The smash product for derived categories in stable homotopy theory (Q436116)

Many decades ago, when we all were much younger, one got for several reasons interested in what was called ``higher categories''. In the approach of the present paper it started with the fact that certain multiplications where not strictly associative but only associative up to homotopy. In the course of time one came down with \(E_n\)-ring spectra \(R\), where \(E_n,\;n \in \mathbb N \cup \{ \infty\}\), denotes a hierarchy of structures. In a previous paper [The multiplication on \(BP\), \url{arXiv:1101.0023}] the present author together with \textit{M. Basterra} investigated the Brown-Peterson spectrum \(BP\), reaching the conclusion that \(BP\) is an \(E_4\) spectrum. Whether however \(BP\) is an \(E_\infty\) spectrum is not known.NEWLINENEWLINE Let \(R\) be a ring spectrum, then the homotopy category of the category of \(R\)-modules (the latter carrying the structure of a homotopy) is called the derived category \(\mathcal D_R\). There is a balanced product between the derived category of right and left \(R\)-modules \(\wedge_R\). An \(E_1\) ring spectrum has a derived category of left modules. An \(E_\infty\) ring spectrum has a derived category having a symmetric monoidal product.NEWLINENEWLINE The main issue of the present paper is the investigation of properties of \(E_n\) for low \(n\):NEWLINENEWLINE (1) Let \(R\) be an \(E_2\) ring spectrum.NEWLINENEWLINEThe derived category of left modules \(\mathcal D_R\) is equivalent to the derived category of right modules \(\mathcal D_{R^{op}}\) having a closed monoidal product NEWLINE\[NEWLINE\wedge_R: \mathcal D_R \times \mathcal D_R \longrightarrow \mathcal D_RNEWLINE\]NEWLINE extending the balanced product.NEWLINENEWLINE (2) If \(R\) is a \(E_3\) ring spectrum, \(\wedge_R\) has a braiding.NEWLINENEWLINE (3) If \(R\) is a \(E_4\) spectrum \(\mathcal D_R\) is a closed symmetric monoidal category.NEWLINENEWLINE The paper contains a wealth of technical details; the reviewer would wish for some more words about the motivation and probably about applications.