to extend Proposition 0.1.3.1 to the case where X0 is not smooth. To understand the issues involved, let us begin by how the tangent bundle is defined in the special case where X0 “SpecB is an affine algebraic variety over C. Recall that to any homomorphism of commutative rings φ:AÑB, one can introduce aB-module ΩB{A called themodule of K¨ahler differentials of B relative to A: it is generated as anB-module by symbols tdbubPB which are subject only to the relations
dpb`b1q “db`db1 dpbb1q “bdb1`b1db dφpaq “0.
In the special case where B is asmooth A-algebra, theB-module ΩB{A is locally free of finite rank. In particular, if X0 “SpecB is smooth affine algebraic variety over C, then theB-module ΩB{Cis locally free of finite rank, and therefore corresponds to an algebraic vector bundleTX˚0 on X0 whose dual is (by definition) the tangent bundleTX0.
To a pair of ring homomorphisms φ:AÑB and ψ:B ÑC, one can associate a short exact sequence of K¨ahler differentials
CbBΩB{AÑΩC{AÑΩC{BÑ0. (1) This sequence is generally not exact on the left unless the ring homomorphism ψ is smooth.
To remedy this defect, Andr´e and Quillen (independently) introduced the theory known as Andr´e-Quillen homology(generalizing earlier work of Lichtenbaum and Schlessinger), which has the following features:
• To every homomorphism of commutative rings φ:A Ñ B and every B-module M, one associates a sequence of B-modules tDnpB{A;Mquně0, called the Andr´e-Quillen homology groups ofB relative to A with coefficients in M.
• The theory of Andr´e-Quillen homology generalizes the theory of K¨ahler differentials:
for every homomorphism of commutative rings φ:AÑ B and every B-module M, there is a canonical isomorphismD0pB{A;Mq »MbBΩB{A.
• Ifφ:AÑB is a smooth homomorphism of commutative rings, then the Andr´e-Quillen homology groups DnpB{A;Mqvanish for ną0 and anyB-moduleM.
• To every composable pair of commutative ring homomorphisms φ : A Ñ B and ψ:B ÑC and everyC-moduleM, one can associate a long exact sequence
¨ ¨ ¨ ÑDn`1pC{B;Mq ÑDnpB{A;Mq ÑDnpC{A;Mq ÑDnpC{B;Mq Ñ ¨ ¨ ¨, extending the short exact sequence (1) in the special case M “C.
The Andr´e-Quillen homology groupsD˚pB{A;Mqare obtained from a more fundamental invariantLalgB{A, called thecotangent complex of B overA, a chain complex of (projective) B-modules which determines Andr´e-Quillen homology via the formula D˚pB{A;Mq “ H˚pMbBLalgB{Aq. This invariant was generalized in the work of Illusie (see [?] and [100]), which associates to every morphism f :X ÑS of schemes an object LalgX{S of the derived category DpXq, specializing to the chain complex LalgB{Ain the case whereX “SpecB and S “SpecAare affine. With this generalization, Proposition 0.1.3.1 extends as follows:
Proposition 0.1.3.2. Let X0 be an algebraic variety over C (not necessarily smooth).
Then:
paq For every first-order deformation X of X0, there is a canonical bijection tAutomorphisms of X restricting to the identity on X0u »Ext0DpX
0qpLalgX{SpecC,OXq.
pbq There is a canonical bijection
tIsomorphism classes of first-order deformations of X0u »Ext1DpX
0qpLalgX{SpecC,OXq.
Illusie’s work on the cotangent complex was an important precursor to the theory of spectral algebraic geometry developed in this book. To understand this point, let us begin by recalling that the module of K¨ahler differentials ΩB{A can be characterized by a universal property: for anyB-module M, there is a canonical bijection
tB-module homomorphisms ΩB{AÑMu
θ0
t A-algebra sections of the projection map B‘M ÑB u
which carries a map λ : ΩB{A Ñ M to the A-algebra homomorphism sλ : B Ñ B‘M given by sλpbq “b`λpdbq. The entire cotangent complex LalgB{A can be characterized by an analogous universal property. To simplify the discussion, let us assume thatA andB are Q-algebras; in this case, we will denote the cotangent complex LalgB{A simply byLB{A (see Remark 0.1.3.7 below). IfM is a chain complex of B-modules, then the direct sum B‘M can be regarded as anE8-algebra over B, with homotopy groups given by
π˚pB‘Mq “
#B‘H0pMq if˚ “0 H˚pMq if˚ ‰0.
We then have a canonical bijection
t Maps from LB{A intoM in the derived category DpBq u
θ
tHomotopy classes of sections of the projectionq :B‘M ÑBu
0.1. WHY SPECTRAL ALGEBRAIC GEOMETRY? 29 where we interpret q as a map ofE8-algebras overA. Note thatLB{A is characterized by this universal property: that is, we can defineLB{A to be an object which corepresents the functor
M ÞÑ tHomotopy classes of sections of the projectionq :B‘M ÑBu.
This supplies a definition ofLB{Awhich makes sense foranymorphism ofE8-ringsφ:AÑB. This leads to the theory of topological Andr´e-Quillen homology (see [145], [15], [16]). Like the theory of ordinary Andr´e-Quillen homology, it can be “relativized” to non-affine settings:
to every morphism of spectral schemes f : X Ñ S, one can associate an object LX{S of the triangulated category hQCohpXq called the relative cotangent complex of f. The construction pf :XÑSq ÞÑLX{S has the following features:
p1q Suppose thatX and S are ordinary schemes over SpecQ. Then the relative cotangent complex LX{S (in the setting of spectral schemes) agrees with the cotangent complex LalgX{S of Illusie (for the case of positive and mixed characteristic, see Remark 0.1.3.7 below).
p2q Let f :pX,OXq Ñ pS,OSq be a morphism of spectral schemes, let X0 “ pX, π0OXq andS0 “ pS, π0OSqbe their underlying ordinary schemes, and let A be the abelian category of quasi-coherent sheaves onX0. Then the “degree zero” part ofLX{S agrees, as an object ofA, with the sheaf of K¨ahler differentials ΩX0{S0 of the underlying map f0 :X0 ÑS0.
One of the advantages of working in the setting of spectral algebraic geometry is that there is a much larger class of geometric objects for which the cotangent complex is well-behaved:
piq If X is a spectral C-scheme of finite presentation, then the cotangent complex LX{SpecCPQCohpXq is perfect: that is, it is dualizable as an object of QCohpXq. piiq If X is an ordinary C-scheme of finite presentation, then the cotangent complex
LX{SpecC is a perfect complex if and only ifX is a local complete intersection (see [?]).
To reconcile piqandpiiq, we remark that if X is an ordinary scheme, then the assumption that X is of finite presentation as aC-scheme does not imply that it is of finite presentation as a spectral C-scheme. However, the converse does hold: more generally, if pX,OXq is a spectral scheme of finite presentation over C(as a spectral scheme), then the underlying ordinary scheme X0 “ pX, π0OXq is of finite presentation overC (as a scheme). In this case, the cotangent complexLX{SpecCis often a much more natural and useful object than LX0{SpecC.
Example 0.1.3.3 (Quot Schemes). LetX be a projective algebraic variety over Cand let F be a quasi-coherent sheaf on X. For every commutative C-algebra R, letXR denote the fiber productXˆSpecCSpecR and letFR denote the pullback of F toXR. We let FpRq denote the collection of isomorphism classes of exact sequences
0ÑF1 ÑFRÑF2Ñ0
in the abelian category of quasi-coherent sheaves onXRfor which F2 is flat overR. The construction RÞÑFpRq is representable by aC-scheme Quot (see [?]). Given aC-valued point ηPFpCq classifying an exact sequence
0ÑF1 ÑF ÑF2 Ñ0,
the Zariski cotangent space of Quot at the pointηis the dual of the vector space HomApF1,F2q, where A is the abelian category of quasi-coherent sheaves onX. This vector space can be described as the 0th homology of the cotangent fiber η˚LQuot{SpecC. However, the scheme Quot is usually highly singular, so the entire cotangent fiber η˚LQuot{SpecC can be difficult to describe.
The situation is better if we work in the setting of spectral algebraic geometry. The definitions of XR,FR, andFpRq make sense more generally whenR is anE8-algebra over C, and the resulting functor onE8-rings is representable by a spectral scheme Quot` having Quot is its underlying scheme (this object was introduced in [41]). The structure of the cotangent complex LQuot`{SpecC has an immediate description in terms of the functor F (onE8-algebras). For example, if ηPFpCq is as above, then the complexη˚LQuot`{SpecC
is dual to RHompF1,F2q: in particular, the nth homology group of η˚LQuot`{SpecC can be identified with theC-linear dual of ExtnApF1,F2q.
Example 0.1.3.3 illustrates a general phenomenon: if Z0 is a scheme representing which represents a functorF on the category of commutative rings, there is often a natural way to extend the definition ofFpRqto the case whereR is anE8-ring in such a way that extended functor is representable by a spectral scheme Z. Roughly speaking, we can think of the specification of Z as given by “equippingZ0 with a deformation theory” (we will make this idea precise in Part??; see Theorem 18.1.0.1), according to the rough heuristic
Spectral algebraic geometry“Algebraic Geometry`Deformation Theory.
From the above perspective, the subject of this book is a natural outgrowth of deformation theory, which allows us to think about invariants like the cotangent complex in a more flexible and general setting. However, the ideas of spectral algebraic geometry lead to new and useful ways to think about deformation-theoretic questions even for smooth algebraic varieties. For example, consider the following variant of Proposition 0.1.3.1:
0.1. WHY SPECTRAL ALGEBRAIC GEOMETRY? 31 Proposition 0.1.3.4. Let X0 be a smooth algebraic variety over C. Then there is a canonical “obstruction class map”
ρ:t First order deformations ofX0 u ÑH2pX0;TX0q
with the following property: ifX is a first-order deformation ofX0, then the obstruction class ρpXq PH2pX0;TX0q vanishes if and only ifX can be extended to a second-order deformation of X0: that is, if and only if there exists a pullback square of schemes
X //
X
SpecCrs{p2q //SpecCrs{p3q in which the vertical maps are flat.
Proposition 0.1.3.4 is very useful: it allows us to convert an a priorinonlinear problem (deforming an algebraic variety) into a linear one (checking that a certain cohomology class vanishes). This linear problem is often difficult, but in some cases it is trivial: for example, if X0 is a smooth curve, then H2pX0;TX0q » 0, so Proposition 0.1.3.4 implies that any first-order deformation can be extended to a second-order deformation.
One might object that Proposition 0.1.3.4 is not as satisfying as Proposition 0.1.3.1.
Note that Proposition 0.1.3.1 provides concrete geometric interpretations for the cohomology groups H0pX0;TX0q and H1pX0;TX0q. Proposition 0.1.3.4 tells us that the cohomology group H2pX0;TX0q isrelated to the problem of extending first-order deformations ofX0 to second-order deformations, but it does not tell us what a general element of H2pX0;TX0q is. The language of spectral algebraic geometry provides a remedy for this situation. More precisely, it allows us to formulate a generalization of Proposition 0.1.3.1 which supplies a geometric interpretation for all of the cohomology groups HnpX0;TX0q and which has Proposition 0.1.3.4 as a consequence.
Fix a smooth C-scheme X0 as in Proposition 0.1.3.4. For every commutativeC-algebra R equipped with an augmentation ρ:RÑC, let us define adeformation of X0 over R to be a flat R-scheme XR together with an isomorphism X0 »SpecCˆSpecRXR. The theory of spectral algebraic geometry allows us to consider a more general notion of deformation: if R is anE8-algebra over C(still equipped with an augmentationρ:R ÑC), then we can contemplatespectral schemesXRequipped with a flat mapXRÑSpecRand an equivalence X0»SpecCˆSpecRXR (ifR is an ordinary commutative ring, this reduces to the previous definition: the flatness ofXRover SpecR guarantees thatXRis an ordinary scheme). We then have the following generalization of Proposition 0.1.3.1, which we will discuss in Part V (see Proposition 19.4.3.1 and Corollary 19.4.3.3):
Proposition 0.1.3.5. Let X0 be a smooth algebraic variety over the field C of complex numbers and let R denote the direct sumC‘M, where M is the chain complex of complex vector spaces which is concentrated in degree ně0 and is isomorphic to Cin degree n. Let us regard R as an E8-algebra overC. Then:
paq If XR is any deformation of X0 over R, then there is a canonical bijection
tAutomorphisms of XR that are the identity on X0u{homotopy»HnpX0;TX0q.
pbq There is a canonical bijection
tDeformations of X0 over Ru{homotopy equivalence »Hn`1pX0;TX0q.
Remark 0.1.3.6. In the special case n“0, theE8-ringR“C‘M can be identified with the ordinary ring of dual numbersCrs{p2q, and Proposition 0.1.3.5 reduces to Proposition 0.1.3.1. Proposition 0.1.3.5 implies that all of the cohomology groups HmpX0;TX0q arise naturally when studying deformations of X0 over “shifted” versions of the ring of dual numbers.
To relate Propositions 0.1.3.4 and 0.1.3.5, we note that the Crs{p3q can be regarded as a an extension of Crs{p2q by the square-zero idealp2q »C. This implies that there is a pullback diagram of E8-algebras
Crs{p3q //
C
Crs{p2q φ //R,
where R is defined as in Proposition 0.1.3.5 in the special casen“1. We can interpret this pullback square geometrically as supplying a pushout diagram of affine spectral schemes
SpecCrs{p3qoo SpecC
SpecCrs{p2q
OO
SpecR.
OO
oo
It follows that if X is a first-order deformation of X0, then X extends to a second-order deformation if and only if the fiber product XR “ X ˆSpecCrs{p2q SpecR is a trivial deformation of X0 over R (that is, if and only if it is equivalent to X0 ˆSpecCSpecR).
This is equivalent to the vanishing of a certain element ρpXq PH2pX0;TX0q, whereρ is the
“obstruction map” given by the composition
t Deformations of X0 overCrs{p2q u Ñ t Deformations of X0 overRu
» H2pX0;TX0q.
0.2. PREREQUISITES 33 where the first map is given by extension of scalars along φ and the second follows from the identification of Proposition 0.1.3.5.
Remark 0.1.3.7. Let φ:AÑB be a homomorphism of commutative rings, which we can also regard as a morphism of E8-rings. In general, the algebraic Andr´e-Quillen homology of φ(denoted byLalgB{A in the above discussion and throughout this book) is different from the topological Andr´e-Quillen homology of φ (denoted by LB{A in the above discussion and throughout this book), though they are rationally equivalent (and therefore coincide whenever B is a Q-algebra). The deformation theory of spectral schemes is controlled by topological Andr´e-Quillen homology. For applications in positive and mixed characteristic, it is often more appropriate to use the theory of derived algebraic geometry, in which deformations are controlled by algebraic Andr´e-Quillen homology. We will give a detailed exposition of derived algebraic geometry and its relationship to spectral algebraic geometry in Part VII.
0.2 Prerequisites
Throughout this book, we will make extensive use of the language of 8-categories developed in [138] and [139]. The reader will also need some familiarity with stable homotopy theory and the theory of structured ring spectra, which are developed from the8-categorical perspective in [139] (for a different approach to the same material, see [60]). For the reader’s convenience, we include a brief (and incomplete) expository account of some of the relevant material below. For a more detailed account (which includes precise definitions and proofs), we refer the reader to [138] and [139]. Since we will need to refer to these texts frequently in this book, we adopt the following conventions:
pHT Tq We will indicate references to [138] using the letters HTT.
pHAq We will indicate references to [139] using the letters HA.
For example, Theorem HTT.6.1.0.6 refers to Theorem 6.1.0.6 of [138].
The other main prerequisite for reading this book is some familiarity with classical algebraic geometry. To some extent, this is logically unnecessary: the theory of spectral algebraic geometry is developed “from scratch” in this book, and most of our references to the classical theory are purely for motivation. Moreover, we have made an effort to keep this book as self-contained as possible as far as algebraic geometry and commutative algebra are concerned: we have generally opted to include proofs of standard results (particularly in cases where the use of “derived” methods can shed some additional light) except in a few cases which would take us too far afield. Nevertheless, a reader who is not familiar with the classical theory of schemes will almost surely find this book impenetrable (if he or she has even made it this far).
0.2.1 Homotopy Theory and Simplicial Sets For every integer ně0, let |∆n|denote the topologicaln-simplex, given by
|∆n| “ tpx0, . . . , xnq PRn`1ě0 :x0`x1` ¨ ¨ ¨ `xn“1u.
If X is a topological space, we let SingnpXq denote the set of continuous maps from |∆n| into X. These sets play a role in defining many important invariants of X: for example, the singular homology groups of X are obtained from the chain complex of free abelian groups
¨ ¨ ¨ ÑZrSing2XsÝÝÝÝÝÝÑd0´d1`d2 ZrSing1XsÝdÝÝÝ0´dÑ1 ZrSing0Xs
where eachdk: SingnXÑSingn´1Xis the map which assigns to a simplex the face opposite itskth vertex. To describe the structure given by the setstSingnpXquně0 and the face maps dk in a more systematic way, it will be useful to introduce a bit of terminology.
Definition 0.2.1.1. For each integerně0, we letrnsdenote the finite linearly ordered set t0ă1ă ¨ ¨ ¨ ănu. We define a category ∆as follows:
• The objects of ∆are sets of the form rnsforně0.
• A morphism from rmsto rnsin∆consists of a nondecreasing functionα:rms Ñ rns. We will refer to ∆as thecategory of combinatorial simplices. A simplicial setis a functor S‚ :∆op Ñ Set, where Set denotes the category of sets. In this case, we will denote the value ofS‚ on the objectrns P∆ bySn, and refer to it as the set of n-simplices ofS. We let Set∆ denote the category Funp∆op,Setq of simplicial sets.
For each n ě0, it is useful to think of the set rns “ t0 ă 1 ă ¨ ¨ ¨ ă nu as the set of vertices of the topologicaln-simplex|∆n|. Every map of setsα:rms Ñ rnsextends uniquely to a linear map ρ:|∆m| Ñ |∆n|, given in coordinates by the construction
px0, . . . , xmq ÞÑ p ÿ
αpiq“0
xi, . . . , ÿ
αpiq“n
xiq.
IfX is a topological space, then composition with ρ determines a map SingnXÑSingmX.
In particular, we can regard the construction prns P∆q ÞÑ pSingnXPSetq as a simplicial set. We refer to this simplicial set as the singular simplicial set of X and denote it by Sing‚X.
The construction X ÞÑ Sing‚X determines a functor Sing‚ from the categoryTop of topological spaces to the categorySet∆ of simplicial sets. This functor admits a left adjoint
Set∆ÑTop S‚ÞÑ |S‚|
0.2. PREREQUISITES 35