Completions of ℤ<i>∕</i>(<i>p</i>)–Tate cohomology of periodic spectra

Ando, Matthew; Morava, Jack; Sadofsky, Hal

doi:10.2140/gt.1998.2.145

Cited by 34 publications

(49 citation statements)

References 25 publications

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“…It is shown in [25], [35] that these elements survive in the Adams-Novikov spectral sequence to elements of R(p k ). Similarly, we have shown that β k ∈ R [2] BP (α k ). Again, in [25], [35] it is shown that these elements survive to elements of R(α k ) for p ≥ 5.…”

Section: Proposition 102 the Bp ∧ Bp -Root Invariant Of α I/j Is Gisupporting

confidence: 58%

“…The Tate spectrum computations of [11], [10], [27], [16], [2], and [15] indicate that the Z/p-Tate spectrum of a v n -periodic cohomology theory is v n -torsion. The root invariant is defined using the Tate spectrum of the sphere spectrum, so the results of the papers listed above provide even more evidence that the root invariant of a v n -periodic element should be v n -torsion.…”

Section: Introductionmentioning

confidence: 99%

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Root invariants in the Adams spectral sequence

Behrens

2005

Trans. Amer. Math. Soc.

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Abstract. Let E be a ring spectrum for which the E-Adams spectral sequence converges. We define a variant of Mahowald's root invariant called the 'filtered root invariant' which takes values in the E 1 term of the E-Adams spectral sequence. The main theorems of this paper are concerned with when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invariants and differentials and compositions in the E-Adams spectral sequence. These theorems are compared to some known computations of root invariants at the prime 2. We use the filtered root invariants to compute some low-dimensional root invariants of v 1 -periodic elements at the prime 3. We also compute the root invariants of some infinite v 1 -periodic families of elements at the prime 3.

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Section: Proposition 102 the Bp ∧ Bp -Root Invariant Of α I/j Is Gisupporting

confidence: 58%

Section: Introductionmentioning

confidence: 99%

Root invariants in the Adams spectral sequence

Behrens

2005

Trans. Amer. Math. Soc.

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“…This is discussed in more detail, from the homotopy-theoretic point of view, in our earlier paper [AMS98] with Hal Sadofsky; this paper is a kind of continuation, concerned with analytic aspects of these phenomena. We show that Witten's construction in rational cohomology produces K-theoretic genera because of the exponential exact sequence…”

mentioning

confidence: 96%

A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space

Ando¹,

Morava²

2001

Topology, Geometry, and Algebra: Interactions and New Directions

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Abstract. Let E be a circle-equivariant complex-orientable cohomology theory. We show that the fixedpoint formula applied to the free loop space of a manifold X can be understood as a Riemann-Roch formula for the quotient of the formal group of E by a free cyclic subgroup. The quotient is not representable, but (locally at p) its p-torsion subgroup is, by a p-divisible group of height one greater than the formal group of E.

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“…Then the formal group law F n is of height n on the closed point F and of height n − 1 on the generic point K. By the result of Lazard [10], the formal group laws over a separably closed field of characteristic p > 0 are classified up to isomorphism by their height. Hence there is an isomorphism between F n and the Honda group law H n−1 of height n − 1 over the separable closure K sep of K. In [1] Ando, Morava and Sadofsky showed that there is a unique isomorphism between F n and H n−1 over K sep which satisfies certain conditions motivated from a geometric point of view. We would like to consider the above situation with the action of the nth Morava stabilizer group G n .…”

Section: Introductionmentioning

confidence: 99%

On degeneration of one-dimensional formal group laws and applications to stable homotopy theory

Torii¹

2003

ajm

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Abstract. In this note we study a certain formal group law over a complete discrete valuation ring F[[u n−1 ]] of characteristic p > 0 which is of height n over the closed point and of height n − 1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law H n−1 of height n − 1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group S n−1 of H n−1 . We show that the automorphism group Sn of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of S n−1 to the cohomology of Sn with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed.

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Completions of ℤ∕(p)–Tate cohomology of periodic spectra

Cited by 34 publications

References 25 publications

Root invariants in the Adams spectral sequence

Root invariants in the Adams spectral sequence

A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space

On degeneration of one-dimensional formal group laws and applications to stable homotopy theory

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