## Algebraic & Geometric Topology

### $I$–adic towers in topology

Samuel Wüthrich

#### Abstract

A large variety of cohomology theories is derived from complex cobordism $MU∗(−)$ by localizing with respect to certain elements or by killing regular sequences in $MU∗$. We study the relationship between certain pairs of such theories which differ by a regular sequence, by constructing topological analogues of algebraic $I$–adic towers. These give rise to Higher Bockstein spectral sequences, which turn out to be Adams spectral sequences in an appropriate sense. Particular attention is paid to the case of completed Johnson–Wilson theory $Ê(n)$ and Morava $K$–theory $K(n)$ for a given prime $p$.

#### Article information

Source
Algebr. Geom. Topol., Volume 5, Number 4 (2005), 1589-1635.

Dates
Revised: 9 November 2005
Accepted: 15 November 2005
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796492

Digital Object Identifier
doi:10.2140/agt.2005.5.1589

Mathematical Reviews number (MathSciNet)
MR2186112

Zentralblatt MATH identifier
1107.55007

#### Citation

Wüthrich, Samuel. $I$–adic towers in topology. Algebr. Geom. Topol. 5 (2005), no. 4, 1589--1635. doi:10.2140/agt.2005.5.1589. https://projecteuclid.org/euclid.agt/1513796492

#### References

• \itemsep 1.3pt plus 1pt
• A Baker, $A\sb \infty$ structures on some spectra related to Morava $K$-theories, Quart. J. Math. Oxford Ser. (2) 42 (1991) 403–419
• A Baker, A Lazarev, On the Adams spectral sequence for $R$-modules, \agtref120019173199
• A Baker, B Richter, $\Gamma$-cohomology of rings of numerical polynomials and $E_\infty$-structures on $K$-theory.
• A Baker, U Würgler, Liftings of formal groups and the Artinian completion of $v\sb n\sp {-1}{\rm BP}$, Math. Proc. Cambridge Philos. Soc. 106 (1989) 511–530
• A Baker, U Würgler, Bockstein operations in Morava $K$-theories, Forum Math. 3 (1991) 543–560
• J M Boardman, Stable operations in generalized cohomology, from: “Handbook of algebraic topology”, North-Holland, Amsterdam (1995) 585–686
• J M Boardman, Conditionally convergent spectral sequences, from: “Homotopy invariant algebraic structures (Baltimore, MD, 1998)”, Contemp. Math. 239, Amer. Math. Soc., Providence, RI (1999) 49–84
• A K Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257–281
• J D Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. Math. 136 (1998) 284–339
• S Eilenberg, J C Moore, Foundations of relative homological algebra, Mem. Amer. Math. Soc. No. 55 (1965) 39
• A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, American Mathematical Society, Providence, RI (1997)
• P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, from: “Structured ring spectra”, London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press, Cambridge (2004) 151–200
• J P C Greenlees, J P May, Completions in algebra and topology, from: “Handbook of algebraic topology”, North-Holland, Amsterdam (1995) 255–276
• M J Hopkins, N J Kuhn, D C Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000) 553–594
• M Hovey, J H Palmieri, N P Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997)
• M Hovey, N P Strickland, Morava $K$-theories and localisation, Mem. Amer. Math. Soc. 139 (1999)
• J Hunton, The Morava $K$-theories of wreath products, Math. Proc. Cambridge Philos. Soc. 107 (1990) 309–318
• D Husemoller, J C Moore, Differential graded homological algebra of several variables, from: “Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69)”, Academic Press, London (1970) 397–429
• C U Jensen, Les foncteurs dérivés de $\varprojlim$ et leurs applications en théorie des modules, Springer-Verlag, Berlin (1972)
• I Kriz, Morava $K$-theory of classifying spaces: some calculations, Topology 36 (1997) 1247–1273
• I Kriz, K P Lee, Odd-degree elements in the Morava $K(n)$ cohomology of finite groups, Topology Appl. 103 (2000) 229–241
• S Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin (1995)
• S Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, New York (1998)
• H Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge (1989)
• J P May, $E\sb{\infty }$ ring spaces and $E\sb{\infty }$ ring spectra, with contributions by F Quinn, N Ray, and J Tornehave, Lecture Notes in Mathematics 577, Springer-Verlag, Berlin (1977)
• H R Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, J. Pure Appl. Algebra 20 (1981) 287–312
• D C Ravenel, Morava $K$-theories and finite groups, from: “Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981)”, Contemp. Math. 12, Amer. Math. Soc., Providence, R.I. (1982) 289–292
• D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121, Academic Press Inc., Orlando, FL (1986)
• D C Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies 128, Princeton University Press, Princeton, NJ (1992)
• L Smith, On realizing complex bordism modules. Applications to the stable homotopy of spheres, Amer. J. Math. 92 (1970) 793–856
• N P Strickland, Morava $E$-theory of symmetric groups, Topology 37 (1998) 757–779
• N P Strickland, Products on ${\rm MU}$-modules, Trans. Amer. Math. Soc. 351 (1999) 2569–2606
• C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge (1994)
• S Wüthrich, Multiplicative structures on $I$–adic towers, in preparation
• S Wüthrich, Homology of $I$–adic towers.
• S Wüthrich, Homology of powers of regular ideals, Glasg. Math. J. 46 (2004) 571–584