### Topological Quillen localization of structured ring spectra

#### J.E. Harper, Y. Zhang

The aim of this short paper is two-fold: (i) to construct a \$\TQ\$-localization functor on algebras over a spectral operad \$\capO\$, in the case where no connectivity assumptions are made on the \$\capO\$-algebras, and (ii) more generally, to establish the associated \$\TQ\$-local homotopy theory as a left Bousfield localization of the usual model structure on \$\capO\$-algebras, which itself is almost never left proper, in general. In the resulting \$\TQ\$-local homotopy theory, the ``weak equivalences'' are the \$\TQ\$-homology equivalences, where ``\$\TQ\$-homology'' is short for topological Quillen homology, which is also weakly equivalent to stabilization of \$\capO\$-algebras. More generally, we establish these results for \$\TQ\$-homology with coefficients in a spectral algebra \$\capA\$. A key observation, that goes back to the work of Goerss-Hopkins on moduli problems, is that the usual left properness assumption may be replaced with a strong cofibration condition in the desired subcell lifting arguments: Our main result is that the \$\TQ\$-local homotopy theory can be established (e.g., a semi-model structure in the sense of Goerss-Hopkins and Spitzweck, that is both cofibrantly generated and simplicial) by localizing with respect to a set of strong cofibrations that are \$\TQ\$-equivalences.
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Tbilisi Mathematical Journal, Vol. 12(3) (2019), pp. 69-91

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