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Rational and P-Adic Homotopy Theory Derived Algebraic Geometry XIII: Rational and p-adic Homotopy Theory December 15, 2011 Contents 1 Rational Homotopy Theory 4 1.1 Cohomological Eilenberg-Moore Spectral Sequences . 5 1.2 k-Rational Homotopy Types . 9 1.3 Rational Homotopy Theory and E1-Algebras . 15 1.4 Differential Graded Lie Algebras . 18 1.5 Comparison with Formal Moduli Problems . 24 2 E1-Algebras in Positive Characteristic 27 2.1 Norm Maps . 28 2.2 Power Operations on E1-Algebras . 33 2.3 Finitely Constructible Sheaves . 38 2.4 A Universal Coefficient Theorem . 48 2.5 Compactness of Relative Cochain Algebras . 55 2.6 Affine Behavior of p-Constructible Morphisms . 64 3 p-Profinite Homotopy Theory 69 3.1 Pro-Constructible Sheaves and p-Profinite Spaces . 70 3.2 Whitehead's Theorem . 73 3.3 p-Profinite Spaces of Finite Type . 82 3.4 Connectivity in p-Profinite Homotopy Theory . 87 3.5 p-adic Homotopy Theory . 94 3.6 Etale´ Homotopy Theory . 98 1 Introduction The principal goal of algebraic topology is to study topological spaces X by means of invariants associated ∗ to X, such as the homology and cohomology groups H∗(X; Z) and H (X; Z). Ideally, we would like to devise invariants X 7! F (X) which satisfy the following requirements: (a) The invariant F is algebraic in nature: that is, it assigns to each space X something like a group or a vector space, which is amenable to study using methods of abstract algebra. (b) The invariant F is powerful: that is, F (X) contains a great deal of useful information about X. There is a natural tension between these requirements: the simpler an object F (X) is, the less information we should expect it to contain. Nevertheless, there are some invariants which do a good job of meeting both objectives. One example is provided by Sullivan's approach to rational homotopy theory. To every topological space X, Sullivan associates a polynomial deRham complex A(X), which is a commutative differential graded algebra over the field Q of rational numbers. This is an object of a reasonably algebraic nature (albeit not quite so simple as a group or a vector space), which at the same time captures the entire rational homotopy type of X: for example, if X is a simply connected space whose homotopy groups fπnXgn≥2 are finite- dimensional vector spaces over Q, then we can functorially recover X (up to homotopy equivalence) from A(X). As a chain complex, Sullivan's polynomial deRham complex A(X) is quasi-isomorphic with the complex of singular cochains C∗(X; Q). The advantage of A(X) of C∗(X; Q) is that A(X) is equipped with a mul- tiplication which is commutative at the level of cochains, whereas the multiplication on C∗(X; Q) (given by the Alexander-Whitney construction) is only commutative at the level of cohomology. We can therefore think of A(X) as a remedy for the failure of the multiplication on C∗(X; Q) to be commutative at the level of cochains. This is specific to the case of rational coefficients. If p is a prime number and Fp is the finite field ∗ with p elements, then there is no way to replace the chain complex C (X; Fp) by a commutative differential ∗ graded algebra over Fp, which is functorially quasi-isomorphic to C (X; Fp). However, a different remedy ∗ is available over Fp: although the multiplication on C (X; Fp) is not commutative, it is commutative up to ∗ coherent homotopy. More precisely, C (X; Fp) has the structure of an E1-algebra over Fp. Moreover, Man- dell has used this observation to develop a \p-adic" counterpart of rational homotopy theory. For example, he has shown that if X is a simply connected space whose homotopy groups are finitely generated modules ∗ over Zp, then X can be functorially recovered from C (X; Fp), together with its E1-algebra structure (see [54]). Our goal in this paper is to give an exposition of rational and p-adic homotopy theory from the 1- categorical point of view, emphasizing connections with the earlier papers in this series. We will begin with the case of rational homotopy theory. Sullivan's work on the subject was preceded by Quillen, who showed that the homotopy theory of rational spaces can be described in terms of the homotopy theory of differential graded Lie algebras over Q. This result of Quillen was the impetus for later work of many authors, relating differential graded Lie algebras to the study of deformation problems in algebraic geometry. In [49], we made this relationship explicit by constructing an equivalence of 1-categories Liek ' Modulik, where k is any field of characteristic zero. Here Liek denotes the 1-category of differential graded Lie algebras over k, and Modulik the 1-category of formal moduli problems over k. In x1, we will apply this result (in the special case k = Q) to recover Quillen's results. Along the way, we will discuss Sullivan's approach to rational homotopy theory and its relationship with the theory of coaffine stacks developed in [47]. In x2 we will turn our attention to the case of p-adic homotopy theory. We will say that a space X is p-finite if it has finitely many connected components and finitely many nonzero homotopy groups, each of which is a finite p-group (Definition 2.4.1). If X is a p-finite space, then Mandell shows that X can be recovered as the mapping space ∗ ∗ MapCAlg (C (X; Fp); Fp) ' MapCAlg (C (X; Fp); Fp) Fp Fp where CAlgk denotes the 1-category of E1-algebras over k and Fp denotes the algebraic closure of Fp. Here ∗ it is important to work over Fp rather than Fp: the functor X 7! C (X; Fp) is not fully faithful (even when 2 restricted to p-finite spaces). We can explain this point as follows: if k is an arbitrary field of characteristic p, then the homotopy theory of E1-algebras over k is most naturally related not to the homotopy theory of (p-finite) spaces, but to the theory of (p-finite) spaces equipped with a continuous action of the absolute Galois group Gal(k=k). More generally, we will show that if k is a commutative ring in which p is nilpotent, p−fc op p−fc then there is a fully faithful embedding from Shvk into CAlgk (Corollary 2.6.12); here Shvk denotes the 1-category of p-constructible ´etalesheaves (of spaces) on Spec R (see Definition 2.4.1). In order to apply the results of x2 to the study of an arbitrary space X, we need to study the problem of approximating X by p-finite spaces. For this, it is convenient to introduce the notion of a p-profinite space. By definition, a p-profinite space is a Pro-object in the 1-category Sp−fc of p-finite spaces. The collection of p-profinite spaces can be organized into an 1-category SPro(p) = Pro(Sp−fc). In x3 we will study the 1-category SPro(p), and show that it behaves in many respects like the usual 1-category of spaces. Using Corollary 2.6.12, we will construct a fully faithful embedding Pro(p) op S ! CAlgk X 7! C∗(X; k); where k is any separably closed field (Proposition 3.1.16). Moreover, if k is algebraically closed, we can explicitly describe the essential image of this functor (Theorem 3.5.8). We then recover some results of [54] by restricting our attention to p-profintie spaces of finite type (Corollary 3.5.15). Notation and Terminology We will use the language of 1-categories freely throughout this paper. We refer the reader to [43] for a general introduction to the theory, and to [44] for a development of the theory of structured ring spectra from the 1-categorical point of view. We will also assume that the reader is familiar with the formalism of spectral algebraic geometry developed in the earlier papers in this series. For convenience, we will adopt the following reference conventions: (T ) We will indicate references to [43] using the letter T. (A) We will indicate references to [44] using the letter A. (V ) We will indicate references to [45] using the Roman numeral V. (VII) We will indicate references to [46] using the Roman numeral VII. (VIII) We will indicate references to [47] using the Roman numeral VIII. (IX) We will indicate references to [48] using the Roman numeral IX. (X) We will indicate references to [49] using the Roman numeral X. (XI) We will indicate references to [50] using the Roman numeral XI. (XII) We will indicate references to [51] using the Roman numeral XII. For example, Theorem T.6.1.0.6 refers to Theorem 6.1.0.6 of [43]. If C is an 1-category, we let C' denote the largest Kan complex contained in C: that is, the 1- category obtained from C by discarding all non-invertible morphisms. We will say that a map of simplicial sets f : S ! T is left cofinal if, for every right fibration X ! T , the induced map of simplicial sets FunT (T;X) ! FunT (S; X) is a homotopy equivalence of Kan complexes (in [43], we referred to a map with this property as cofinal). We will say that f is right cofinal if the induced map Sop ! T op is left cofinal: that is, if f induces a homotopy equivalence FunT (T;X) ! FunT (S; X) for every left fibration X ! T .
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