IDEMPOTENTS AND LANDWEBER EXACTNESS IN BRAVE NEW ALGEBRA J.P. MAY Abstract.We explain how idempotents in homotopy groups give rise to spli* *t- tings of homotopy categories of modules over commutative S-algebras, and we observe that there are naturally occurring equivariant examples invol* *ving idempotents in Burnside rings. We then give a version of the Landweber e* *xact functor theorem that applies to MU-modules. In 1997, not long after [6] was written, I gave an April Fool's talk on how to prove that BP is an E1 ring spectrum or equivalently, in the language of [6], a commutative S-algebra. Unfortunately, the problem of whether or not BP is an E1 ring spectrum remains open. However, two interesting remarks emerged and will be presented here. One concerns splittings along idempotents and the other concerns the Landweber exact functor theorem. One of the nicest things in [6] is its one line proof that KO and KU are com- mutative S-algebras. This is an application of the following theorem [6, VIII.2* *.2], or rather the special case that follows. Theorem 1. Let R be a cell commutative S-algebra, A be a cell commutative R- algebra, and M be a cell R-module. Then the Bousfield localization ~ : A -! AM * * of A at M can be constructed as the inclusion of a subcomplex in a cell commutativ* *e R- algebra. In particular, the commutative R-algebra AM is a commutative S-algebra by neglect of structure. The cell assumptions can always be arranged by use of the cofibrant replaceme* *nt constructions in [6], so they result in no loss of generality. The theorem spec* *ializes as follows to algebraic localizations at elements of R* = ß*(R) [6, VIII.4.2]. Theorem 2. Let R be a cell commutative S-algebra and X a set of elements of R*. The localization ~ : R -! R[X-1] that induces the algebraic localization R* -! R*[X-1] can be constructed as the unit of a cell commutative R-algebra. The connective real K-theory spectrum ko is a commutative S-algebra by multi- plicative infinite loop space theory [11], and KO is the localization ko[fi-1] * *obtained by inverting the Bott class. Therefore KO is a commutative ko-algebra and thus a commutative S-algebra. That's the one line. Complex K-theory works similarly. As a matter of algebra, idempotents give localizations. Since MU arises in nature as an E1 ring spectrum, that being the paradigmatic example that led to the definition [10], one might try to prove that BP is a Bousfield localization* * of MU and thus a commutative MU-algebra. That is April Fool's nonsense, but the basic idea has a correct version with other applications, as we shall explain. Essent* *ially ____________ Date: July 4, 2001. 1991 Mathematics Subject Classification. Primary 55N20, 55N91, 55P43. The author was partially supported by the NSF. 1 2 J.P. MAY the same idea occurred independently to Schwänzl, Vogt, and Waldhausen, who gave quite different applications [13, 14]. Definition 3. Let R be a cell commutative S-algebra and let e 2 R0 be an idem- potent element. As a matter of algebra, R*[e-1] = eR*. Define eR to be the cell commutative R-algebra R[e-1] of Theorem 2. Theorem 4. Let 1 = e1+ . .+.en where the ei are orthogonal idempotents in R*. Then the canonical map " : R -! e1R x . .x.enR of commutative R-algebras is a weak equivalence. Therefore the category of R- modules is Quillen equivalent to the product of the categories of eiR-modules. Proof.The first statement is obvious. The second statement follows from the next two results. The first is implicit in [6, III.4.2 and VII.4.8] and explicit in * *[9, I.3.6] and the second is proven by an easy formal argument. Theorem 5. If f : R -! Q is a weak equivalence of commutative S-algebras, then the extension of scalars functor f* : MR - ! MQ and the pullback of structure functor f* : MQ -! MR specify a Quillen equivalence of model categories. Theorem 6. If R is a product of commutative S-algebras Ri with projections "i: R -! Ri, then the functor that sends an R-module M to the tuple ("i*M) is the left adjoint of a Quillen equivalence from MR to the product of the categories * *MRi. The right adjoint sends (Ni) to the product of the R-modules "*iNi. Theorem 4 shows that the homotopy theory of R-modules entirely decomposes into the homotopy theories of the modules over the eiR. The ring spectra that algebraic topologists usually work with have no non-trivial idempotents. Howeve* *r, interesting examples do arise naturally in algebraic K-theory, as observed in [* *13]. Remark 7. If R is connective, we have a map R -! HR0 that induces an isomor- phism on ß0 [6, IV.3.1]. Here, if X R0 and we apply the functor (-) ^R HR0 to ~ : R -! R[X-1], we obtain a model for the localization ~ : HR0 ~=R ^R HR0 -! R[X-1] ^R HR0 ~=(HR0)[X-1] ~=H(R0[X-1]). In particular, for an idempotent e 2 R0, eR ^R HR0 is equivalent to H(eR0). This observation is the starting point of [13, 14]. Interesting examples also arise in equivariant algebraic topology. The resul* *ts above generalize directly to the equivariant setting of commutative SG -algebras and their modules [6, 9, 12], where G is a compact Lie group and SG is the sphe* *re G-spectrum. Here, for a commutative SG -algebra R, we take R* = ß*(RG ). In particular, (SG )* is the equivariant stable homotopy groups of spheres and (SG* * )0 is isomorphic to the Burnside ring A(G). The ring A(G), and more so its localizati* *ons at subrings of the rationals, usually does have non-trivial idempotents [5, 8]. The splittings of Theorem 4 give model theoretic refinements of splittings in equivariant stable homotopy theory that are discussed in [8, V] and [12, XVIIx6* *]. Those sources describe splittings of homology and cohomology theories, and it is now apparent that these splittings arise from splittings of corresponding equiv* *ariant stable categories. The splittings involve change of group functors, and these * *are discussed model theoretically in the contexts both of SG -modules and of orthog- onal G-spectra in [9]. Briefly, by [9, VI.1.2], for an inclusion ' : H G, the* *re is IDEMPOTENTS AND LANDWEBER EXACTNESS IN BRAVE NEW ALGEBRA 3 a Quillen adjoint pair (G+ ^H (-), '*) relating HM to GM . Let W H = NH=H and let " : NH -! W H be the quotient homomorphism. By [9, 3.12], there is also a Quillen adjoint pair relating NHM to W HM . This remains true after local- ization at a prime or rationalization. Thus we can split localized stable categ* *ories along idempotents and identify the pieces as equivalent to stable categories ov* *er subquotient groups. We now turn to a completely different topic, but one that also arises natural* *ly from consideration of spectra constructed from MU, namely the Landweber exact functor theorem. In fact, that result has the following more structured version* * in the category of MU-modules. We say that an MU*-module M* is Landweber exact if, for each prime p, the set {vi|i 0}is a regular sequence for M*. Here v0 =* * p and the vifor i > 0 are indecomposable elements of degree 2pi- 2 with Chern numbers divisible by p. Theorem 8. If M* is a Landweber exact MU*-module, then there is an MU- module M such that ß*(M) = M* and, for any finite cell MU-module X, ß*(X) MU* M* ~=ß*(X ^MU M). As a matter of algebra, Landweber [7, 2.6] proved the following result. Let M* * U denote the category of comodules over MU*(MU) that are finitely presented as MU-modules. Theorem 9 (Landweber). The functor (-) MU* M* on the category M U is exact if and only if the MU*-module M* is Landweber exact. By the following two results, MU-modules naturally gives rise to objects of M* * U . Lemma 10. If X is a finite cell MU-module, then ß*(X) is a finitely presented MU*-module. Proof.This is proven by exactly the same induction on the number of cells as in* * the classical special case X = MU ^ Y , where Y is a finite CW spectrum. Of course, in that case ß*(X) = MU*(Y ). For example, the proof is clear from the algebraic argument given by Adams [1, pp. 132-133]. Lemma 11. If X is an R-module, where R is a commutative S-algebra such that R*R is R*-flat, then the Hurewicz map gives X* a structure of R*R-comodule. Proof.This is proven by diagram chasing as in Adams [1]. It is the starting poi* *nt of the development of an Adams spectral sequence in brave new algebra [3]. The main point is that R*R R* X* ~=ß*((R ^ R) ^R X) ~=ß*(R ^R X). Of course, this applies with R = MU. The previous three results imply the following conclusion. Proposition 12. Let M* be a Landweber exact MU*-module. Then the functor ß*(X) MU* M* specifies a homology theory on finite cell MU-modules X. Applying Adams' variant [2] of Brown's representability theorem, which applies since MU* is countable [6, III.2.13], we obtain the MU-module M promised in Theorem 8. The construction of M is non-uniquely functorial: given a map f* : M* -! N* of Landweber exact MU-modules, there is a map f : M - ! N of MU-modules that realizes f*, but f will not be unique unless the relevant lim1 groups vanish. 4 J.P. MAY Example 13. Recall that KU* = Z[u, u-1], where deg (u) = 2, and give it the MU*-module structure specified by the ring homomorphism MU* -! KU* that sends [M2n] to T d(M2n)un. We know by the methods of [6, Vx4] that KU is an MU-module and in fact an MU-ring spectrum. There results an isomorphism ß*(X) MU* KU* -! KU*(X) for finite cell MU-modules X. Alternatively, granting that there is a unique ri* *ng spectrum KU with the cited homotopy groups, we can construct KU as an MU- module by Theorem 8 and then show that it is an MU-ring spectrum by the meth- ods of [6, Vx4]. The resulting map T d : MU -! KU is a map of MU-ring spectra. The calculation of T d* in terms of the Todd genius is evident from the present approach, but is not clear from the approach of [6, Vx4]. In any case, this giv* *es a generalization to MU-modules of the Conner-Floyd theorem that MU-theory determines KU-theory. Of course, if BP is a commutative S-algebra, then the Landweber exact functor theorem will admit a precisely analogous and more useful version for BP*-module* *s. References [1]J.F. Adams. Lectures on generalized cohomology. Springer Lecture Notes in Ma* *thematics Vol 99, 1969, pp 1-138. [2]J.F. Adams. A variant of E.H. Brown's representability theorem. Topology 10(* *1971), 185-198. [3]A. Baker and A. Lazarev. On the Adams spectral sequence for R-modules. Prepr* *int, 2001. [4]P.E. Connor and E.E. Floyd. The relation of cobordism to K-theories. Springe* *r Lecture Notes in Mathematics Vol. 28. 1966. [5]T. tom Dieck. Idempotent elements in the Burnside ring. J. Pure and Applied * *Algebra. 10(1977), 239-247. [6]A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by * *M. Cole). Rings, modules, and algebras in stable homotopy theory. Mathematical Surveys and Mon* *ographs Vol. 47. 1997. American Mathematical Society. [7]P. S. Landweber. Homological properties of comodules over MU*(MU) and BP*(BP* *). Amer- ican J. Math. 98(1976), 591-610. [8]L. G. Lewis, J. P. May, and M. Steinberger, with contributions by J. E. McCl* *ure. Equivariant stable homotopy theory. Lecture Notes in Mathematics Vol. 1213. Springer-Verl* *ag, 1986. [9]M. A. Mandell and J.P. May Equivariant orthogonal spectra and S-modules. Mem* *oirs Amer. Math. Soc. To appear. [10]J. P. May (with contributions by F. Quinn, N. Ray, and J. Tornehave). E1 -r* *ing spaces and E1 -ring spectra. Springer Lecture Notes in Mathematics Vol. 577. 1977. [11]J.P. May. Multiplicative infinite loop space theory. J. Pure and Applied Al* *gebra 26(1983), 1-69. [12]J. P. May, et al. Equivariant homotopy and cohomology theory. CBMS Regional* * Conference Series in Mathematics, Number 91. American Mathematical Society. 1996. [13]R. Schwänzl, R.M. Vogt, and F. Waldhausen. Adjoining roots of unity to E1 r* *ing spectra in good cases - a remark. in Homotopy invariant algebraic structures. Contemp. M* *ath Vol. 239, 1999, 245-249. Amer. Math. Soc. [14]R. Schwänzl, R.M. Vogt, and F. Waldhausen. Topological Hoschschild homology* *. J. London Math. Soc.(2) 62(2000), 345-356. Department of Mathematics, University of Chicago, Chicago, IL 60637 E-mail address: may@uchicago.edu