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Logarithmic Structures on Commutative Hk-Algebra Spectra

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Logarithmic Structures on Commutative Hk-Algebra Spectra Logarithmic structures on commutative Hk-algebra spectra Dissertation zur Erlangung des Doktorgrades an der Fakultät für Mathematik, Informatik und Naturwissenschaften der Universität Hamburg vorgelegt am Fachbereich Mathematik von Julia Schulz Hamburg 2018 Als Dissertation angenommen vom Fachbereich Mathematik der Universität Hamburg Auf Grund der Gutachten von Prof. Dr. Birgit Richter und PD Dr. Steffen Sagave Datum der Disputation: 14.09.2018 Vorsitzender der Prüfungskommission: Prof. Dr. Ingo Runkel Contents Introduction1 1 The category J¯ 7 1.1 Definitions . .7 1.2 Well-structured index categories . 10 1.3 The classifying space BJ¯ ........................... 19 2 Pre-log structures on E∞ dgas 25 2.1 Preliminaries on diagram spaces . 25 2.2 Preliminaries on symmetric spectra . 30 2.3 Commutative Hk-algebra spectra are Quillen equivalent to E∞ dgas . 32 2.4 Symmetric spectra and J¯-spaces . 35 2.5 Definition of pre-log structures . 54 2.6 Units, log structures and logification . 57 2.7 An approach via diagram chain complexes . 61 3 Group completion in commutative diagram spaces 71 3.1 Useful results about comma categories and left Bousfield localizations . 71 3.2 Diagram spaces are Quillen equivalent to spaces over the classifying space of the indexing category . 78 3.3 The diagram spaces EI K and BI K ...................... 80 3.4 K-spaces are Quillen equivalent to I-spaces over BI K ........... 84 3.5 Commutative K-spaces are Quillen equivalent to commutative I-spaces over BI K .................................... 87 3.6 Localized model structures on commutative diagram spaces . 92 3.7 The group completion model structure on commutative diagram spaces . 97 3.8 More examples of pre-log cdgas . 103 4 Logarithmic topological Hochschild homology of log cdgas 109 4.1 The cyclic and replete bar constructions . 109 4.2 General repletion . 111 4.3 Logarithmic topological Hochschild homology . 116 4.4 Formally log THH-étale morphisms . 120 Outlook 131 References 132 Introduction This thesis is devoted to establish a theory of logarithmic structures on E∞ differential graded k-algebras (E∞ dgas) where k is a commutative ring with unit. The concept of logarithmic structures has its origin in algebraic geometry [Kat89]. The relevance of logarithmic structures for homotopy theory became apparent in the work of Hesselholt and Madsen who used logarithmic structures for the description of algebraic K-theory of local fields [HM03]. Motivated by the aim to extend structural results about the algebraic K-theory of commutative rings to the algebraic K-theory of commutative ring spectra, Rognes transferred this notion to homotopy theory [Rog09]. A pre-log structure on a commutative ring A is a commutative monoid M together with a map of commutative monoids α: M → (A, ·) from M to the underlying multiplica- tive commutative monoid of A. The triple (A, M, α) is called a pre-log ring. The datum (M, α) is a log structure on A if the map α−1(A×) → A× from the sub commutative monoid α−1(A×) ⊆ M of elements mapping to the units A× of A is an isomorphism. A log ring is a commutative ring A equipped with a log structure (M, α). An easy example is the trivial log ring (A, A×,A× ,→ A). As a homotopical generalization of log rings, Rognes introduced log ring spectra where commutative symmetric ring spectra play the role of commutative rings. Exploiting the Quillen equivalence between commutative I-spaces and E∞ spaces [SS12], for a commutative symmetric ring spectrum A, there is a commutative I-space ΩI (A) defined m by m 7→ Ω (A(m)), representing the underlying multiplicative E∞ space of A. In this way, commutative I-spaces may be viewed as a homotopical counterpart of commutative monoids, and Rognes related them to commutative symmetric ring spectra via a Quillen adjunction. Further, for a commutative symmetric ring spectrum A, there is a sub I I commutative I-space GL1 (A) of Ω (A) that models the grouplike E∞ space of units I I of A. One drawback is that both commutative I-spaces Ω (A) and GL1 (A) do not carry any information about the negative dimensional homotopy groups of A. Consequently, I I the functors Ω and GL1 do not distinguish between a commutative symmetric ring spectrum and its connective cover so that they cannot detect periodicity phenomena in stable homotopy theory. For example, the connective cover map of complex topological I ∼ I K-theory ku → KU induces a weak equivalence GL1 (ku) −→ GL1 (KU). Sagave and Schlichtkrull managed to overcome this problem in [SS12] by employing the more elaborate index category J . The latter is given by Quillen’s localization construction Σ−1Σ on the category of finite sets and bijections Σ. Hence, the classifying space BJ is 0 homotopy equivalent to Q(S ), which is the underlying additive E∞ space of the sphere spectrum S. The categories I and J are examples of well-structured index categories which is a suitable framework to obtain model structures on (structured) diagram spaces. Sagave and Schlichtkrull prove that for a well-structured index category K satisfying some assumptions, the model category of commutative K-spaces is Quillen equivalent to E∞ spaces over the classifying space BK. So commutative J -spaces are Quillen equivalent to 1 0 E∞ spaces over Q(S ) that Sagave and Schlichtkrull describe as graded E∞ spaces. For a commutative symmetric ring spectrum A, the commutative J -space ΩJ (A) is built from m all spaces Ω 2 (A(m1)). This makes it possible to specify a sub commutative J -space J J GL1 (A) of Ω (A) from which we can recover all units in the graded ring π∗(A). A pre-log structure on a commutative symmetric ring spectrum A is a commutative J -space M together with a map of commutative J -spaces α: M → ΩJ (A). The re- sulting pre-log ring spectrum (A, M, α) is a log ring spectrum if the base change map −1 J J J J α (GL1 (A)) → GL1 (A) of the structure map α along the inclusion GL1 (A) → Ω (A) is a weak equivalence. In joint work Rognes, Sagave and Schlichtkrull introduced logarithmic topological Hochschild homology which is an extension of ordinary topological Hochschild homology [RSS15]. The logarithmic topological Hochschild homology of appropriate pre-log ring spectra participates in interesting localization homotopy cofibre sequences that are similar to localization sequences for algebraic K-theory. This is significant for achieving results on algebraic K-theory of commutative ring spectra by means of localization techniques and trace maps from algebraic K-theory to topological Hochschild homology [RSS15]. For instance, topological K-theory spectra yield convenient logarithmic ring spectra which can be regarded as objects sitting in between the connective and the periodic versions of the respective topological K-theory spectra ([RSS15],[RSS18],[Sag14]). Moreover, the tamely ramified extension of the inclusion of the connective Adams summand ` into the p-local connective topological complex K-theory spectrum ku(p) is formally étale with respect to logarithmic topological Hochschild homology [RSS18]. The log ring spectra considered so far either come from log rings or involve topo- logical K-theory spectra. The goal of this thesis is to provide a framework to gain new examples through algebraic objects. Richter and Shipley constructed a chain of Quillen equivalences connecting commutative Hk-algebra spectra to E∞ dgas [RS17]. Using this, we develop a concept of log structures in the algebraic setting. Pre-log structures on E∞ dgas An intermediate model category in Richter and Shipley’s chain of Quillen equivalences between commutative Hk-algebra spectra and E∞ dgas is the category of commutative symmetric ring spectra in simplicial k-modules C(SpΣ(smod)) (see Subsection 2.3). We relate this model category to commutative J¯-spaces. Proposition (Proposition 2.30). There is a Quillen adjunction ΛJ¯ CSJ¯ C(SpΣ(smod)). (0.1) ΩJ¯ Here the category J¯ (see Definition 1.7) arises from the category J by defining an equivalence relation on the morphism sets of the latter. The idea for the category J¯ 2 results from the fact that the action of the symmetric group Σn on the pointed space n of the n-sphere S permutes coordinates, while the action of Σn on the n-sphere chain n complex S (k) is just the sign action. The category J¯ is a well-structured index category, too (see Proposition 1.13), and the induced map of grouplike E∞ spaces BJ → BJ¯ models the first Postnikov section of the sphere spectrum S. Σ We employ the category C(Sp (smod)) as a model for E∞ dgas. Given the above Quillen adjunction (0.1), we define pre-log structures on E∞ dgas as follows. Definition (Definition 2.32). Let A be an object in C(SpΣ(smod)).A pre-log structure on A is a pair (M, α) consisting of a commutative J¯-space M and a map of commutative J¯-spaces α: M → ΩJ¯(A). If (M, α) is a pre-log structure on A, we call the triple (A, M, α) a pre-log cdga. In consideration of Sagave and Schlichtkrull’s definition of pre-log ring spectra, the following proposition confirms that the above definition of pre-log cdgas is reasonable. Proposition (Proposition 2.36). For a positive fibrant object A in C(SpΣ(smod)) and J¯ (m1, m2) in J¯ such that m1 ≥ 1, the space Ω (A)(m1, m2) is weakly equivalent to the m space Ω 2 (U(A)(m1)) where U denotes the forgetful functor to commutative symmetric ring spectra in pointed simplicial sets. Making use of this result, we see that a homology class in the graded homology ring of an E∞ dga gives rise to a pre-log cdga.
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