The nilpotence height of 𝑃_{𝑡}^{𝑠} for odd primes

Karaca, Ismet

doi:10.1090/S0002-9947-99-01906-6

The nilpotence height of $P_t^s$ for odd primes
HTML articles powered by AMS MathViewer

by Ismet Karaca

Trans. Amer. Math. Soc. 351 (1999), 547-558

DOI: https://doi.org/10.1090/S0002-9947-99-01906-6

PDF | Request permission

Abstract:

K. G. Monks has recently shown that the element $P^{s}_{t}$ has nilpotence height $2[\frac {s}{t}] + 2$ in the mod $2$ Steenrod algebra. Here the method and result are generalized to show that for an odd prime $p$ the element $P^{s}_{t}$ has nilpotence height $p[\frac {s}{t}] + p$ in the mod $p$ Steenrod algebra.

References

J. F. Adams and H. R. Margolis, Modules over the Steenrod algebra, Topology 10 (1971), 271–282. MR 294450, DOI 10.1016/0040-9383(71)90020-6
J. F. Adams and H. R. Margolis, Sub-Hopf-algebras of the Steenrod algebra, Proc. Cambridge Philos. Soc. 76 (1974), 45–52. MR 341487, DOI 10.1017/s0305004100048714
Donald W. Anderson and Donald M. Davis, A vanishing theorem in homological algebra, Comment. Math. Helv. 48 (1973), 318–327. MR 334207, DOI 10.1007/BF02566125
Dan Arnon, Monomial bases in the Steenrod algebra, J. Pure Appl. Algebra 96 (1994), no. 3, 215–223. MR 1303282, DOI 10.1016/0022-4049(94)90099-X
D.P. Carlisle and R.M.W. Wood, On an Ideal Conjecture in the Steenrod Algebra, preprint (1994).
Donald M. Davis, The antiautomorphism of the Steenrod algebra, Proc. Amer. Math. Soc. 44 (1974), 235–236. MR 328934, DOI 10.1090/S0002-9939-1974-0328934-1
D.M. Davis, On the height of $Sq^{2^{n}}$, preprint (1985).
Andrew M. Gallant, Excess and conjugation in the Steenrod algebra, Proc. Amer. Math. Soc. 76 (1979), no. 1, 161–166. MR 534410, DOI 10.1090/S0002-9939-1979-0534410-8
Brayton Gray, Homotopy theory, Pure and Applied Mathematics, Vol. 64, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. An introduction to algebraic topology. MR 0402714
Leif Kristensen, On a Cartan formula for secondary cohomology operations, Math. Scand. 16 (1965), 97–115. MR 196740, DOI 10.7146/math.scand.a-10751
H. R. Margolis, Spectra and the Steenrod algebra, North-Holland Mathematical Library, vol. 29, North-Holland Publishing Co., Amsterdam, 1983. Modules over the Steenrod algebra and the stable homotopy category. MR 738973
John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171. MR 99653, DOI 10.2307/1969932
Kenneth G. Monks, Nilpotence in the Steenrod algebra, Bol. Soc. Mat. Mexicana (2) 37 (1992), no. 1-2, 401–416. Papers in honor of José Adem (Spanish). MR 1317590
Kenneth G. Monks, The nilpotence height of $P^s_t$, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1297–1303. MR 1301039, DOI 10.1090/S0002-9939-96-03150-4
Judith H. Silverman, Conjugation and excess in the Steenrod algebra, Proc. Amer. Math. Soc. 119 (1993), no. 2, 657–661. MR 1152292, DOI 10.1090/S0002-9939-1993-1152292-8
Judith H. Silverman, Hit polynomials and the canonical antiautomorphism of the Steenrod algebra, Proc. Amer. Math. Soc. 123 (1995), no. 2, 627–637. MR 1254854, DOI 10.1090/S0002-9939-1995-1254854-8
Philip D. Straffin Jr., Identities for conjugation in the Steenrod algebra, Proc. Amer. Math. Soc. 49 (1975), 253–255. MR 380796, DOI 10.1090/S0002-9939-1975-0380796-3
G. Walker and R. M. W. Wood, The nilpotence height of $\textrm {Sq}^{2^n}$, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1291–1295. MR 1307571, DOI 10.1090/S0002-9939-96-03203-0
G.Walker and R.M.W. Wood, The Nilpotence Height of $P^{p^{n}}$, preprint (1995).
R. M. W. Wood, A note on bases and relations in the Steenrod algebra, Bull. London Math. Soc. 27 (1995), no. 4, 380–386. MR 1335290, DOI 10.1112/blms/27.4.380
R.M.W. Wood, Differential Operators and the Steenrod Algebra, preprint (1995).