SIMPLICIAL COMMUTATIVE Fp-ALGEBRAS ... - Semantic Scholar

SIMPLICIAL COMMUTATIVE Fp -ALGEBRAS THROUGH THE LOOKING-GLASS OF Fp -LOCAL SPACES JAMES M. TURNER Dedicated to Michael Boardman in honor of his 60th birthday Abstract. We propose a dictionary approach to studying the homotopy theory of simplicial augmented commutative Fp -algebras using the homotopy theory of connected Fp -local spaces as our guide. We indicate how standard topological tools translate to the setting of simplicial algebras. We further indicate how theorems translate as well. For example, we recall a theorem of P. Goerss giving an algebraic version of the HiltonMilnor theorem which fits in our framework. We next propose how a theorem of J.-P. Serre on Fp -local spaces with bounded homotopy groups translates into our algebraic setting and relate it to a conjecture of D. Quillen on the vanishing of André-Quillen homology. We also describe what a simplicial algebra version of a theorem of D. Kan and W. Thurston should look like.

“Oh, Kitty, how nice it would be if we could only get through into the LookingGlass House! I’m sure it’s got, oh! such beautiful things in it! . . . Why it’s turning into a sort of mist now, I declare! It’ll be easy to get through.” – Alice [6] Introduction In [1] and [14], M. André and D. Quillen constructed the first complete (co-)homology theory for commutative algebras. The approach they took involved applying simplicial homotopy theory, a la [13], to the category of simplicial commutative algebras. As a result, a (co-)homology can be constructed from first principles as the derived functors of abelianization, which is defined for any simplicial commutative algebra and, in particular, for any commutative algebra viewed as a constant simplicial algebra. Thus methods from simplicial homotopy theory should prove beneficial in discerning results for AndréQuillen homology. In this paper, we attempt to spell out how simplicial homotopy theory can be used to analyse simplicial objects over the category Ap of supplemented Fp -algebras, that is, commutative unitary Fp -algebras augmented over the field Fp . We denote the resulting simplicial category by sAp . This is a pointed category with a closed simplicial model Date: June 15, 1998. 1991 Mathematics Subject Classification. Primary: 13D03, 18G30, 18G55; Secondary: 55P60, 55P99, 55S05. Key words and phrases. simplicial commutative Fp -algebras, Fp -local spaces, André-Quillen homology. 1

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structure. To describe the weak equivalences, a further invariant is used, namely homotopy π∗ , which is defined by first thinking of a simplicial algebra as a simplicial group. A map f : A → B in sAp is thus a weak equivalence provided π∗ f is an isomorphism. We will return to describing the rest of the model category structure shortly. The perspective we will be developing in this paper is that results in the homotopy theory of simplicial supplemented Fp -algebras can be viewed as analogues of results in the homotopy theory of connected Fp -local spaces. To make such a perspective fly, we need to perform a kind of role reversal. We will view the André-Quillen homology of a simplicial supplemented Fp -algebra as the appropriate analogue of homotopy groups for pointed Fp -local spaces. By duality, we take as the appropriate analogue of Fp -homology for spaces the homotopy groups of simplicial supplemented Fp -algebras. With this perspective in place, we describe the appropriate analogues of the technical tools often used to analyse the homotopy of Fp -local spaces to the homotopy theory of simplicial supplemented Fp -algebras. We then proceed with a study of how to translate theorems, describing the homotopy theory of Fp -local spaces, to theorems describing the homotopy theory of simplicial supplemented Fp -algebras. We begin by recalling a theorem of P. Goerss giving an algebraic version of the Hilton-Milnor theorem, which describes the homotopy type for the loop space of a wedge of spheres. We next translate a theorem of J.-P. Serre which classifies a large class of Fp -local spaces with bounded homotopy and Fp -homology. We connect this to a conjecture of D. Quillen which classifies certain commutative algebras with bounded André-Quillen homology. Bolstered by our successes thus far, we close by describing what the appropriate analogue would be of a theorem of D. Kan and W. Thurston which shows that every connected space possesses the same homology of a K(π, 1). The validity of such an analogue would place strong limits on our ability to further weaken the conditions in Quillen’s conjecture. The reader will find the contents of this paper totally bereft of proofs. This is intended as it is hoped it will make for a fun filled journey, with lots of interesting sights, as Alice soon discovered. Acknowledgements. My thanks to Haynes Miller for suggesting this approach to studying simplicial commutative algebras and for several related conversations. My thanks also to Michele Intermont for her thorough reading and comments on an earlier draft. 1. Homotopy and Homology of Simplicial Supplemented Fp -Algebras The homotopy theory for the category sAp that we will deal with comes from the closed model structure as described in [13], [12], and [8]. Specifically, a map f : A → B in sAp is called 1. a weak equivalence iff π∗ f is an isomorphism, 2. a fibration iff the induced map A → π0 A ×π0 B B is a surjection, and 3. a cofibration iff f has the left lifting property with respect to any map which is at once a weak equivalence and a fibration.

SIMPLICIAL COMMUTATIVE Fp -ALGEBRAS AND Fp -LOCAL SPACES

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Note: Cofibrations in sAp can be explicitly determined. See [13] and [12]. 2 From this model structure, the homotopy category Ho(sAp ) can be constructed from sAp by inverting weak equivalences, or, equivalently, by taking the subcategory of cofibrant objects (i.e. simplicial supplemented Fp -algebras A for which the unit Fp → A is a cofibration). Note that every object of sAp is fibrant, that is, given A ∈ sAp then A → π0 A is a surjection. Philosophically, homology is appropriately constructed as the derived functors of abelianization. The subcategory Ab(Ap ) of abelian group objects in Ap consists of those algebras A such that (IA)2 = 0. There is a functor (−)+ : V → Ab(Ap ), where V is the category of vector spaces, which is an equivalence of categories. The induced functor (−)+ : V → Ap has a left adjoint Q : Ap → V given by QA = IA/(IA)2 . This latter functor is the abelianization functor for simplicial supplemented Fp -algebras, commonly called the indecomposables functor. Prolonging this functor to the simplicial categories Q : sAp → sV, it satisfies Quillen’s criterion to induce a functor on the homotopy categories LQ : Ho(sAp ) → Ho(sV). (weak equivalences in sV are also π∗ -isomorphisms). This is called the total left derived functor of Q and we define the André-Quillen homology of a simplicial supplemented Fp -algebra A to be H∗Q (A) = π∗ LQ(A).

2. The Looking-Glass: Parallels with the homotopy of Fp -local spaces As we noted in the introduction, the viewpoint we are taking is that results about the homotopy theory of simplicial supplemented Fp -algebras can be produced by analogy from results about the homotopy theory of pointed Fp -local spaces. To get this parallel off the ground, we take André-Quillen homology as the proper analogue of homotopy groups. Below is our proposed looking-glass for translating other results from Fp -local spaces to results about simplicial supplemented Fp -algebras.

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connected Fp -Local Spaces homotopy groups π∗ Fp -homology H∗ (−; Fp ) cartesian W product × wedge fibration sequences long exact sequences in homotopy Eilenberg-MacLane spaces K(π, n) loop spaces ΩX n-connected covers Postnikov towers algebra structure on H ∗ (−; Fp ) Whitehead product on π∗ classifying spaces Bπ Fp -local spheres Spn Serre spectral sequence for homology Poincaré series for homology

Simplicial supplemented Fp -algebras André-Quillen homology H∗Q homotopy groups π∗ tensor product ⊗ augmented product ×Fp cofibration sequences long exact sequences in homology sphere algebras S(V, n) suspension algebras ΣA homology approximations skeletal filtrations Γ-algebra structure on π∗ Whitehead product on HQ∗ (constant simplicial) Fp -algebras A abelian group objects K(n)+ Serre spectral sequence for homotopy Poincaré series for homotopy

Several other analogies can be drawn from Fp -local spaces for simplicial Fp -algebras. Many of the ones listed above are spelled out in [12], [8], and [16]. For the purposes of this paper, we will limit ourselves to just these. Our goal is to indicate how this looking glass can be used to produce theorems about simplicial supplemented Fp -algebras from theorems about Fp -local spaces. A similar type of looking glass approach was created by L. Avramov and S. Halperin between rational homotopy and local algebra. See particularly [3] and [5]. In fact, their work is one of the main inspirations for our present approach. 3. Algebraic Hilton-Milnor Theorem One of the first results established utilizing this looking-glass approach was an algebraic version of the classic Hilton-Milnor theorem. Established by P. Goerss [9], it was needed in order to determine the primary operations for André-Quillen cohomology [8]. Recall that the topological Hilton-Milnor theorem [17] says that there is a homotopy equivalence _ Y ' ΩSpn(w) Ω(Spn1 Spn2 ) → w

Here each w is an element of the Hall basis for the free Lie algebra generated by {x1 , x2 }. For each such w, n(w) = n1 j1 (w) + n2 j2 (w) + `(w) − 1 where ji (w) = # of times that xi appears in w for i = 1, 2, and `(w) = j1 (w) + j2 (w).


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If A is a simplicial supplemented Fp -algebra, let ΣA be the suspension of A defined by (ΣA)s := Fp ⊗As Bs As where B• (−) is the simplicial bar construction on algebras. By our looking-glass, the Hilton-Milnor theorem translates into Theorem 3.1. [9] For n1 , n2 ≥ 0, there is a weak equivalence in sAp '

Σ(K(n1 )+ ×Fp K(n2 )+ ) → ⊗w ΣK(n(w))+ . Of course, just as the original Hilton-Milnor theorem is concerned with more general wedges of spaces, the algebraic Hilton-Milnor theorem can be generalized to products of abelian group objects in sAp . 4. Algebraic Serre Theorem The next topological theorem that we pass through our looking-glass is the following theorem originally proved, when p = 2, by J.-P. Serre in [15]. Theorem 4.1. Let X be an nilpotent Fp -local space with finite-type and bounded Fp homology. Then the following are equivalent 1. πs X = 0 for s 0 and 2. πs X = 0 for s > 1. Here we are using the terminology bounded, for a graded module M, to mean that Ms = 0 for s 0. In order to frame Serre’s theorem in our algebraic setting, we call a simplicial supplemented Fp -algebra A a homology complete intersection provided HsQ (A) = 0 for s > 1. The origin of this terminology comes from commutative algebra. There, there is a notion of local complete intersection. Specifically, given a supplemented Noetherian Fp -algebra B, let BI be the localization of B at the augmentation ideal I. The Cohen Presentation Theorem (see [7]) says that there is an epimorphism of algebras f : Fp [[x1 , . . . , xm ]] → BÎ onto the I-adic completion of BI . Then B is a local complete intersection (at the ideal I) provided Ker(f) is generated by a regular sequence. The following can be scried from [1]. Theorem 4.2. Given a supplemented Noetherian Fp -algebra B then B is a local complete intersection if and only if B is a homology complete intersection. Now we say that a supplemented Fp -algebra B is finitely generated provided there is a map of algebras Fp [x1 , . . . , xm ] → B which is onto. We further say that B is a finite Cohen extension provided there is a map Fp [[x1 , . . . , xn ]] → B so that the fibre, at the maximal ideal (x1 , . . . , xn ), is finitely

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generated. Thus, for example, every complete local supplemented Fp -algebra is a finite Cohen extension, by the Cohen Presentation Theorem. Now for a simplicial supplemented Fp -algebra A, we say that π0 A is a simplicially inherited finite Cohen extension provided there is a simplicial algebra model P for the power series algebra Fp [[x1 , . . . , xn ]] and a map P → A of simplicial algebras for which the fibre of the induced map Fp [[x1 , . . . , xn ]] → π0 A, at the maximal ideal (x1 , . . . , xn ), is finitely generated. Using our looking glass, we now translate Serre’s theorem as follows (see [16]): Theorem 4.3. Let A be a simplicial supplemented Fp -algebra with H∗Q (A) of finitetype, π0 A a simplicially inherited finite Cohen extension, and π∗ A bounded. Then A is a homology complete intersection if and only if HsQ A = 0 for s 0. Note: Unfortunately, it is not known whether it is sufficient to assume that π0 A is a finite Cohen extension. The problem occurs that while a map Fp [x1 , . . . , xn ] → π0 A of algebras lifts to a map Fp [x1 , . . . , xn ] → A of simplicial algebras, the same cannot be said for a map Fp [[x1 , . . . , xn ]] → π0 A of algebras. Thus we make the assumption that such a lift exists, up to homotopy, a priori. As an immediate corollary of Theorem 4.3, we have Corollary 4.4. Let A be a supplemented Noetherian Fp -algebra. Then the following are equivalent: 1. HsQ A = 0 for s 0, 2. A is a homology complete intersection, and 3. A is a local complete intersection. Remark: Corollary 4.4 is a special case of a conjecture posed by D. Quillen in [14]. The following general form of this conjecture is a consequence of a recent result proved by L. Avramov in [4]: Theorem 4.5. Let R be commutative Noetherian ring, A a commutative Noetherian R-algebra having finite flat dimension over R, ℘ ⊆ A a prime ideal of A, and ` the field A℘ /℘. Then the following are equivalent 1. Hn (A|R; `) = 0 for n 0, 2. Hn (A|R; `) = 0 for n > 2, and 3. A is a local complete intersection at ℘. Here H∗ (S|R; M) is the André-Quillen homology for a commutative R-algebra S with coefficients in an S-module M (see [1] and [14]). For a simplicial supplemented Fp algebra A, the two notions of André-Quillen homology are related by H Q (A) ∼ = Hn+1 (A|Fp ; Fp ) n

for all n ≥ 0. When R = Fp , this theorem can be shown to be a consequence of our results [16]. Presently, research is being directed at determining simplicial generalizations of this theorem. 2


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Recently, J. Grodal, in [10], proved a generalization of Serre’s theorem. To describe it, we define the height of an element x ∈ H ∗ (X; Fp ) of positive degree to be the integer ht(x) so that ht(x) ≤ n provided xn+1 = 0. If no such integer n exists, we say x has infinite height. Theorem 4.6. Let X be a connected Fp -local space such that π1 X is a finite p-group, π∗ X is non-trivial and bounded, and H ∗ (X; Fp ) of finite-type. Then there exists an element of positive degree in H ∗ (X; Fp ) with infinite height. Evoking our looking glass again, let A be a simplicial supplemented Fp -algebra. We define for x ∈ π∗ A, with deg(x) > 1, its height to be the integer ht(x) so that ht(x) ≤ n provided (γp )n+1 x = 0 (γp is the divided pth power [8]). If no such integer n exists, we say x has infinite height. We thus make the following Conjecture 4.7. Let A be a simplicial supplemented Fp -algebra with H∗Q (A) being of finite-type and bounded and with π0 A a simplicially inherited finite Cohen extension. Then either a. A is a homology complete intersection, or b. π∗ A possesses an element of degree > 1 with infinite height. For example, if A is connected and π∗ A 6= 0, then π∗ ΣA is a non-trivial simplyconnected Hopf algebra with divided powers. Such divided power algebras possess elements of infinite height, by a theorem of M. André [2]. 5. Algebraic Kan-Thurston Theorem In [11], D. Kan and W. Thurston proved the following Theorem 5.1. Let X be a connected space. Then there exists a discrete group G and a map BG → X such that the following holds: a. The induced map G → π1 X is a surjection and ∼ = b. the induced map H∗ (BG; Z) → H∗ (X; Z) is an isomorphism. As noted by our looking glass, (constant simplicial) supplemented Fp -algebras corresponds to classifying spaces. Furthermore, we also take the perspective that AndréQuillen homology is also the appropriate analogue of integral homology. From this, we make the following Conjecture 5.2. Let A be a simplicial supplemented Fp -algebra. Then there exists a (constant simplicial) supplemented Fp -algebra Λ and a map Λ → A such that the following holds: a. The induced map Λ → π0 A is a surjection and ∼ = b. the induced map H∗Q (Λ) → H∗Q (A) is an isomorphism. If this conjecture is true, its importance cannot be understated. To get a feel for its strength, let V be a simplicial vector space and let S(V ) the simplicial free supplemented Fp -algebra generated by V . Then, since S(V ) is cofibrant, H Q (S(V )) = π∗ QS(V ) ∼ = π∗ V. ∗

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Thus 5.2 would allow us to construct supplemented Fp -algebras having any predescribed André-Quillen homology. For example, let K(n) be a simplicial vector space with π∗ K(n) ∼ = Fp < xn > where deg(xn ) = n. Let S(n) = S(K(n)) so that H∗Q (S(n)) ∼ = Fp < xn >. By 5.2, there would exist a supplemented Fp -algebra Λ(n) such that H Q (Λ(n)) ∼ = H Q (S(n)) ∼ = Fp < xn > ∗

∗

again concentrated in degree n. Notice, by Corollary 4.4, each Λ(n), for n ≥ 2, cannot be Noetherian, but their existence would indicate that the Noetherian condition in Corollary 4.4 may not be weakened further. Examples: The polynomial algebra Fp [x] is an obvious canidate for Λ(0). If we let F : Fp [x] → Fp [x] be the Frobenius map (or pth -power map), given by F x = xp , then forming the colimit of F F Fp [x] → Fp [x] → . . . −∞ gives a supplemented Fp -algebra Fp [xp ] with the following properties: −∞ a. Fp [x] → Fp [xp ] is a flat morphism and −∞

F [xp

]

b. T ors p (Fp , Fp ) = 0 for s > 0. −∞ From b., it can be shown that H∗Q (Fp [xp ]) = 0. Thus applying a. along with the flat base change and transitivity axioms for André-Quillen homology, we can conclude −∞ that Fp ⊗Fp [x] Fp [xp ] is a model for Λ(1). (My thanks to Haynes Miller for showing 2 me this example.) These examples actually indicate how a proof of Conjecture 5.2 can be constructed based upon the proof of Theorem 5.1 [11]. Specifically, perform an appropriate induction on the simplices by replacing the action of attaching a new simplex with the action of “attaching” a supplemented Fp -algebra B via a map B → C(B) where C(B) is the cone on B with the following properties: a. C(B) is flat over B and C(B) b. T ors (Fp , Fp ) = 0 for s > 0. Conjecture 5.3. For any B in Ap , the cone C(B) exists. Closing Remarks. As pointed out before, the looking-glass approach between topology and algebra is not new. L. Avramov et.al. developed just such an approach to studying local algebra using rational homotopy as a guide (see [3] and [5]). In fact, it was this approach which was central to L. Avramov’s proof of Quillen’s conjecture [4]. The difference between the approach L. Avramov et.al. adopt and ours is that they stress differentially graded algebras whereas we focus upon simplicial methods. The goals nonetheless are the same, namely to use homotopical and homological techniques to study commutative algebra. In each approach, the focus is to analyse André-Quillen homology which has consistently proven to be a useful device for studying commutative algebras. To effectively obtain such information, it proves useful to have a robust notion of homotopy. In the homological approach adopted by L. Avramov et.al., homotopy is


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constructed from T or∗U (Fp , Fp ) for differentially graded augmented Fp -algebras U. Our approach is to consider commutative Fp -algebras as special cases of simplicial commutative Fp -algebras A so that the homotopy groups π∗ A are defined directly. Each approach has its merit and take their cues from different areas of the homotopy of spaces. At present, effort is being made to see how the approaches relate and where they intersect. Of course, results about commutative algebras and their homology need not exclusively come as parallels of topological results. In fact, novel results about simplicial commutative algebras and their homotopy and homology may point to new results in topology. References [1] M. André, Homologie des Algèbres Commutatives, Die Grundlehren der Mathematischen Wissenschaften 206, Springer-Verlag, 1974. [2] , Hopf algebras with divided powers, J. Algebra 18, (1971) 19-50 [3] L. Avramov, Local algebra and rational homotopy Homotopie algébrique et algèbre locale (J.-M. Lemaire, J.-C. Thomas, eds.), Astérisque, vol. 113-114, Soc. Math. France, Paris, 1984, pp. 15-43. , Locally complete intersection homomorphisms and a conjecture of Quillen on the [4] vanishing of cotangent homology, to appear in the Annals of Math. [5] L. Avramov and S. Halperin, Through the looking glass: A dictionary between rational homotopy theory and local algebra, Algebra, algebraic topology, and their interactions (J.-E. Roos),Lecture Notes Math., vol. 1183, Springer, Berlin, 1986, pp. 1-27. [6] Lewis Carroll, Alice’s Adventures in Wonderland & Through the Looking-Glass, Signet Classic 1960. [7] D. Eisenbud, Commutative Algebra With a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag, 1995. [8] P. Goerss, On the André-Quillen cohomology of commutative F2 -algebras, Astérique 186(1990). , A Hilton-Milnor theorem for categories of simplicial algebras, Amer. J. Math,. [9] 111(1989), 927–971. [10] J. Grodal, The transcendence degree of the mod p cohomology of finite Postnikov systems, M.I.T. preprint 1997. [11] D. Kan and W. Thurston, Every connected space has the homology of a K(π, 1), Topology 15 (1976), pp. 253-258. [12] H. Miller, The Sullivan conjecture on maps from classifying spaces, Annals of Math. 120 (1984), 39–87. [13] D. Quillen, Homotopical Algebra, Lecture Notes in Mathematics 43, Springer-Verlag, 1967. [14] , On the (co)homology of commutative rings, Proc. Symp. Pure Math. 17(1970), 65–87. [15] J.-P. Serre, Cohomologie modulo 2 des espaces d’Eilenberg-MacLane, Comment. Math. Helv. 27 (1953), 198-231 [16] J.M. Turner, Simplicial commutative algebras with vanishing André-Quillen homology, preprint 1998. [17] G. Whitehead, Elements of Homotopy Theory, Graduate Texts in Math. 61, Springer-Verlag, 1995. Department of Mathematics, Calvin College, 3201 Burton Street, S.E., Grand Rapids, MI 49546 E-mail address: [email protected]