theory arises in this way. More precisely, this construction yields a bijection
t Spectra u{equivalence » tCohomology theories u{isomorphism .
Spectra are generalized abelian groups: Let E be a spectrum. Then the 0th space Ω8E is an example of an E8-space: that is, it can be equipped with an addition law
`: Ω8EˆΩ8E ÑΩ8E
which is unital, commutative, and associative up to coherent homotopy. Moreover, the constructionE ÞÑΩ8E restricts to an equivalence from the 8-category Spcn of connective spectra to the 8-category CAlggppSq of grouplike E8-spaces (an E8-space Ais said to begrouplikeif the addition on Aexhibits the set of connected components π0A as an abelian group).
Spectra are the universal stable 8-category: The 8-category Sp is stable, admits small colimits, and contains a distinguished object S (the sphere spectrum). Moreover, it is universal with respect to these properties: if C is any stable 8-category which admits small colimits and LFunpSp,Cq denotes the full subcategory of FunpSp,Cq spanned by those functors which preserve small colimits, then the constructionF ÞÑ FpSq induces an equivalence of 8-categories e: LFunpSp,Cq ÑC. In particular, for each object CPC, there is an essentially unique functorF : SpÑC which preserves small colimits and satisfies FpSq “C.
Let X be a spectrum. We will say that X is discrete if the homotopy groups πnX vanish forn‰0. In this case,X is determined (up to canonical equivalence) by the abelian group π0X. More precisely, the constructionX ÞÑπ0X induces an equivalence from the full subcategory Sp♥ ĎSp spanned by the discrete spectra to the ordinary category of abelian groups (which we can regard as an8-category by taking its nerve). We can use an inverse of this equivalence to identify the category of abelian groups with the full subcategory Sp♥ĎSp. IfA is an abelian group, then the image of Aunder this identification is called the Eilenberg-MacLane spectrum of A. As an infinite loop space, it is given by the sequence tKpA, nqu; hereKpA, nq denotes the Eilenberg-MacLane space characterized by the formula
π˚KpA, nq “
#A if˚ “n 0 otherwise.
The corresponding cohomology theory is ordinary cohomology with coefficients in A.
Throughout this book, we will will often abuse notation by identifying an abelian group A with its corresponding Eilenberg-MacLane spectrum.
It follows from the universal property of the8-category Sp that there is an essentially unique functorb: SpˆSpÑSp which preserves small colimits separately in each variable and satisfies SbS “ S. We will refer to this functor as the smash product. Using the universal property of the 8-category Sp, one can show that the smash product functor endows Sp with the structure of asymmetric monoidal8-category: that is, the functorbis commutative, associative, and unital up to coherent homotopy.
Definition 0.2.3.13. For any symmetric monoidal8-categoryC, we let CAlgpCqdenote the 8-category of commutative algebra objects of C. Roughly speaking, an object of CAlgpCq is given by an object A P C equipped with a multiplication m : A bA Ñ A which is commutative, associative, and unitalup to coherent homotopy. In the special case whereC is the8-category of spectra (equipped with the symmetric monoidal structure given by the smash product), we will denote CAlgpCq simply by CAlg. We will refer to the objects of CAlg as E8-rings and to the8-category C asthe 8-category of E8-rings.
Remark 0.2.3.14. LetE be a spectrum, so that E determines a cohomology theory which assigns to each space X a graded abelian group E˚pXq. If E is an E8-ring, then the associated cohomology theory is multiplicative: that is, it assigns to each spaceX a graded ring E˚pXq which is commutative in the graded sense (meaning thatxy “ p´1qmnyxfor x P EmpXq and y P EnpXq). However, the converse fails dramatically: there are many examples of multiplicative cohomology theories which cannot be represented byE8-rings.
Roughly speaking, one expects a cohomology theoryE˚ to be represented by anE8-ring if it is can be equipped with a multiplicative structure for which commutativity and associativity can be seen (at least up to coherent homotopy) at the level of cochains, rather than merely at the level of cohomology.
Let E be an E8-ring. Then the collection of homotopy groups π˚E “ E´˚ptxuq has the structure of a graded-commutative ring. In particular,π0E is a commutative ring and each πnE can be regarded as a module over π0E. We will say that E is connective if it is connective when regarded as a spectrum (that is, the homotopy groups πnE vanish for nă0) and we will say thatE isdiscrete if it is discrete when regarded as a spectrum (that is, the homotopy groups πnE vanish forn‰0). We let CAlgcn denote the full subcategory of CAlg spanned by the connective E8-rings, and we let CAlg♥ denote the full subcategory of CAlg spanned by the discrete E8-rings.
Remark 0.2.3.15. The constructionAÞÑΩ8Ainduces an equivalence from the8-category of connective spectra to the 8-category of grouplike E8-spaces. We can phrase this more informally as follows: giving a connective spectrum is equivalent to giving a space X which behaves like an abelian group up to coherent homotopy. This heuristic can be extended toE8-rings: a connective E8-ring A can be thought of as a spaceX which behaves like a commutative ring up to coherent homotopy.
0.3. OVERVIEW 53 Remark 0.2.3.16. The construction A ÞÑ π0A determines an equivalence from the 8 -category CAlg♥ to the ordinary category of commutative rings (regarded as an8-category via its nerve). We will generally abuse notation by using this equivalence to identify CAlg♥ with the category of commutative rings, so that every commutative ringR is regarded as an E8-ring. In terms of the heuristic of Remark 0.2.3.15, this corresponds to regardingR as a space equipped with the discrete topology.
By virtue of Remark 0.2.3.16, we can regard the theory of E8-rings as a generalization of classical commutative algebra. Moreover, it is a robust generalization: all of the basic results, constructions, and ideas that are needed to set up the foundations of classical algebraic geometry have analogues in the setting of E8-rings, which we will make use of throughout this book.
0.3 Overview
This book is divided into nine parts, each of which is devoted to exploring some facet of the relationship between algebraic geometry and structured ring spectra. Our first goal is to establish foundations for the subject. We begin in Part I by introducing “spectral” versions of various algebro-geometric objects (such as schemes, algebraic spaces, and Deligne-Mumford stacks) and studying how these objects are related to their classical counterparts. We also explain how to associate to every spectral schemeX (or, more generally, any spectral Deligne-Mumford stack) a stable 8-category QCohpXq of quasi-coherent sheaves on X, which is closely related to the abelian categories of quasi-coherent sheaves which appear in classical algebraic geometry.
Part II is concerned with proper morphisms in the setting of spectral algebraic geometry.
In some sense, there is very little to say here: a morphism f : pX,OXq Ñ pY,OYq of spectral schemes is proper if and only if the underlying morphism of ordinary schemes f0 :pX, π0OXq Ñ pY, π0OYq is proper. However, some aspects of the theory work more smoothly in the spectral setting. For example, many important foundational results about proper morphisms between Noetherian schemes (for example, the direct image theorem, the theorem on formal functions, the Grothendieck existence theorem, and Grothendieck’s formal GAGA principle) admit generalizations to the setting of spectral algebraic geometry which do not require any Noetherian assumptions (see Theorem 5.6.0.2, Lemma 8.5.1.1, Theorem 8.5.0.3, and Corollary 8.5.3.4).
The subject of Part III is the following general question: to what extent can an algebro-geometric object X can be recovered from the stable 8-category QCohpXq? We address this question by proving several “Tannaka reconstruction” type results which assert that, in many circumstances, we can recoverX as a kind of “spectrum” of QCohpXq (much like an affine schemepY,OYqcan be recovered as the spectrum of its coordinate ring ΓpY;OYq). We
also show that there is a close relationship between stable8-categoriesC equipped with an action of QCohpXq and sheaves of stable8-categories onX (categorifying the relationship between quasi-coherent sheaves on an affine schemepY,OYqand modules over the coordinate ring ΓpY;OYq.
A standard heuristic principle of deformation theory asserts that over a field κ of characteristic zero, one can describe a formal neighborhood of any algebro-geometric object X near a point x P X in terms of a differential graded Lie algebra. In Part IV, we will formulate this principle precisely by introducing an 8-category Moduliκ of formal moduli problemsoverκ and constructing an equivalence of Moduliκ with an8-category of differential graded Lie algebras overκ. We also study variants of this principle in the setting of noncommutative geometry (which are valid in any characteristic). Part IV is mostly independent of the first three parts (they are relevant mainly because they provide examples of formal moduli problems which can be analyzed using the formalism of Part IV).
In Part??we study representability problems in the setting of spectral algebraic geometry.
Suppose we are given a functorh: CAlgcnÑS, where CAlgcn denotes the8-category of connective E8-rings and S denotes the 8-category of spaces. We might then ask if there exists a spectral schemeX (or some other sort of algebro-geometric object) whichrepresents the functor h, in the sense that there exist homotopy equivalenceshpAq »MappSpecA, Xq depending functorially onA(such anX is uniquely determined up to equivalence, as we will see in Part I). In the setting of classical algebraic geometry, this sort of question can often be addressed using Artin’s representability theorem, which gives necessary and sufficient conditions for a functor to be representable by an algebraic stack which is locally of finite presentation over a (sufficiently nice) commutative ring R. The main goal of Part ??is to formulate and prove an analogous statement in the spectral setting.
The basic objects of study in classical and spectral algebraic geometry can be described in a very similar way: they are given by pairs pX,OXq, where X is a topological space (or some variant thereof: in the theory of Deligne-Mumford stacks, it is convenient to allowX to be a topos; when studying higher Deligne-Mumford stacks, it is convenient to allowX to be an 8-topos) and OX is a “structure sheaf” on X. The difference between classical and spectral algebraic geometry lies in what sort of sheafOX is: in the classical case,OX
is a sheaf of commutative rings; in the spectral case, it is a sheaf ofE8-rings. In Part VI, we introduce general formalism of “8-topoi with structure sheaves” which is intended to capture the spirit of these types of definitions in a broad degree of generality. Part VI does not depend on any of the earlier parts of this book (logically, it could precede Part I; however, most readers will probably find it easier to digest the theory of CAlg-valued sheaves than the general sheaf theory of Part VI).
In Part VII, we study several variants of spectral algebraic geometry:
• derived differential topology, whose basic objects (derived manifolds) are analogous
0.3. OVERVIEW 55