POSTNIKOV TOWERS OF COMMUTATIVE S-ALGEBRAS

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1 POSTNIKOV TOWERS OF COMMUTATIVE S-ALGEBRAS DANIEL HESS Contents 1. Introduction 1 2. Classical André-Quillen Homology & Cohomology Motivation: Homology in Model Categories Abelianization for Augmented Algebras The Construction of André-Quillen Homology & Cohomology Properties of André-Quillen Homology & Cohomology 9 3. The Stable Homotopy Category The Boardman Category of Spectra Ring Spectra & The Category of S-modules Topological André-Quillen Homology & Cohomology Postnikov Towers of Commutative S-algebras 20 Appendix A. Model Categories 24 A.1. Derived Functors 26 References Introduction Spectra, which are the main objects of study in stable homotopy theory, are similar to both topological spaces and abelian groups in many respects. Just as for spaces, it is possible to define ordinary homology, cohomology, and homotopy groups for spectra as well as wedge sums, homotopy (co)fibers of maps of spectra, etc. There is also a theory of Postnikov towers for spectra in which k-invariants are given by classes in ordinary cohomology. On the other hand, just as abelian groups may be viewed as modules over the integers, Z, spectra may be viewed as modules over the sphere spectrum, S. This is was first made precise in 1993 by Elmendorff et al. in [8], where it is also shown that many formal algebraic constructions make perfect sense in the category of S-modules. For instance, analogous to the tensor product Z of abelian groups is the smash product, S. Using the smash product, associative/commutative S-algebras can be defined as associative/commutative monoids in the category of S-modules. This is completely analogous to the categorical definition of a ring as a monoid in the category of Z-modules. Under the correspondence between spectra and generalized cohomology theories [10, 4.E], commutative S-algebras correspond to cohomology theories that have a commutative multiplication and power operations. Because such cohomology theories are particularly useful within algebraic topology, it is natural to ask whether or not a given S-module has the structure of a commutative S-algebra. This is often not an easy question to answer; Date: November 23,

2 2 DANIEL HESS for example, the question of whether or not the Brown-Peterson spectrum, BP, admits a commutative S-algebra structure has been open for roughly 40 years. I. Kriz [13] attempted to settle this question in 1993 by developing a theory of Postnikov towers for commutative S-algebras. Here, k-invariants are not classes in ordinary cohomology, but rather in topological André-Quillen cohomology. This and its companion homology theory are topological analogues of André-Quillen (co)homology, which is a (co)homology theory for commutative rings or, more generally, for commutative algebras over any fixed commutative ring that was independently constructed by M. André [2] and D. Quillen [21, 23] in the late 1960s/early 1970s. In his paper, Kriz showed that there is a forgetful map from topological André-Quillen cohomology back to ordinary cohomology and that an S-module admits a commutative S-algebra structure if and only if all of its ordinary k-invariants have lifts along this map to topological André-Quillen cohomology. Calculations in topological André-Quillen (co)homology are therefore essential to the study of commutative S-algebras. Unfortunately, Kriz s construction of this (co)homology theory using a stabilization procedure made it difficult to work with on a theoretical level. This problem was taken care of when M. Basterra [5] gave a new construction that closely follows Quillen s construction of classical André-Quillen (co)homology via his theory of model categories (see Appendix A). The main purpose of the present paper is to describe the work of both Quillen and Basterra, as well as Kriz s application of the theory. Quillen s work will be discussed in 2, while in 3 we lay the necessary foundations in stable homotopy theory to discuss Basterra s work in 4. The topic of 5 will be Kriz s theory of Postnikov towers for commutative S-algebras. We will conclude the paper by posing a question that points at a way in which the k-invariants of commutative S-algebras might be understood in terms of ordinary k-invariants. Conventions: Throughout this paper, all spaces are based CW-complexes and are assumed to be path-connected. All maps between spaces are based and continuous. In addition, all rings are unital, associative, and commutative unless otherwise specified. Notation: For any category C, we will write C(X, Y ) for the set/class of morphisms X Y in C. If M is a model category, then hm will denote its homotopy category, as in [5]. 2. Classical André-Quillen Homology & Cohomology 2.1. Motivation: Homology in Model Categories. Our definition of André-Quillen homology, which was first given by Quillen in [23] and will be discussed in 2.3, is an example of a more general notion of homology for objects in model categories admitting an abelianization functor. This is the notion of Quillen homology and is defined in terms of the total left derived functor of abelianization. To explain this in more detail, we begin with the following definition: Definition 2.1. An object X of a small category C is an abelian object if, for any object Y of C, the set C(Y, X) has a natural abelian group structure. If C has finite products and a terminal object 1, then this is equivalent to having morphisms m: X X X e: 1 X i: X X (multiplication) (unit or inclusion of the identity element ) (inversion) that are subject to category-theoretic analogues of the usual axioms of an abelian group:

3 POSTNIKOV TOWERS OF COMMUTATIVE S -ALGEBRAS 3 Multiplication is associative: If id X : X X denotes the identity map on X, then m (m id X ) = m (id X m). Multiplication is commutative: If τ : X X X X denotes the twist map that switches the two factors of X, then m τ = m. e is a two-sided identity: If proj 1, proj 2 : X X X denote projection onto the first and second factors, respectively, then m (id X e) = proj 1 and m (e id X ) = proj 2. i is a two-sided inverse: If : X X X denotes the diagonal map and ε denotes the unique map X 1, then m (i id X ) = e ε and m (id X i) = e ε. We let C ab denote the subcategory of abelian objects of C. Example 2.1. If C = Set, then C ab = Ab, the category of abelian groups. Example 2.2. If C = Top, then C ab = TopAb, the category of topological abelian groups. The left adjoint to the forgetful functor C ab C, if it exists, is the abelianization functor Ab: C C ab. If C and C ab are, in addition, both model categories and Ab preserves cofibrations and acyclic cofibrations, then Ab is a left Quillen functor (see Appendix A, Definition A.3). The total left derived functor LAb then exists and we can make the following definition: Definition 2.2. The Quillen homology of a given object C of a model category C is the object LAb(C) hc. The abelian groups π LAb(C) are the Quillen homology groups of C. Example 2.3. Let C = Top and let (X, e) be a based topological space. In this case, Ab(X) = AG(X), the free topological abelian group on X with identity e. This space is the quotient of SP (X X), the free topological abelian monoid on X X (also with identity e), by the equivalence relation x x + x + τ(x ) where x, x SP (X X) and τ is the map SP (X X) SP (X X) induced by the map τ : X X X X interchanging the two summands of X X. If Y is a cofibrant replacement of X (e.g., a CW-approximation of X), then it follows that LAb(X) is AG(Y ). The Dold-Thom theorem [6] states that π AG(Y ) = H (Y ; Z). Since H (Y ; Z) = H (X; Z), it then follows that π LAb(X) = H (X; Z). Example 2.4. Let C = TopGp, the category of topological groups. If G TopGp, then Ab(G) TopAb is the usual abelianization of G (with the quotient topology). In this case, π LAb(G) = H +1 (G; Z), the homology of G with coefficients in the trivial Z[G]-module Z. A proof of this fact is given in [9, Ex. 4.6] Abelianization for Augmented Algebras. With the goal of defining André-Quillen homology in mind, we now fix a commutative ring R and begin our study of the Quillen homology of commutative R-algebras. As the following result shows, however, we cannot simply work with the category Alg R of such R-algebras: Proposition 2.1. (Alg R ) ab is equivalent to the trivial category. Proof. The terminal object in Alg R is the trivial R-algebra, {0}. As a result, if X is an abelian object of Alg R, then the fact that e: {0} X must be a two-sided unit forces X = {0}.

4 4 DANIEL HESS Therefore, to construct a nontrivial homology theory for R-algebras, we must instead fix a nontrivial R-algebra A and work in the category Alg R /A of R-algebras B equipped with an augmentation map ε: B A. We say that such an algebra is augmented over A. This forces the terminal object of Alg R /A to be A and, because of this, the subcategory of abelian objects in Alg R /A is much more interesting than that of Alg R. Proposition 2.2. (Alg R /A) ab is equivalent to the category Mod A of A-modules. The remainder of this section will be divided into two halves. In the first, we will prove Proposition 2.2; in the second, we will determine the value of the composite on any augmented R-algebra. Alg R /A Ab (Alg R /A) ab = ModA The Structure of (Alg R /A) ab. For the proof of Proposition 2.2, we will need the following method of constructing an object of Alg R /A out of a given A-module: Definition 2.3. Let M be an A-module. The square-zero extension of A by M is an object A M Alg R /A defined as follows: The underlying R-module of A M is isomorphic to A M; The product of (a, x), (b, y) A M is (a, x)(b, y) := (ab, ay + bx); The R-algebra structure is given by r(a, x) := (ra, rx); The augmentation A M A is induced by the projection proj 1 : A M A. As motivation for the terminology, note that given (0, x), (0, y) A M, their product is (0, 0); in particular, any element of M has square equal to zero in A M. As we will see in the statement of Lemma 2.1 below, square-zero extensions are closely related to the following concept. Definition 2.4. Let B be a commutative R-algebra and let M be a B-module. An R-linear derivation of B with coefficients in M is an R-module homomorphism d: B M (here the R-module structure on M is given by restricting the B-module structure on M) such that, for any b, b B, d(bb ) = bd(b ) + b d(b). The set of all R-linear derivations of B with coefficients in M will be written Der R (B, M) and is an abelian group under pointwise addition. Lemma 2.1. For any B Alg R /A, there is a natural bijection Alg R /A(B, A M) Der R (B, M). Since Der R (B, M) is naturally an abelian group, it follows that A M is an abelian object of Alg R /A. Proof. Any f Alg R /A(B, A M) must, by definition, be a map of augmented algebras; in other words, the following diagram commutes: B f A M ε A proj 1

5 POSTNIKOV TOWERS OF COMMUTATIVE S -ALGEBRAS 5 As a homomorphism on underlying R-modules, the commutativity of this diagram forces f = ε d f for some R-module homomorphism d f : B M. This is because the direct sum A M is the product of A and M in Mod R. We claim that d f is an R-linear derivation of B with coefficients in M (which is viewed as a B-module via ε) and that the map Φ: Alg R /A(B, A M) Der R (B, M) defined by sending f to d f is a bijection. To verify the first of these claims, choose b, b B. On one hand, because f is an R-algebra homomorphism, we have f(bb ) = f(b)f(b ) = (ε(b), d f (b))(ε(b ), d f (b )) = (ε(b)ε(b ), ε(b)d f (b ) + ε(b )d f (b)). On the other hand, f(bb ) = (ε(bb ), d f (bb )). Therefore d f (bb ) = ε(b)d f (b ) + ε(b )d f (b), and so d f is a derivation. To construct an inverse to Φ, let d Der R (B, M) be any R-linear derivation. From this, we can construct an R-module homomorphism f d : B A M by setting f d (b) = (ε(b), d(b)) A M A similar computation to the one above shows that f d is actually a map B A M. The map Der R (B, M) Alg R /A(B, A M) sending d to f d is clearly inverse to Φ, and the result follows. Remark 2.1. As an abelian object in Alg R /A, the multiplication on A M is a map m: (A M) A (A M) A M. As a result, the formula for the product of two elements in A M is m((a, x), (a, y)) = (a, x + y) for a A and x, y M, which differs from the formula given in Definition 2.3. The unit e: A A M is induced by the R-module homomorphism A A M sending a to (a, 0). The inverse i: A M A M sends (a, x) to (a, x). We now prove Proposition 2.2, which, now that Lemma 2.1 has been proved, will amount to showing that every abelian object of the category Alg R /A is necessarily a square-zero extension of A by some A-module M. Proof of Proposition 2.2. Let X be an abelian object of Alg R /A. Recall that this means that we have homomorphisms m: X A X X e: A X i: X X in Alg R /A satisfying analogues of the usual axioms of abelian groups. Let M be the kernel of the augmentation ε: X A. We can consider X, and hence M, as an A-module via e. Moreover, because the composite ε e: A A is the identity (it is the unique map from the terminal object in Alg R /A to

6 6 DANIEL HESS itself), we have an isomorphism X = A M as A-modules. Explicitly, this isomorphism is given by ϕ: A M X (a, x) e(a) + x. We claim that ϕ determines an R-algebra isomorphism X = A M. To see this, note that m(ϕ(a, x), ϕ(a, y)) = m(e(a) + x, e(a) + y) = m(e(a), e(a)) + m(x, 0) + m(0, y) = e(a) + x + y = ϕ(a, x + y) = ϕ m((a, x), (a, y)) where the second equality follows from the fact that m is a homomorphism and the third from the fact that for any two elements x, y M, we have m(x, 0) = m(x, e(0)) = x and m(0, y) = m(e(0), y) = y since e is a two-sided identity and m is a map of augmented R-algebras. The multiplication on X therefore agrees with the multiplication on A M, and so X = A M as R-algebras. In light of the above and Lemma 2.1, we see that there is an equivalence Mod A (AlgR /A) ab M A M. From here on, we will identify the categories (Alg R /A) ab and Mod A with each other The Abelianization Functor. Having determined the structure of (Alg R /A) ab, we now turn to the question of determining the A-module associated to the abelianization of any B Alg R /A. To begin answering this, we need two definitions. Definition 2.5. Let X be an object of Alg A /A. The augmentation ideal of X, written I A (X), is defined to be the kernel of the augmentation X A. This is naturally a nonunital A-algebra. Definition 2.6. Let N be a non-unital A-algebra. The module of indecomposables of N, written Q A (N), is defined to be the cokernel of the multiplication N A N N. Note that for any B Alg R /A, the tensor product A R B is an A-algebra augmented over A. We may therefore consider the A-module Q A (I A (A R B)). It will eventually be shown in Proposition 2.3 that this is the A-module associated to Ab(B) under the equivalence of Proposition 2.2. As a first step towards proving this, we observe that and make the following definition: Q A (I A (A R B)) = Q A (I A (A B B R B)) = A B Q B (I B (B R B)) Definition 2.7. The B-module of Kähler differentials of the R-algebra B is given by Ω R (B) := Q B (I B (B R B)). The module of Kähler differentials is important for us because of its relationship to derivations and square-zero extensions. To spell this out, we note that there is a universal derivation δ : B Ω R (B) whose value on an element b B is the class of b 1 1 b in Ω R (B). In addition, there is an isomorphism Der R (B, M) = Mod B (Ω R (B), M) which

7 POSTNIKOV TOWERS OF COMMUTATIVE S -ALGEBRAS 7 follows from checking that, given any R-linear derivation d: B M, there is a unique B-module homomorphism ψ : Ω R (B) M making the following diagram commute: B d M δ Ω R (B) (See [12] for details.) We will make use of this fact in the proof of Proposition 2.3, whose statement is given below. Proposition 2.3. There is an adjoint pair! ψ A ( ) Ω R ( ): Alg R /A Mod A A ( ). It follows from this that the abelianization of B Alg R /A is the A-module A B Ω R (B). Proof. Recall that, for an A-module M, there is a natural isomorphism Alg R /A(B, A M) = Der R (B, M). As calculated in the proof of Proposition 2.2, the image of M under the forgetful functor U : Mod A = (AlgR /A) ab Alg R /A is the square-zero extension A M. This allows us to rewrite the above isomorphism as Der R (B, M) = Alg R /A(B, U(M)) = Mod A (Ab(B), M). The second isomorphism follows from the fact that the defining property of the functor Ab: Alg R /A (Alg R /A) ab = ModA is that it is left adjoint to U. Comparing this to the natural isomorphism : gives the result. Mod A (A B Ω R (B), M) = Der R (B, M) 2.3. The Construction of André-Quillen Homology & Cohomology. The results of 2.2 show that Quillen homology in the category Alg R /A is defined in terms of the total left derived functor of A ( ) Ω R ( ). Because Alg R /A is not an abelian category, we must make use of Dold and Puppe s theory of nonadditive derived functors [7] to compute this. The main idea behind their theory is to let simplicial objects in Alg R /A play the same role that chain complexes would in deriving a functor on an abelian category, such as Mod R. Roughly, a simplicial object in a category C is constructed from objects of C in a way that is reminiscent of how a simplicial complex is constructed from n-simplices. The precise definition is as follows: Definition 2.8. The simplex category, denoted by, is the category whose objects are nonempty totally ordered sets [n] = {0 < 1 < < n} (n 0) and whose morphisms are order preserving maps [n] [m]. These maps are generated by the face maps δ i : [n] [n + 1], the unique order preserving map missing only the element i [n + 1]; degeneracy maps σ j : [n + 1] [n], the unique order preserving map hitting j [n] exactly twice and every other element exactly once.

8 8 DANIEL HESS Definition 2.9. A simplicial object in a category C is a functor F : op C. This may be viewed a sequence C = {C n } n 0 of objects of C (where C n = F [n]) with face maps F (δ i ) = d i : C n+1 C n and degeneracy maps F (σ j ) = s j : C n C n+1. This is often written as a diagram of the form C 0 C 1 where there is an arrow for each face map and the degeneracy maps are omitted. We will write sc for the category of simplicial objects in C. Example 2.5. For any object C in a category C, there is the constant simplicial object const (C). This has terms const n (C) = C for all n 0 and all face and degeneracy maps given by the identity on C. If C is a small category, then the functor const : C sc is a fully faithful embedding. If C = Alg R or Mod R for any commutative ring R, then sc is a model category. The model structure is defined by transferring the model structure on simplicial sets to sc (see Appendix A, Examples A.4 and A.5). In particular, it makes sense to consider the homotopy groups of a simplicial R-algebra or a simplicial R-module. These homotopy groups can be computed with the aid of the following theorem, which also motivates the idea that simplicial objects in C should be thought of as chain complexes of objects of C. Theorem 2.1 (The Dold-Kan Correspondence). The category smod R is equivalent to the category of nonnegatively graded chain complexes of R-modules. Under this equivalence, the homotopy groups of a simplicial R-module M correspond to the homology groups of the associated chain complex Ch(M ). This is the chain complex whose terms are the same as those of M and whose boundary maps n : M n M n 1 are given by the alternating sum of face maps n i=0 ( 1)i d i. Proof. See [17]. To construct André-Quillen (co)homology, we will need the following analogue of a free resolution. Definition Let A be an R-algebra. A simplicial resolution P of A over R is an augmented simplicial R-algebra such that for all n 0, A P 0 P 1 P n is the free R-algebra on a set X n ; each degeneracy map on P n sends X n into X n+1, and π 0 (P ) = A while π k (P ) = 0 for all k 1. In [9, Prop. 4.21], it is shown that simplicial resolutions exist and are unique up to homotopy equivalence. In the language of model categories, a simplicial resolution of A over R is the same as a cofibrant replacement of const (A) in salg R. Using this and the fact that const : Alg R /A salg R /const (A) is a fully faithful embedding, we can compute the total left derived functor of A ( ) Ω R ( ) on Alg R /A by applying the functor A ( ) Ω R ( ) levelwise to a simplicial resolution of A. As a result, we obtain the following simplicial A-module.

9 POSTNIKOV TOWERS OF COMMUTATIVE S -ALGEBRAS 9 Definition Let A Alg R /A and let P be a simplicial resolution of A over R. The cotangent complex of A is the simplicial A-module L R (A) := A P Ω R (P ). We can alternatively view L R (A) as an object of the derived category D(Mod A ) by taking its associated chain complex, Ch(L R (A)). The cotangent complex L R (A) should be viewed as the Quillen homology object of A in Alg R /A. The associated Quillen (co)homology groups are the following: Definition Let A Alg R /A and let M Mod A. The André-Quillen homology and the André-Quillen cohomology of A relative to R with coefficients in M, denoted AQ (A, R; M) and AQ (A, R; M), respectively, are defined as AQ (A, R; M) := H Ch(M A L R (A)) and AQ (A, R; M) := H Ch(Mod A (L R (A), M)). Here, the simplicial A-modules M A L R (A) and Mod A (L R (A), M) are constructed by applying the functors M A ( ) and Mod A (, M) levelwise to L R (A). André-Quillen (co)homology groups can be described in low degrees as follows: AQ 0 (A, R; M) = Ω R (A) A M AQ 0 (A, R; M) = Der R (A, M) Classes in AQ 1 (A, R; M) correspond to isomorphism classes of square-zero extensions of A by M [21, Prop. 3.12]. Example 2.6. Let A = R[x], a polynomial algebra over R. Since this is a free R-algebra, const (R[x]) R[x] is a simplicial resolution of R[x] over R. It follows that L R (R[x]) = R[x] const (R[x]) Ω R (const (R[x])) = const Ω R (R[x]). Since Ω R (R[x]) = R[x] as R[x]-modules, we see that Ch(L R (R[x])) is the chain complex Therefore, for any R[x]-module M, R[x] 0 R[x] id R[x] 0 R[x] 0. AQ 0 (R[x], R; M) = M and AQ k (R[x], R; M) = 0 for k 1. In general, if A is any smooth R-algebra, then AQ (A, R; M) will be concentrated in degree zero [12, Thm. 9.6] Properties of André-Quillen Homology & Cohomology. We conclude 2 by listing some properties of André-Quillen (co)homology. Each of these shows that AQ and AQ do deserve to be called homology/cohomology theories. The first is a naturality property. Theorem 2.2. If R A A is a commutative diagram of ring homomorphisms and M is any A -module, then there is a map A A L R (A) L R (A ) that induces maps AQ (A, R; M ) AQ (A, R ; M ) AQ (A, R ; M ) AQ (A, R; M ) R

10 10 DANIEL HESS Moreover, these maps preserve composition; in other words, if R R R A f A g A is a commutative diagram of rings, then the map on André-Quillen (co)homology induced by g f is the composite of the maps induced by g and f. Proof. See [23, pg. 73]. Using the result of Theorem 2.2, we can state the following: Theorem 2.3. [23, Thm. 5.1] Let R B A be a sequence of maps of commutative rings. Then there is a distinguished triangle in the derived category D(Mod A ) of the form that gives rise to long exact sequences and L R (B) B A L R (A) L B (A) Σ(L R (B) B A) AQ 1 (A, B; M) AQ 0 (B, R; M) AQ 0 (A, R; M) AQ 0 (A, B; M) 0 0 AQ 0 (A, B; M) AQ 0 (A, R; M) AQ 0 (B, R; M) AQ 1 (A, B; M) for any A-module M. Finally, we have the following flat base change and additivity result: Theorem 2.4. [23, Thm. 5.3] Suppose that A and B are R-algebras with Tor R k (A, B) = 0 for k 1. Then there are isomorphisms L B (A R B) = A R L R (B) L R (A R B) = (L R (A) R B) (A R L R (B)) in the derived category D(Mod A R B). In this case, if M is any A R B-module, then there are isomorphisms AQ (A R B, B; M) = AQ (A, R; M) AQ (A R B, R; M) = AQ (A, R; M) AQ (B, R; M). The same isomorphisms also hold in André-Quillen cohomology. 3. The Stable Homotopy Category In this section, we begin to set the stage for our discussion of Basterra s topological analogue of André-Quillen (co)homology in 4. As mentioned in the introduction, this is a (co)homology theory for commutative S-algebras the commutative rings of stable homotopy theory. The story of this subject begins with the following definition.

11 POSTNIKOV TOWERS OF COMMUTATIVE S -ALGEBRAS 11 Definition 3.1. Let (X, x 0 ) be a based space and let s 0 S 1 be a chosen basepoint of S 1. The suspension of X is the smash product ΣX := X S 1 = (X S 1 )/(X {s 0 } {x 0 } S 1 ) loop space of X is the space ΩX := Map(S 1, X) of maps (S 1, s 0 ) (X, x 0 ) with the compact-open topology. We will use the notation Σ = ( ) S 1 and Ω = Map(S 1, ) to denote the suspension and loop functors Top Top. An important property of these functors is that they are an adjoint pair up to homotopy; in other words, for any two spaces X and Y, there is an isomorphism of groups htop (ΣX, Y ) = htop (X, ΩY ). This is simply a consequence of the more general fact that if Z is some fixed space, then the functors ( ) Z and Map(Z, ) are an adjoint pair up to homotopy. 1 In [1], Adams refers to an invariant in homotopy theory as stable if it is essentially independent of suspension. To make this idea more precise, let F : htop GrAb be a functor from the homotopy category of spaces to the category of graded abelian groups. For a graded abelian group A, we will temporarily abuse notation and let (ΣA) be the shift suspension of A, the graded abelian group whose (k + 1) st graded piece is given by (ΣA) k+1 = A k. We say that F is a stable invariant if F (ΣX) = ΣF (X). Roughly, F is a stable invariant if it sees ΣX as a shifted copy of X. For example, reduced homology is a stable invariant: for any space X and integer k 0, there is an isomorphism H k (X) = H k+1 (ΣX). In contrast to this, homotopy groups are not stable invariants; for example, π 0 (S 0 ) = Z/2 while π 1 (S 1 ) = Z. However, the following theorem implies that homotopy groups are stable invariants when restricted to a certain range. We will refer to this range as the stable range because of this. Before the statement of the theorem, recall that a space X is said to be s-connected if π k (X) is trivial for 0 k s and that the dimension of a CW-complex Y, written dim(y ), is the largest value of k (which could be + ) such that H k (Y ) 0. Theorem 3.1 (Freudenthal Suspension Theorem). If X is s-connected and dim(y ) 2s, then the map Σ: htop (Y, X) htop (ΣY, ΣX) induced by suspension is an isomorphism. To see the connection to homotopy groups, fix an integer k 0, let Y = S k, and let X be an s-connected space. Consider the direct system ( ) π k (X) Σ π k+1 (ΣX) Σ π k+2 (Σ 2 X) Σ Since X is s-connected, Σ n X is at least (s + n)-connected; as a consequence of this and Theorem 3.1, the maps π k+n (Σ n X) π k+n+1 (Σ n+1 X) in ( ) are all isomorphisms once n is large enough to make the inequality k + n 2(s + n) hold. In other words, they are all isomorphisms once n k 2s. Definition 3.2. The colimit of ( ), which, by the above, is isomorphic to π k+n (Σ n X) for any n k 2s, is the k th stable homotopy group of the space X, denoted π st k (X). 1 This is essentially a homotopy-theoretic version of the tensor-hom adjunction in algebra.

12 12 DANIEL HESS Note that πk st (X) is always an abelian group and that stable homotopy groups are, in fact, stable invariants: πk st(x) = πk+1 st (ΣX). In addition to being able to interpret both homotopy and reduced homology groups as stable invariants, another important advantage of working in the stable range is that cofiber and fiber sequences coincide. This is implied by the following corollary of Theorem 3.1. Corollary 3.1. Suppose that X i Y j Z is a cofiber sequence and that U satisfies dim(u) 2c, where c is the minimum of the connectivities of X, Y, and Z. Then the sequence htop (U, X) i htop (U, Y ) j htop (U, Z) (where the maps given by postcomposition with i and j) is exact. In other words, the image of i is equal to the preimage under j of the map U Z sending U to the basepoint of Z. Proof. See [16, pg. 8-10]. The idea of working in the stable range plays a central role in modern algebraic topology and is made precise by working in the stable homotopy category instead of the ordinary homotopy category of spaces. This is a category whose objects are called spectra that is constructed as the homotopy category of some model category C obtained by stabilizing the category Top. Before discussing the problem of constructing C, we list some properties that hc ought to have: 1. For each space X, there should be a spectrum Σ X in hc (to be thought of as a free stabilization of X) and our definition of homotopy in C should imply that π k (Σ X) = πk st(x). Moreover, the association X Σ X should be functorial and admit a right adjoint Ω : hc htop. 2. The suspension functor Σ should be an equivalence in hc and be defined in a way that ensures that Σ Σ and Σ Σ are naturally isomorphic. 3. The set hc(e, F ) of homotopy classes of maps between any two spectra E, F hc should have a natural abelian group structure. Composition of functions should also be bilinear. Intuitively, this property implies that spectra should be thought of as homotopy-theoretic versions of abelian groups. 4. Continuing with the theme of the previous property and the result of Corollary 3.1, the category hc should have properties similar to that of the derived category of abelian groups. In particular, like the category of abelian groups, C should have a biproduct, the wedge sum (analogous to the direct sum ); a zero object = Σ (pt) (analogous to the abelian group 0); and a symmetric monoidal product 2, the smash product, with unit S := Σ S 0 (analogous to the tensor product with unit Z). that all pass to the homotopy category of C. 3 In addition to this, hc should be a triangulated category with the shift operation given by Σ and distinguished triangles given by fiber/cofiber sequences E F G ΣE 2 In other words, a unital, associative, and commutative product. 3 As in [5], we will write L for the passage of to hc; this is inspired by the notion L used for the derived tensor product.

13 POSTNIKOV TOWERS OF COMMUTATIVE S -ALGEBRAS Within hc, it should be possible to define mapping spectra Map(E, F ) as well as homotopy fibers, F (f), and homotopy cofibers, C(f), of maps of spectra f : E F that fit into fiber/cofiber sequences of the form F (f) E f F ΣF (f) and E f F C(f) ΣE, respectively. Moreover, the functor Map(E, ) should be right adjoint to ( ) E. It turns out that it is indeed possible to construct a stable homotopy category with the above properties. A 1991 theorem of Lewis [14], however, states that it is impossible to construct the point-set level category C so that it has all of the properties that one might hope for. Fortunately, there are several possible choices of C, each having different properties, but determining the same stable homotopy category. J.M. Boardman constructed the first relatively modern version of this category, called the Boardman category of spectra, which we will denote by Sp, in his 1964 PhD thesis. The objects of this category are easily defined and it enjoys several nice properties, but, as we will see, it has a flaw that makes it unsuitable for the applications we have in mind. We discuss this category in 3.1 mainly for the purpose of developing intuition. In 3.2 we will discuss a suitable choice for C, the category of S-modules of [8] The Boardman Category of Spectra. The objects and morphisms in the category Sp are defined in [1, Pt. III] by Adams as follows: Definition 3.3. A spectrum is a sequence E = {E n } n Z of based spaces together with maps σ n : ΣE n E n+1 (alternatively, maps ω n : E n ΩE n+1 ) for all n. Example 3.1. For any space X, there is the suspension spectrum Σ X with n th term { (Σ Σ n X if n 0 X) n = if n < 0 and the obvious structure maps. A particularly important suspension spectrum is the sphere spectrum S := Σ S 0. Example 3.2. For any abelian group A, there is the Eilenberg-Mac Lane spectrum HA with n th term { K(A, n) if n 0 (HA) n = if n < 0 and with the structure maps for n 0 given by weak equivalences K(A, n) ΩK(A, n+1). Example 3.3. Given any spectrum E and space X, there is a spectrum E X whose n th term is E n X. The structure maps of E X are obtained by applying the functor ( ) X to the structure maps of E. With Example 3.3 in mind, note that if X is a space, then the spectrum (Σ X) S 1, which we would like to call the suspension of Σ X, has n th term (Σ n X) S 1 = Σ n+1 X = (Σ X) n+1. Taking this as inspiration, we define the suspension of a general spectrum as follows: Definition 3.4. Let k be an integer. The k th suspension of a spectrum E, denoted Σ k E, is the spectrum whose n th term is given by (Σ k E) n = E n+k. The morphisms in Sp may be thought of in terms of the following definition.

14 14 DANIEL HESS Definition 3.5. A function f : E F between spectra E and F is a collection of maps f n : E n F n for each n Z such that the diagram Σf n ΣE n ΣF n σ n τ n E n+1 F n+1 fn+1 commutes. (Here, τ n is the n th structure map of F.) A map of spectra is an eventually defined function in the sense of [1, pg ]. As one might hope, homotopies in Sp can be defined by imitating the usual definition in Top. To do this, let I + = [0, 1] { } be the union of the unit interval with a disjoint basepoint. Note that there are maps i 0, i 1 : E E I + which send E to E ({0} { }) and E ({1} { }), respectively. Definition 3.6. We say that two maps f, g : E F of spectra are homotopic if there is a map h: E I + F such that h i 0 = f and h i 1 = g. As in Top, homotopy is an equivalence relation. We can also define homotopy groups of spectra by imitating the usual definition in Top. Definition 3.7. The k th homotopy group of the spectrum E is π k (E) := Sp(Σ k S, E)/, where f g if and only if f and g are homotopic. Remark 3.1. By [1, Pt. III, Thm. 3.7], the set of homotopy classes of maps between any two spectra has an abelian group structure. In particular, π k (E) is always an abelian group. Also note that Definition 3.7 makes sense for negative integers. Spectra with no nontrivial negative dimensional homotopy groups are called connective. Remark 3.2. For any two spectra E and F, it is possible to define a spectrum of maps E F, written Map(E, F ). We can use this together with Definition 3.7 to define the ordinary homology and cohomology groups of a spectrum E with coefficients in the abelian group A as H (E; A) := π (E HA) H (E; A) := π Map(E, HA). The spectrum Map(E, F ) satisfies Σ Map(E, F ) = Map(E, ΣF ). A consequence of this is that, for any integer n, we have H n (E; A) = hsp(e, Σ n HA). Note that this is similar to the fact that, if X is a space, then H n (X; A) = htop (X, K(A, n)). To see that Definition 3.7 is correct, we note that, by [1, Pt. III, Prop. 2.8], the abelian group π k (E) can be alternatively viewed as the colimit of the sequence π k (E 0 ) Σ π k+1 (ΣE 0 ) π k+1 (E 1 ) σ 0 Σ π k+2 (ΣE 1 ) In particular, if we take E = Σ X, then we do indeed have π k+2 (E 2 ) π k (Σ X) = lim π k+n (Σ n X) = πk st (X). In addition to the above, it can be shown that hsp has all five desired properties of the stable homotopy category listed just before 3.1. In a bit more detail: σ 1

15 POSTNIKOV TOWERS OF COMMUTATIVE S -ALGEBRAS 15 From the above definitions, it is straightforward to check that the process of taking suspension spectra determines a functor Σ : htop hsp. The functor Ω : hsp htop that is right adjoint to Σ sends a connective spectrum E to the colimit of the sequence ω E 0 Ωω 0 1 ΩE1 Ω 2 Ω E 2 ω 2 2 Note that if each ω n is a homeomorphism (or weak equivalence), then Ω E = E 0, the infinite loop space associated to the spectrum E. The wedge sum of two spectra E and F is the spectrum E F = {E n F n }. This coproduct is also a product by [1, Pt. III, Prop. 3.11]. The smash product E F of two spectra is defined in [1, Pt. III, 4] and its unit is S. It is important to point out here that this turns out to only be a symmetric monoidal product after passage to hsp. The homotopy cofiber of a map f : E F is the spectrum C(f) := {Y n fn (I + X n )} (see [1, pg. 154] for details on this construction). In hsp, C(f) can be defined via the pushout square E f F C(f). Cofiber sequences in hsp coincide with fiber sequences by [1, Pt. III, Prop. 3.10]. As a result, the homotopy fiber of f could be defined as F (f) := Σ 1 C(f); alternatively, F (f) can be defined via the following pullback square in hsp: F (f) E f F Ring Spectra & The Category of S-modules. The interpretation of spectra as homotopy-theoretic versions of abelian groups is important in stable homotopy theory and naturally leads to the following question: Can rings, modules, and algebras be constructed in the stable homotopy category as well? As one might hope, this can be done using the smash product in hsp in a way that is completely analogous to the usual definitions in ordinary algebra. Definition 3.8. An (associative) ring spectrum is a monoid in hsp. In other words, a ring spectrum is a spectrum R together with a multiplication map µ: R R R and a two-sided unit map η : S R such that the following diagrams commute up to homotopy: R R R id µ R R S R η id R R id η R S µ id R R µ R µ µ R

16 16 DANIEL HESS If, in addition, the diagram R R swap factors R R µ R commutes, so that R is a commutative monoid in hsp, then we say that R is a commutative ring spectrum. Given a ring spectrum R, it is possible to define R-module spectra and R-algebra spectra in the stable homotopy category in a similar way. In order to study these objects seriously, it is necessary to make sense of them on the point-set level this is analogous to the idea that the study of ordinary rings, modules, and algebras should not be done solely on the level of derived categories. Unfortunately, the fact that the smash product is not a symmetric monoidal product in Sp itself makes this difficult. For example, as pointed out in [3, pg. 8], a particular problem that arises is the following: If R is a ring spectrum and M, N Sp pass to R-module spectra in the stable homotopy category, then the cofiber of a map M N in Sp could fail to be an R-module spectrum in hsp. To fix this, the category Sp must be replaced with a point-set level category of spectra that has a symmetric monoidal smash product before passage to the stable homotopy category. A suitable replacement is the category M S of S-modules, introduced in 1993 and described in [8]. Its smash product is denoted by S and its unit is S. The term S-module is justified by the fact that each S-module M comes equipped with a specified isomorphism λ: S S M M that makes the diagrams µ S S S S M λ id S S M M λ 1 S S M id λ S S M λ M λ commute. In addition to the product S, the category M S also has an internal hom Map S (, ). As would be expected from the analogy with ordinary algebra, there is an adjunction ( ) S N Map S (, N) (see [8, Thm. 1.6, pg. 37]). The upshot of all of this is that the product S allows us to define associative/commutative ring objects and modules over them in M S with the following simple definitions: Definition 3.9. An S-algebra is a monoid in M S. Similarly, a commutative S-algebra is a commutative monoid in M S. We will write C S for the category of commutative S-algebras. Remark 3.3. It is shown in [8, pg ] that S-algebras and commutative S-algebras are, up to weak equivalence, the same as A -ring spectra and E -ring spectra, respectively. These older concepts, defined in 1972 by J.P. May (see [20] for details), are ring spectra that are associative and commutative up to coherent higher homotopies. In other words, they are as close as one can get to an associative or commutative ring in homotopy theory without requiring strict associativity or commutativity. Just as there is a notion of module over any commutative ring in ordinary algebra, there is a notion of module over any commutative S-algebra in stable homotopy theory: id λ M

17 POSTNIKOV TOWERS OF COMMUTATIVE S -ALGEBRAS 17 Definition Let R be a commutative S-algebra. An R-module is an S-module M together with S-module maps η : R M and α: R S M M such that the following associativity and unitality diagrams commute: R S R S M µ id R S M S S M η id R S M id α R S M α M α Here, as in Definition 3.8, η and µ are the unit and multiplication maps, respectively, of R. We will denote the category of R-modules by M R. As in [8], the smash product and mapping R-module can be defined for two R-modules M and N using the fact that M S is bicomplete (i.e., all small limits and colimits exist; see [8, Prop. 1.4, pg. 36] for a proof of this fact). If α and β denote the R-module structure maps of M and N, respectively, then the smash product M R N is defined as the coequalizer of M S R S N α β M S N in M S and the function R-module Map R (M, N) is defined as the equalizer of Map S (M, N) γ Map S (R S M, N) δ in M S, where γ = Map(α, id) and δ is the adjoint to the composite R S (M S Map S (M, N)) id ɛ R S N β N in which ɛ: M S Map S (M, N) N is the topological analogue of the evaluation map M Z Mod Z (M, N) N = m f f(m). Both R and Map R have properties that are analogous to those of R and Hom R in ordinary algebra. For more on this, see [8, Ch. III]. Finally, we can define R-algebras for any commutative S-algebra R just as in Definition 3.9 by replacing the category M S with the category M R. Definition Let R be a commutative S-algebra. An R-algebra is a monoid in M R. Similarly, a commutative R-algebra is a commutative monoid in M R. We will write C R for the category of commutative R-algebras. α M. 4. Topological André-Quillen Homology & Cohomology With all of the algebraic structure of the category of commutative S-algebras in-hand, we are now able to construct a topological analogue of André-Quillen (co)homology. Although Kriz originally referred to this as E -(co)homology, it is more commonly called topological André-Quillen (co)homology. For the sake of brevity, we will often use the term TAQ (co)homology instead. As in [5], this construction will essentially be done by mimicing the constructions of 2 with the commutative ring R replaced by a fixed commutative S-algebra R. To begin, we adapt Definitions 2.5 and 2.6 to the topological setting. Throughout this section, we let A be a commutative R-algebra and let Σ (pt) = denote the zero object in M A.

18 18 DANIEL HESS Definition 4.1. Let X be an object of C A /A. The augmentation ideal of X, written I A (X), is defined via the pullback square I A (X) in M A, where ε denotes the augmentation map on X. commutative A-algebra. X ε A This is naturally a non-unital Definition 4.2. Let N be a non-unital commutative A-algebra. The module of indecomposables of N, written Q A (N), is defined via the pushout square N A N µ N Q A (N) in M A, where µ denotes the multiplication map on N. If B C R /A, then A R B may be viewed as an A-algebra augmented over A. Since the functors I A ( ) and Q A ( ) pass to homotopy categories to give functors RI A ( ) and LQ A ( ) (see [5, 2, 3]), we can now consider LQ A (RI A (A L R B)) hm A. Assuming that A is cofibrant and B is both cofibrant and fibrant (in Basterra s paper [5], the terms q-cofibrant and q-fibrant are used instead), we have LQ A (RI A (A L R B)) LQ A (RI A (A L B B L R B)) A L B LQ B (RI B (B L R B)) where the last equivalence here uses [5, Lemmas 4.4, 4.5]. Definition 4.3. The TAQ homology spectrum of the R-algebra B is given by TAQ R (B) := LQ B (RI B (B L R B)) hm B In analogy with the classical case, A L B TAQ R(B) should be viewed as the abelianization of B in hm A. To justify this, we need to state and prove a topological analogue of Proposition 2.3. For its statement, we first need to define the square-zero extension of the R-algebra A by an A-module M in this new setting. This is the object A M of C R /A whose underlying R-module is A M with multiplication given by the map (A M) R (A M) = (A R A) (A R M) (M R A) (M R M) A M which sends A R A to A via the multiplication on A, both A R M and M R A to M via the A-module structure on M, and M R M to. Just as in Definition 2.3, the projection A M A gives an augmentation A M A. Proposition 4.1. There is an adjoint pair 4 A L ( ) TAQ R( ): hc R /A hm A A ( ) It follows from this that the abelianization of B hc R /A is the A-module A L B TAQ R(B). : 4 As mentioned in [5, pg. 119], these functors are not adjoint before passing to homotopy categories. This is the reason why we immediately did this after Definition 4.2.

19 POSTNIKOV TOWERS OF COMMUTATIVE S -ALGEBRAS 19 Proof. See [5, Prop. 3.2]. This is in direct analogy to the classical setting and shows that, if we set B = A and consider it as an object of C R /A with augmentation the identity map A A, then TAQ R (A) is the correct topological analogue of the cotangent complex 5 L R (A). We can therefore use the spectrum TAQ R (A) to define TAQ homology and cohomology by imitating Definition Definition 4.4. Let A C R /A and let M M A. The topological André-Quillen homology and the topological André-Quillen cohomology of A relative to R with coefficients in M, denoted TAQ (A, R; M) and TAQ (A, R; M), respectively, are defined as TAQ (A, R; M) := π (TAQ R (A) A M) and TAQ (A, R; M) := π (Map A (TAQ R (A), M)). It can be shown, as in [5], that TAQ (co)homology satisfies properties that are completely analogous to the properties of classical AQ (co)homology that are listed in 2.4. In addition, thanks to Proposition 4.1, there is a useful interpretation of TAQ cohomology classes in terms of maps into a square-zero extension: for any k 0, TAQ k (A, R; M) = π k (Map A (TAQ R (A), M)) = hm S (Σ k S, Map A (TAQ R (A), M)) = hm S (S, Σ k Map A (TAQ R (A), M)) = hm S (S, Map A (TAQ R (A), Σ k M)) = hm A (TAQ R (A), Σ k M) = hc R /A(A, A Σ k M) From this viewpoint, we can easily define a forgetful map TAQ k (A, R; M) H k (A; M) by postcomposition with the projection A Σ k M Σ k M. Example 4.1. Let X be a cofibrant R-module. The free commutative R-algebra on X, written P R (X), is the wedge sum P R (X) = i 0 X Ri /Σ i, where the symmetric group Σ i acts on the i-fold smash power X Ri by permuting the factors. In this case, we have TAQ R (P R (X)) P R (X) R X [4, Prop. 1.6]. Therefore, if M is any P R (X)-module, TAQ (P R (X), R; M) = π (TAQ R (P R (X)) PR (X) M) = π (X R M) The latter group is also known as X R (M), the X-homology groups of M as an R-module. Example 4.2. In [24], B. Shipley shows that if R is a Q-algebra, then any commutative HR-algebra A can be modeled by a commutative differential graded algebra (cdga) over R. If we write cdga(a) for this cdga, then a result of M. Mandell [15, Cor. 7.9] implies that if π 0 (A) = R, then TAQ (A, HR; HR) = AQ (cdga(a), R; R). 5 We say cotangent complex here instead of Kähler differentials because we are already working on the level of homotopy categories.

20 20 DANIEL HESS 5. Postnikov Towers of Commutative S-algebras A remarkable property of TAQ cohomology is that it is the natural home for k-invariants of connective commutative S-algebras. To begin explaining why this is, we recall how the Postnikov tower of a connective spectrum E is constructed. Theorem 5.1. A connective spectrum E admits a Postnikov tower. In other words, there is a tower of spectra. τ 2 E τ 1 E k 2 k 1 Σ 4 Hπ 3 (E) Σ 3 Hπ 2 (E) in which: E τ 0 E k 0 Σ 2 Hπ 1 (E) the map E τ n E induces an isomorphism on π in the range [0, n]; π (τ n E) = 0 for n + 1; and the spectrum τ n+1 E is constructed as the pullback in M S of the map Σ n+2 Hπ n+1 (E) along the map k n for all n. Proof. Choose a map f 0 : E Hπ 0 (E) =: τ 0 E inducing an isomorphism on π 0. cofiber of this map, Cf 0, has homotopy groups { π k (Cf 0 ) 0 if k = 0, 1 = π k 1 (E) if k 2 The By attaching cells to kill all homotopy groups of Cf 0 above dimension two, we obtain a map c 0 : Cf 0 Σ 2 Hπ 1 (E). The fiber of the composite c k 0 : τ 0 E Cf 0 0 Σ 2 Hπ 1 (E) which we will denote by τ 1 E, has homotopy groups { π k (τ 1 E) π k (E) if k = 0, 1 = 0 if k 2 and fits into a pullback square of the form τ 1 E τ 0 E k0 Σ 2 Hπ 1 (E)

21 POSTNIKOV TOWERS OF COMMUTATIVE S -ALGEBRAS 21 The maps f 0 : E τ 0 E and E induce a map f 1 : E τ 1 E that is a lift of f 0 along τ 1 E τ 0 E. Moreover, f 1 induces an isomorphism on π for = 0, 1. To construct the next stage in the tower, τ 2 E, proceed as above with f 1 playing the role of f 0. Notice that the construction of the Postnikov tower of a spectrum is essentially the same as the construction for spaces, with the spectra Σ n+2 Hπ n+1 (E) taking place of the Eilenberg-Mac Lane spaces K(π n+1 (E), n + 2). Each map k n : τ n E Σ n+2 Hπ n+1 (E) can be viewed as a cohomology class [k n ] H n+2 (τ n E; π n+1 (E)). As in the classical case, we will refer to this as the n th k-invariant of E. If E is replaced with a commutative S-algebra R, then we have the following construction: Theorem 5.2. A connective commutative S-algebra R admits a Postnikov tower. In other words, there is a tower of commutative S-algebras. τ 2 R τ 1 R k 2 k 1 τ 2 R Σ 4 Hπ 3 (R) τ 1 R Σ 3 Hπ 2 (R) R τ 0 R k 0 τ 0 R Σ 2 Hπ 1 (R) in which: the map R τ n R induces an isomorphism on π in the range [0, n]; π (τ n R) = 0 for n + 1; and the commutative S-algebra τ n+1 R is constructed as the pullback in C R /τ n R of the map i 1 : τ n R τ n R Σ n+2 Hπ n+1 (R) (inclusion into the first factor) along the map k n for all n. The proof of this theorem will be extremely similar to that of Theorem 5.1. It will also rely on the results of the following two lemmas: Lemma 5.1. Let R be a commutative S-algebra, let A be a commutative R-algebra, and let M be an A-module. If f : A A M is any map of commutative R-algebras such that the composite A f A M proj 1 A is the identity, then the pullback of the diagram A i 1 A f A M in C R has an underlying R-module that is homotopy equivalent to the fiber of the composite A f A M proj 2 M in M R. Proof. This follows from the fact that the forgetful functor C R M R is a right adjoint (to the functor P R ( )) and hence preserves pullbacks. See [13, pg. 3].