR = {Lin, T. Y. and Lin, I P'{\'e}ng},
     TITLE = {Homological dimensions of locally finite group algebras},
   JOURNAL = {Chinese J. Math.},
  FJOURNAL = {Chinese Journal of Mathematics},
    VOLUME = {5},
      YEAR = {1977},
    NUMBER = {1},
     PAGES = {43--49},
      ISSN = {0379-7570},
   MRCLASS = {16A26},
  MRNUMBER = {MR0450309 (56 \#8605)},
MRREVIEWER = {Donald S. Passman},
}
@article {MR0458426,
    AUTHOR = {Lin, T. Y. and Margolis, H. R.},
     TITLE = {Homological aspects of modules over the {S}teenrod algebra},
   JOURNAL = {J. Pure Appl. Algebra},
  FJOURNAL = {Journal of Pure and Applied Algebra},
    VOLUME = {9},
      YEAR = {1976/77},
    NUMBER = {2},
     PAGES = {121--129},
      ISSN = {0022-4049},
   MRCLASS = {55G10},
  MRNUMBER = {MR0458426 (56 \#16629)},
MRREVIEWER = {S. Cormack},
}
@article {MR0402738,
    AUTHOR = {Lin, T. Y.},
     TITLE = {Duality and {E}ilenberg-{M}ac {L}ane spectra},
   JOURNAL = {Proc. Amer. Math. Soc.},
  FJOURNAL = {Proceedings of the American Mathematical Society},
    VOLUME = {56},
      YEAR = {1976},
     PAGES = {291--299},
      ISSN = {0002-9939},
   MRCLASS = {55E10},
  MRNUMBER = {MR0402738 (53 \#6552)},
MRREVIEWER = {Donald W. Kahn},
}
@article {MR0391086,
    AUTHOR = {Lin, T. Y.},
     TITLE = {Adams type spectral sequences and stable homotopy modules},
   JOURNAL = {Indiana Univ. Math. J.},
  FJOURNAL = {Indiana University Mathematics Journal},
    VOLUME = {25},
      YEAR = {1976},
    NUMBER = {2},
     PAGES = {135--158},
      ISSN = {0022-2518},
   MRCLASS = {55E35 (55H15)},
  MRNUMBER = {MR0391086 (52 \#11908)},
MRREVIEWER = {Raphael Zahler},
}
@article {MR0383402,
    AUTHOR = {Lin, T. Y.},
     TITLE = {On the {$T$}-nilpotent hypothesis of the stable homotopy 
ring
              of spheres},
   JOURNAL = {Bull. Inst. Math. Acad. Sinica},
  FJOURNAL = {Bulletin of the Institute of Mathematics. Academia Sinica},
    VOLUME = {3},
      YEAR = {1975},
    NUMBER = {2},
     PAGES = {259--264},
      ISSN = {0304-9825},
   MRCLASS = {55E45},
  MRNUMBER = {MR0383402 (52 \#4283)},
MRREVIEWER = {J. F. Adams},
}
@article {MR0418092,
    AUTHOR = {Lin, T. Y.},
     TITLE = {On non-connected flat stable homotopy modules},
   JOURNAL = {Chinese J. Math.},
  FJOURNAL = {Chinese Journal of Mathematics},
    VOLUME = {2},
      YEAR = {1974},
    NUMBER = {1},
     PAGES = {148--170},
      ISSN = {0379-7570},
   MRCLASS = {55E10},
  MRNUMBER = {MR0418092 (54 \#6136)},
MRREVIEWER = {Joel M. Cohen},
}
@article {MR0380789,
    AUTHOR = {Lin, T. Y.},
     TITLE = {Homological dimensions of stable homotopy modules and their
              geometric characterizations},
   JOURNAL = {Trans. Amer. Math. Soc.},
  FJOURNAL = {Transactions of the American Mathematical Society},
    VOLUME = {172},
      YEAR = {1972},
     PAGES = {473--490 (1973)},
      ISSN = {0002-9947},
   MRCLASS = {55E45},
  MRNUMBER = {MR0380789 (52 \#1686)},
MRREVIEWER = {M. Mimura},
}