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Derived Algebraic Geometry III: Commutative Algebra

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Derived Algebraic Geometry III: Commutative Algebra Derived Algebraic Geometry III: Commutative Algebra October 21, 2009 Contents 1 1-Operads 4 1.1 Basic Definitions . 5 1.2 Fibrations of 1-Operads . 11 1.3 Algebra Objects . 14 1.4 Cartesian Monoidal Structures . 19 1.5 CoCartesian Monoidal Structures . 27 1.6 Monoidal Envelopes . 35 1.7 Subcategories of O-Monoidal 1-Categories . 41 1.8 1-Preoperads . 45 1.9 O-Operad Families . 50 1.10 Unital 1-Operads . 56 2 Algebras 59 2.1 Limits of Algebras . 60 2.2 Operadic Colimit Diagrams . 63 2.3 Unit Objects and Trivial Algebras . 72 2.4 Strictly Unital 1-Categories . 77 2.5 Operadic Left Kan Extensions . 78 2.6 Free Algebras . 87 2.7 Colimits of Algebras . 92 3 Modules 96 3.1 Coherent 1-Operads and Modules . 97 3.2 Algebra Objects of 1-Categories of Modules . 106 3.3 Limits of Modules . 119 3.4 Colimits of Modules . 130 3.5 Modules for the Associative 1-Operad . 140 3.6 Modules for the Commutative 1-Operad . 153 4 Commutative Ring Spectra 162 4.1 Commutativity of the Smash Product . 162 4.2 E1-Ring Spectra . 168 4.3 Symmetric Monoidal Model Categories . 175 1 Introduction Let K denote the functor of complex K-theory, which associates to every compact Hausdorff space X the Grothendieck group K(X) of isomorphism classes of complex vector bundles on X. The functor X 7! K(X) is an example of a cohomology theory: that is, one can define more generally a sequence of abelian groups n fK (X; Y )gn2Z for every inclusion of topological spaces Y ⊆ X, in such a way that the Eilenberg-Steenrod axioms are satisfied (see [18]). However, the functor K is endowed with even more structure: for every topological space X, the abelian group K(X) has the structure of a commutative ring (when X is compact, the multiplication on K(X) is induced by the operation of tensor product of complex vector bundles). One would like that the ring structure on K(X) is a reflection of the fact that K itself has a ring structure, in a suitable setting. To analyze the problem in greater detail, we observe that the functor X 7! K(X) is representable. That is, there exists a topological space Z = Z×BU and a universal class η 2 K(Z), such that for every sufficiently nice topological space X, the pullback of η induces a bijection [X; Z] ! K(X); here [X; Z] denotes the set of homotopy classes of maps from X into Z. According to Yoneda's lemma, this property determines the space Z up to homotopy equivalence. Moreover, since the functor X 7! K(X) takes values in the category of commutative rings, the topological space Z is automatically a commutative ring object in the homotopy category H of topological spaces. That is, there exist addition and multiplication maps Z × Z ! Z, such that all of the usual ring axioms are satisfied up to homotopy. Unfortunately, this observation is not very useful. We would like to have a robust generalization of classical algebra which includes a good theory of modules, constructions like localization and completion, and so forth. The homotopy category H is too poorly behaved to support such a theory. An alternate possibility is to work with commutative ring objects in the category of topological spaces itself: that is, to require the ring axioms to hold \on the nose" and not just up to homotopy. Although this does lead to a reasonable generalization of classical commutative algebra, it not sufficiently general for many purposes. For example, if Z is a topological commutative ring, then one can always extend the functor X 7! [X; Z] to a cohomology theory. However, this cohomology theory is not very interesting: in degree zero, it simply gives the following variant of classical cohomology: Y n H (X; πnZ): n≥0 In particular, complex K-theory cannot be obtained in this way. In other words, the Z = Z × BU for stable vector bundles cannot be equipped with the structure of a topological commutative ring. This reflects the fact that complex vector bundles on a space X form a category, rather than just a set. The direct sum and tensor product operation on complex vector bundles satisfy the ring axioms, such as the distributive law E ⊗(F ⊕ F0) ' (E ⊗ F) ⊕ (E ⊗ F0); but only up to isomorphism. However, although Z × BU has less structure than a commutative ring, it has more structure than simply a commutative ring object in the homotopy category H, because the isomorphism displayed above is actually canonical and satisfies certain coherence conditions (see [43] for a discussion). To describe the kind of structure which exists on the topological space Z × BU, it is convenient to introduce the language of commutative ring spectra, or, as we will call them, E1-rings. Roughly speaking, an E1-ring can be thought of as a space Z which is equipped with an addition and a multiplication for which the axioms for a commutative ring hold not only up to homotopy, but up to coherent homotopy. The E1-rings play a role in stable homotopy theory analogous to the role played by commutative rings in ordinary algebra. As such, they are the fundamental building blocks of derived algebraic geometry. Our goal in this paper is to introduce the theory of E1-ring spectra from an 1-categorical point of view. An ordinary commutative ring R can be viewed as a commutative algebra object in the category of abelian groups, which we view as endowed with a symmetric monoidal structure given by tensor product of abelian groups. To obtain the theory of E1-rings we will use the same definition, except that we will replace the 2 ordinary category of abelian groups by the 1-category of spectra. In order to carry out the details, we will need to develop an 1-categorical version of the theory of symmetric monoidal categories. The construction of this theory will occupy our attention throughout most of this paper. The notion of a symmetric monoidal 1-category can be regarded as a special case of the more general notion of an 1-operad. The theory of 1-operads bears the same relationship to the theory of 1-categories that the classical theory of colored operads (or multicategories) bears to ordinary category theory. Roughly speaking, an 1-operad C⊗ consists of an underlying 1-category C equipped with some additional data, given by \multi-input" morphism spaces MulC(fXigi2I ;Y ) for Y 2 C, where fXigi2I is any finite collection of objects of C; we recover the usual morphism spaces in C by taking the set I to contain a single element. We will give a more precise definition in x1, and construct a number of examples. Remark 0.0.1. An alternate approach to the theory of 1-operads has been proposed by Cisinski and Moerdijk, based on the formalism of dendroidal sets (see [15] for details). It seems very likely that their theory is equivalent to the one presented here. More precisely, there should be a Quillen equivalence between the category of dendroidal sets and the category POp1 of 1-preoperads which we describe in x1.8. Nevertheless, the two perspectives differ somewhat at the level of technical detail. Our approach has the advantage of being phrased entirely in the language of simplicial sets, which allows us to draw heavily upon the preexisting theory of 1-categories (as described in [46]) and to avoid direct contact with the combinatorics of trees (which play an essential role in the definition of a dendroidal set). However, the advantage comes at a cost: though our 1-operads can be described using the relatively pedestrian language of simplicial sets, the actual simplicial sets which we need are somewhat unwieldy and tend to encode information in a inefficient way. Consequently, some of the proofs presented here are perhaps more difficult than they need to be: for example, we suspect that Theorems 2.5.4 and 3.4.3 admit much shorter proofs in the dendroidal setting. If C⊗ is a symmetric monoidal 1-category (or, more generally, an 1-operad), then we can define an 1-category CAlg(C) of commutative algebra objects of C. In x2, we will study these 1-categories in some detail: in particular, we will show that, under reasonable hypotheses, there is a good theory of free algebras which can be used to construct colimits of diagrams in CAlg(C). If A 2 CAlg(C) is a commutative algebra object of C, we can also associate to A an 1-category ModA(C) of A-module objects of C. We will define and study these 1-categories of modules in x3, and show that (under some mild hypotheses) they inherit the structure of symmetric monoidal 1-categories. The symmetric monoidal 1-category of greatest interest to us is the 1-category Sp of spectra. In x4, we will show that the smash product monoidal structure on Sp (constructed in xM.4.2) can be refined, in an essentially unique way, to a symmetric monoidal structure on Sp. We will define an E[1]-ring to be a commutative algebra object in the 1-category of Sp. In x4.2, we will study the 1-category CAlg(Sp) of E[1]-rings and establish some of its basic properties. In particular, we will show that the 1-category CAlg(Sp) contains, as a full subcategory, the (nerve of the) ordinary category of commutative rings. In other words, the theory of E[1]-rings really can be viewed as a generalization of classical commutative algebra.
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