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 theor* *y of simplicial augmented commutative Fp-algebras using the homotopy theory o* *f 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 tra* *nslate as well. For example, we recall a theorem of P. Goerss giving an algebraic versio* *n of the Hilton- Milnor theorem which fits in our framework. We next propose how a theore* *m of J.-P. Serre on Fp-local spaces with bounded homotopy groups translates into ou* *r algebraic setting and relate it to a conjecture of D. Quillen on the vanishing of * *Andre-Quillen homology. We also describe what a simplicial algebra version of a theore* *m of D. Kan and W. Thurston should look like. "Oh, Kitty, how nice it would be if we could only get through into the Loo* *king- Glass 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. Andre and D. Quillen constructed the first complete (co-)* *homology theory for commutative algebras. The approach they took involved applying simpl* *icial 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 meth* *ods from simplicial homotopy theory should prove beneficial in discerning results f* *or Andre- Quillen 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 resu* *lting simplicial category by sAp. This is a pointed category with a closed simplicia* *l model ___________ Date: June 15, 1998. 1991 Mathematics Subject Classification. Primary: 13D03, 18G30, 18G55; Second* *ary: 55P60, 55P99, 55S05. Key words and phrases. simplicial commutative Fp-algebras, Fp-local spaces, * * Andre-Quillen homology. 1 2 JAMES M. TURNER structure. To describe the weak equivalences, a further invariant is used, name* *ly homo- topy ss*, which is defined by first thinking of a simplicial algebra as a simpl* *icial group. A map f : A ! B in sAp is thus a weak equivalence provided ss*f is an isomorphi* *sm. 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 ho* *motopy theory of simplicial supplemented Fp-algebras can be viewed as analogues of res* *ults in the homotopy theory of connected Fp-local spaces. To make such a perspective fl* *y, we need to perform a kind of role reversal. We will view the Andre-Quillen homolog* *y of a simplicial supplemented Fp-algebra as the appropriate analogue of homotopy grou* *ps 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 the* *ory of simplicial supplemented Fp-algebras. We then proceed with a study of how to tra* *nslate theorems, describing the homotopy theory of Fp-local spaces, to theorems descri* *bing the homotopy theory of simplicial supplemented Fp-algebras. We begin by recalli* *ng 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 t* *ranslate a theorem of J.-P. Serre which classifies a large class of Fp-local spaces with* * bounded ho- motopy and Fp-homology. We connect this to a conjecture of D. Quillen which cla* *ssifies certain commutative algebras with bounded Andre-Quillen homology. Bolstered by our successes thus far, we close by describing what the appropri* *ate 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(ss; 1). The validity of suc* *h an analogue would place strong limits on our ability to further weaken the conditi* *ons in Quillen's conjecture. The reader will find the contents of this paper totally bereft of proofs. Thi* *s 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 ea* *rlier 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 ss*f is an isomorphism, 2. a fibration iff the induced map A ! ss0A xss0BB 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 F p-ALGEBRAS AND Fp-LOCAL SPACES 3 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 f* *rom sAp by inverting weak equivalences, or, equivalently, by taking the subcategory* * of cofi- brant objects (i.e. simplicial supplemented Fp-algebras A for which the unit F* *p ! A is a cofibration). Note that every object of sAp is fibrant, that is, given A 2* * sAp then A ! ss0A is a surjection. Philosophically, homology is appropriately constructed as the derived functor* *s 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 s* *implicial supplemented Fp-algebras, commonly called the indecomposables functor. Prolong* *ing 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 ss*-isomorphisms). This is called the total l* *eft derived functor of Q and we define the Andre-Quillen homology of a simplicial supplemen* *ted Fp-algebra A to be HQ*(A) = ss*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 analo* *gy from results about the homotopy theory of pointed Fp-local spaces. To get this * *parallel off the ground, we take Andre-Quillen homology as the proper analogue of homoto* *py groups. Below is our proposed looking-glass for translating other results from * *Fp-local spaces to results about simplicial supplemented Fp-algebras. 4 JAMES M. TURNER __________________________________________________________________________|||* *|| |_|_connected_Fp-Local_Spaces_______|Simplicial_supplemented_Fp-algebras_|_||* *|||| |_|___________________________________|__________________________________|_| | | homotopy groups ss | Andre-Quillen homology HQ | | |_|___________________*________________|_______________________________*_|_||* *|||| |_|_Fp-homology_H*(-;_Fp)_____________|________________homotopy_groups_ss*_||* *||||| |_|_cartesian_product_x________________|_________________tensor_product__|_||* *|W||| |_|_wedge____________________________|_____________augmented_product_xFp_|_||* *|||| |_|_fibration_sequences_________________|_____________cofibration_sequences_|* *|||||| |_|_long_exact_sequences_in_homotopy____|__long_exact_sequences_in_homology_|* *|||||| |_|_Eilenberg-MacLane_spaces_K(ss;_n)___|___________sphere_algebras_S(V;_n)_|* *|||||| |_|_loop_spaces_X____________________|______________suspension_algebras_A_|||* *|||| |_|_n-connected_covers_________________|__________homology_approximations_|||* *|||| |_|_Postnikov_towers___________________|_________________skeletal_filtrations* *_||||*||| |_|_algebra_structure_on_H_(-;_Fp)______|_________-algebra_structure_on_ss*_|* * | | | Whitehead product on ss | Whitehead product on H* | | |_|________________________*___________|_______________________________Q_|_||* *|||| |_|_classifying_spaces_Bss_______________|(constant_simplicial)_Fp-algebras_A* * | | | | F -local spheres Sn | abelian group objects K(n) | | |_|__p________________p_________________|________________________________+_||* *||||| |_|_Serre_spectral_sequence_for_homologyS|erre_spectral_sequence_for_homotopy* *_||||||| |_|_Poincare_series_for_homology________|______Poincare_series_for_homotopy_|* * | Several other analogies can be drawn from Fp-local spaces for simplicial Fp-a* *lgebras. Many of the ones listed above are spelled out in [12], [8], and [16]. For the p* *urposes 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. Hal* *perin between rational homotopy and local algebra. See particularly [3] and [5]. In f* *act, 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 wa* *s an alge- braic version of the classic Hilton-Milnor theorem. Established by P. Goerss [9* *], it was needed in order to determine the primary operations for Andre-Quillen cohomolog* *y [8]. Recall that the topological Hilton-Milnor theorem [17] says that there is a h* *omotopy equivalence _ Y (Sn1p Sn2p) '! Sn(w)p 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) = n1j1(w) + n2j2(w) + `(w) - 1 where ji(w) = # of times that xi appears in w for i = 1; 2, and `(w) = j1(w) + j2(w): SIMPLICIAL COMMUTATIVE F p-ALGEBRAS AND Fp-LOCAL SPACES 5 If A is a simplicial supplemented Fp-algebra, let A be the suspension of A de* *fined by (A)s := Fp As BsAs where Bo(-) is the simplicial bar construction on algebras. By our looking-gla* *ss, the Hilton-Milnor theorem translates into Theorem 3.1. [9] For n1; n2 0, there is a weak equivalence in sAp (K(n1)+ xFp K(n2)+) '!wK(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 pro* *ducts of abelian group objects in sAp. 4.Algebraic Serre Theorem The next topological theorem that we pass through our looking-glass is the fo* *llowing 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 bounde* *d Fp- homology. Then the following are equivalent 1. sssX = 0 for s 0 and 2. sssX = 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 simplic* *ial supple- mented Fp-algebra A a homology complete intersection provided HQs(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 F* *p-algebra B, let BI be the localization of B at the augmentation ideal I. The Cohen Prese* *ntation Theorem (see [7]) says that there is an epimorphism of algebras f : Fp[[x1; : :;:xm ]] ! ^BI onto the I-adic completion of BI. Then B is a local complete intersection (at t* *he ideal I) provided Ker(f) is generated by a regular sequence. The following can be scr* *ied from [1]. Theorem 4.2. Given a supplemented Noetherian Fp-algebra B then B is a local c* *om- plete intersection if and only if B is a homology complete intersection. Now we say that a supplemented Fp-algebra B is finitely generated provided th* *ere is a map of algebras Fp[x1; : :;:xm ] ! B which is onto. We further say that B is a finite Cohen extension provided ther* *e is a map Fp[[x1; : :;:xn]] ! B so that the fibre, at the maximal ideal (x1; : :;:xn)* *, is finitely 6 JAMES M. TURNER 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 ss0A is a simplic* *ially 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 f* *or which the fibre of the induced map Fp[[x1; : :;:xn]] ! ss0A, 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 HQ*(A) of fin* *ite- type, ss0A a simplicially inherited finite Cohen extension, and ss*A bounded. T* *hen A is a homology complete intersection if and only if HQsA = 0 for s 0. Note: Unfortunately, it is not known whether it is sufficient to assume that ss* *0A is a finite Cohen extension. The problem occurs that while a map Fp[x1; : :;:xn] ! s* *s0A of algebras lifts to a map Fp[x1; : :;:xn] ! A of simplicial algebras, the same ca* *nnot be said for a map Fp[[x1; : :;:xn]] ! ss0A of algebras. Thus we make the assumptio* *n 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 followin* *g are equivalent: 1. HQsA = 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 resu* *lt 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 ` t* *he 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 Andre-Quillen homology for a commutative R-algebra S w* *ith coefficients in an S-module M (see [1] and [14]). For a simplicial supplemente* *d Fp- algebra A, the two notions of Andre-Quillen homology are related by HQn(A) ~=Hn+1(A|Fp; Fp) 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 g* *eneralizations of this theorem. 2 SIMPLICIAL COMMUTATIVE F p-ALGEBRAS AND Fp-LOCAL SPACES 7 Recently, J. Grodal, in [10], proved a generalization of Serre's theorem. To * *describe it, we define the height of an element x 2 H*(X; Fp) of positive degree to be t* *he 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 ss1X is a finite p* *-group, ss*X is non-trivial and bounded, and H*(X; Fp) of finite-type. Then there exis* *ts an element of positive degree in H*(X; Fp) with infinite height. Evoking our looking glass again, let A be a simplicial supplemented Fp-algebr* *a. We define for x 2 ss*A, with deg(x) > 1, its height to be the integer ht(x) so tha* *t ht(x) n provided (flp)n+1x = 0 (flp is the divided pthpower [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 HQ*(A) bein* *g of finite-type and bounded and with ss0A a simplicially inherited finite Cohen ext* *ension. Then either a. A is a homology complete intersection, or b. ss*A possesses an element of degree > 1 with infinite height. For example, if A is connected and ss*A 6= 0, then ss*A is a non-trivial simp* *ly- connected Hopf algebra with divided powers. Such divided power algebras possess* * ele- ments of infinite height, by a theorem of M. Andre [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 ! ss1X 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* * cor- responds to classifying spaces. Furthermore, we also take the perspective that * *Andre- Quillen homology is also the appropriate analogue of integral homology. From th* *is, we make the following Conjecture 5.2. Let A be a simplicial supplemented Fp-algebra. Then there exi* *sts a (constant simplicial) supplemented Fp-algebra and a map ! A such that the following holds: a. The induced map ! ss0A is a surjection and ~= b. the induced map HQ*() ! HQ*(A) is an isomorphism. If this conjecture is true, its importance cannot be understated. To get a fe* *el 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, HQ*(S(V )) = ss*QS(V ) ~=ss*V: 8 JAMES M. TURNER Thus 5.2 would allow us to construct supplemented Fp-algebras having any predes* *cribed Andre-Quillen homology. For example, let K(n) be a simplicial vector space with ss*K(n) ~=Fp < xn > where deg(xn) = n. Let S(n) = S(K(n)) so that HQ*(S(n)) ~= Fp < xn >. By 5.2, there would exist a supplemented Fp-algebra (n) such that HQ*((n)) ~=HQ*(S(n)) ~=Fp < xn > again concentrated in degree n. Notice, by Corollary 4.4, each (n), for n 2, c* *annot 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 (0). If we * *let F : Fp[x] ! Fp[x] be the Frobenius map (or pth-power map), given by F x = xp, t* *hen forming the colimit of Fp[x] F!Fp[x] F!: : : -1 gives a supplemented Fp-algebra Fp[xp ] with the following properties: -1 a. Fp[x] ! Fp[xp ] is a flat morphism and p-1] b. T orFp[xs (Fp; Fp) = 0 for s > 0. -1 From b., it can be shown that HQ*(Fp[xp ]) = 0. Thus applying a. along with * *the flat base change and transitivity axioms for Andre-Quillen homology, we can con* *clude -1 that Fp Fp[x]Fp[xp ] is a model for (1). (My thanks to Haynes Miller for showi* *ng me this example.) 2 These examples actually indicate how a proof of Conjecture 5.2 can be constru* *cted 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 ac* *tion of "attaching" a supplemented Fp-algebra B via a map B ! C(B) where C(B) is the co* *ne on B with the following properties: a. C(B) is flat over B and b. T orC(B)s(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 topo* *logy and algebra is not new. L. Avramov et.al. developed just such an approach to st* *udying 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 techn* *iques to study commutative algebra. In each approach, the focus is to analyse Andre-Q* *uillen homology which has consistently proven to be a useful device for studying commu* *tative algebras. To effectively obtain such information, it proves useful to have a robust not* *ion of homotopy. In the homological approach adopted by L. Avramov et.al., homotopy is SIMPLICIAL COMMUTATIVE F p-ALGEBRAS AND Fp-LOCAL SPACES 9 constructed from T orU*(Fp; Fp) for differentially graded augmented Fp-algebras* * U. Our approach is to consider commutative Fp-algebras as special cases of simplicial * *commuta- tive Fp-algebras A so that the homotopy groups ss*A are defined directly. Each * *approach has its merit and take their cues from different areas of the homotopy of space* *s. At present, effort is being made to see how the approaches relate and where they i* *ntersect. Of course, results about commutative algebras and their homology need not exc* *lu- sively 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. Andre, Homologie des Algebres Commutatives, Die Grundlehren der Mathemat* *ischen Wis- senschaften 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 algebrique et alg* *ebre locale (J.-M. Lemaire, J.-C. Thomas, eds.), Asterisque, vol. 113-114, Soc. Math. France, * *Paris, 1984, pp. 15-43. [4]__________, Locally complete intersection homomorphisms and a conjecture of* * Quillen on the 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 interactio* *ns (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, Gra* *duate Texts in Mathematics 150, Springer-Verlag, 1995. [8]P. Goerss, On the Andre-Quillen cohomology of commutative F2-algebras, Aste* *rique 186(1990). [9]__________, A Hilton-Milnor theorem for categories of simplicial algebras, * *Amer. J. Math,. 111(1989), 927-971. [10]J. Grodal, The transcendence degree of the mod p cohomology of finite Postn* *ikov systems, M.I.T. preprint 1997. [11]D. Kan and W. Thurston, Every connected space has the homology of a K(ss; 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 Andre-Quillen h* *omology, preprint 1998. [17]G. Whitehead, Elements of Homotopy Theory, Graduate Texts in Math. 61, Spri* *nger-Verlag, 1995. Department of Mathematics, Calvin College, 3201 Burton Street, S.E., Grand Ra* *pids, MI 49546 E-mail address: jmt@ziplink.net