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On the homotopy type of the spectrum representing elliptic cohomology


Author: Andrew Baker
Journal: Proc. Amer. Math. Soc. 107 (1989), 537-548
MSC: Primary 55N22; Secondary 11F11
MathSciNet review: 982399
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Abstract: In this paper we analyse the homotopy type at primes $ p > 3$ of the ring spectrum $ E\ell \ell $ representing a version of elliptic cohomology whose coefficient ring $ E\ell {\ell _ * }$ agrees with the ring of modular forms for $ S{L_2}(\mathbb{Z})$. For any prime (=maximal) graded ideal $ \mathcal{P} \triangleleft E\ell {\ell _*}$ containing the Eisenstein function $ {E_{p - 1}}$ as well as $ p$, we show that there is a morphism of ring spectra

$\displaystyle \widehat{E(2)} \to (E\ell \ell )_{\hat{\mathcal{P}}}$

and a corresponding splitting

$\displaystyle (E\ell \ell )_{\hat{\mathcal{P}}} \simeq \mathop \bigvee\limits_i {\Sigma ^{2\theta (i)}}\widehat{E(2)}$

of algebra spectra over $ \widehat{E(2)}$ (the $ {I_2}$-adic completion of $ E(2)$); here $ (\;)_{\hat{\mathcal{P}}}$ denotes the $ \mathcal{P}$-adic completion of the spectrum $ E\ell \ell $. Moreover, there is a multiplicative reduction $ {(E\ell \ell /\mathcal{P})^ * }(\;)$ and we similarly show that there is a splitting of $ K(2)$ algebra spectra

$\displaystyle E\ell \ell /\mathcal{P} \simeq \mathop \bigvee\limits_i {\Sigma ^{2\theta '(i)}}K(2).$

In each case the indexing $ i$ ranges over a finite set.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0982399-6
Keywords: Elliptic cohomology, $ {v_2}$-Periodicity, Morava $ K$-theory
Article copyright: © Copyright 1989 American Mathematical Society