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Topological Hochschild homology of connective complex \(K\)-theory. (English) Zbl 1107.55006

For an odd prime \(p\), let \(ku\) denote the connective complex \(K\)-theory ring spectrum completed at \(p\) and let \(THH(ku)\) be Topological Hochschild Homology of \(ku\) with an \(\mathbb{S}\)-algebra structure defined along the lines of J. E. McClure and R. E. Staffeldt [Am. J. Math 115, No.1, 1-45 (1993; Zbl 0770.55010)]. Let \(V(0)\) be the mod \(p\) Moore spectrum and let \(V(1)\) be the cofiber of the periodic multiplication \(\Sigma^{2n-2}V(0)\to V(0)\). The main result of this paper is a calculation of \(V(1)_*THH(ku)\), where \(V(1)_*X=\pi_*(V(1)\wedge X)\); these are also called the \((p,\nu_1)\) homotopy groups of \(X\). Theorem 1.1 states that \(V(1)_*THH(ku)\) contains a class \(\mu_2\) of degree \(2p^2\) which generates a polynomial subalgebra \(\mathbb{F}_p[\mu_2]\). And \(V(1)_*THH(ku)\) is a free module of rank \(4(p-1)^2\) over \(\mathbb{F}_p[\mu_2]\). The localization away from \(\mu_2\), \(A_*=\mu_2^{-1}V(1)_*THH(ku)\), is a Frobenius algebra over the graded field \(k_*=\mathbb{F}_p[\mu_2^{\pm 1}]\), i.e., there is an isomorphism of graded \(A_*\)-modules \[ A_*\cong Hom_{k_*}(A_*,\Sigma^{2p^2+2p-2}k_*). \] Now let the units \(\mathbb{Z}_p^{\times}\) of the \(p\)-adics act as \(p\)-adic Adams operations on \(ku\) and let \(\Delta\) be the cyclic subgroup of order \(p-1\) of \(\mathbb{Z}_p^{\times}\). Then there is a weak equivalence \(l\simeq ku^{h\Delta}\), where \(l\) is the \(p\)-completed Adams summand of \(ku\) and \(ku^{h\Delta}\) are the homotopy fixed points. Using this model for the Adams summand, \(l\) becomes a subalgebra of the \(\mathbb{S}\)-algebra \(ku\). Theorem 10.2 in the paper provides the following descent results: For an odd prime \(p\) the inclusion \(l\to ku\) induces weak equivalences of \(p\)-completed spectra \[ THH(ku)^{h\Delta}\simeq THH(l),\qquad TC(ku;p)^{h\Delta}\simeq TC(l,p),\qquad K(ku)^{h\Delta}\simeq K(l). \]

MSC:

55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
19L41 Connective \(K\)-theory, cobordism

Citations:

Zbl 0770.55010
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