Homological dimensions of ring spectra (Q360561)

The present paper is a systematic exploration of different notions of dimension for \(S\)-algebras (or \(A_\infty\)-ring spectra, as they are often called) and modules over them.NEWLINENEWLINEWe will concentrate in this review on the notion of projective dimension. Let \(E\) be an \(S\)-algebra. Then a (right) \(E\)-module \(M\) is of projective dimension \(0\) if it is projective, i.e. if \(\pi_*M\) is a projective \(\pi_*E\)-module. Inductively, the authors define \(M\) to be of projective dimension \(\leq n+1\) if there is a cofiber sequence NEWLINE\[NEWLINE N \to P \to \widetilde{M}, NEWLINE\]NEWLINE where \(P\) is projective, \(N\) of projective dimension \(\leq n\) and \(\widetilde{M}\) is a retract of \(M\). They define the (right) global dimension \(\mathrm{gl.dim}\, E\) of \(E\) to be the supremum over all projective dimensions of \(E\)-modules.NEWLINENEWLINEAn equivalent characterization uses the notion of a ghost: A map \(f: M \to N\) of \(E\)-modules is a ghost if \(\pi_*(f)= 0\). The (right) global dimension of \(E\) is \(\leq n\) if and only if every composite of \((n+1)\) ghosts between \(E\)-modules is zero. This was proven in quite a general context in [\textit{J. D. Christensen}, Adv. Math. 136, No. 2, 284--339 (1998; Zbl 0928.55010)].NEWLINENEWLINEAn important theme of the present paper is the comparison of \(\mathrm{gl.dim}\, E\) with the dimension of \(\pi_* E\). Indeed, the authors show NEWLINE\[NEWLINE\mathrm{depth} \pi_*E \leq \mathrm{gl.dim}\, E \leq \mathrm{gl.dim}\, \pi_*E NEWLINE\]NEWLINE for a commutative \(S\)-algebra \(E\). The second inequality is an equality if \(\pi_*E\) is noetherian of finite global dimension.NEWLINENEWLINEThese theorems immediately give some examples. In particular, the authors deduce that the global dimension of \(MU\) is infinite, while the global dimension of complex K-theory \(KU\) is 1 as \(\pi_*KU = \mathbb{Z}[u,u^{-1}]\) has global dimension 1. More interesting are the cases, where the coefficients have infinite global dimension, but the spectrum itself has finite one. The authors show that if \(E\) is an \(S\)-algebra, \(X\) a finite spectrum with non-trivial reduced rational cohomology and \(E\wedge X\) has the structure of an \(E\)-algebra with finite global dimension, then also \(E\) has finite global dimension. It follows that real K-theory \(KO\) and the spectra of topological modular forms \(tmf\) and \(TMF\), localized at an arbitrary prime, have finite global dimension. Concrete calculations of these global dimensions are more difficult. The authors demonstrate this for \(KO\), where they have to use the Bousfield structure theory for \(KO\)-modules [\textit{A. K. Bousfield}, J. Pure Appl. Algebra 66, No. 2, 121--163 (1990; Zbl 0713.55007)] to deduce that \(\mathrm{gl.dim}\, KO\) is either 2 or 3.NEWLINENEWLINEThis paper touches a number of other topics, including the comparison of the global dimension to the so-called ghost dimension, Gorenstein conditions and the behavior of global dimension under Morita equivalences. Regarding the last topic, the authors present an example by Hügel of two Eilenberg-MacLane spectra \(HR\) and \(HS\) whose derived categories are equivalent, but whose global dimensions disagree. On the other hand, the authors show that \textit{finiteness} of global dimension \textit{is} Morita invariant.