Stable Homotopy Theory: Overview

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1 Research Statement Romie Banerjee December 6, 2012 Part I Stable Homotopy Theory: Overview Algebraic topology is the study of spaces using homotopy invariant algebraic structures called generalized (co)homology theories. The central object of study in stable homotopy theory is the stable homotopy category which is roughly the derived category of (co)homology theories modulo stable equivalences. Objects in this category are called spectra. The spectrum HZ/p representing ordinary mod-p homology and it ring of endomorphisms End(HZ/p) recovers the p-completion of the stable homotopy category. This study, under the guise Adams Spectral Sequence has been the key tool for accessing the enormous amount of data in the stable category for a long time. However the deeper arithmetic structure hidden in the (HZ/p, End HZ/p) was not revealed until Quillen identified the pair associated to complex cobordism and its endomorphisms with the moduli of one-dimensional formal groups and isogenies. Using the work of Lazard, Lubin, Serre, Tate and others on the structure of the moduli of formal groups Morava, Miller, Ravenel and Wilson exploited the connection established by Quillen to express the pair (MU, End MU) Z (p) as a resolution of the Adams pair. Futhermore they used the height filtration associated to formal groups to define the chromatic filtration of the MU-pair and consequently a filtration of the stable homotopy category. The subsequent layers of the chromatic filtration can be understood completely from automorphism groups of finite height formal groups. These groups, also known as the Morava Stabilizer groups, are p-adic Lie groups which can be described as the group of units in the maximal order of certain division algebras over Q p. Computational insights thus arise from understanding the stable homotopy category one height at a time. This is achieved through the study of the representations of the Morava stabilizers by E -ring spectra. The height one case was completely understood by Adams using variants of the spectrum of topological K-theory. The height two case is the subject of intense ongoing research by Behrens, Hopkins, Goerss, et.al. using the spectrum of topological modular forms. Real Johnson-Wilson theories 1 Introduction Atiyah [2] developed K-theory with Reality KR, which is in a sense a mixture of the K-theory of real vector bundles KO, the K-theory of self-conjugate bundles KSC, and the K-theory of G-vector bundles over G-spaces K G. A real vector bundle over a space X with involution(x x) is a complex vector bundle E over X which also has an involution such that the projection E X commutes with the involution and the map of fibers E x E x is anti-linear. 1

2 The spectrum KR representing real K-theory is an example of a Z/2-equivariant real-oriented spectrum. The universal example of real-oriented spectra was constructed by Hu and Kriz [12] as genuine Z/2 equivariant spectrum MU R whose underlying non-equivariant spectrum is MU and whose equivariant structure comes from the action of complex conjugation. Recently MU R has been used by Hill-Hopkins-Ravenel in a crucial way to solve the Kervaire invariant problem. The v n -periodic analogues of the the v 1 -periodic K-theory are the Johnson-Wilson theories E(n). The Z/2-equivariant Johnson-Wilson spectrum ER(n) was first constructed by Hu and Kriz in [12] from MU R. The homotopy fixed points spectrum ER(n) = F (EZ/2 +, ER(n)) Z/2 (1) The spectrum ER(n) is 2 n+2 (2 n 1)-periodic compared to the 2(2 n 1)-periodic E(n). ER(1) is KO (2) and E(1) is KU (2). 2 Doctoral Research 2.1 Non-immersions of projective spaces Kitchloo and Wilson [15] have used ER(2)-theory to solve certain non-immersion problems of real projective spaces. There is a stable cofibration connecting E(n) and ER(n), Σ λ(n) ER(n) ER(n) E(n) (2) where the first map is multiplication by a homotopy element x. This leads to a Bockstein spectral sequence for x-torsion. It is known that x 2n+1 1 = 0 so there can be only 2 n+1 1 differentials. For the case of our interest n = 2 there are only 7 differentials. From [13] we know that if there is an immersion of RP b to R c then there is an axial map RP b RP 2L c 2 RP 2L b 2. (3) For b = 2n and c = 2k Don Davis shows in [7] that there is no such map when n = m + α(m) 1 and k = 2m α(m), where α(m) is the number of ones in the binary expression of m by finding an obstruction to James s map (2) in E(2)-cohomology. Kitchloo and Wilson get new non-immersion results by computing obstructions in ER(2)-cohomology. In this paper we extend Kitchloo-Wilson s results by computing the ER(2)-cohomology of the odd projective space RP 16K+9. This will give us newer non-immersion results. The main results are the following. Theorem A 2-adic basis of ER(2) 8 (RP 16K+9, ) is given by the elements 2. If (m, α(m)) (6,2) or (1,0) mod 8, α k u j, (k 0, 1 j 8K + 4) v 4 2α k u j, (k 1, 1 j 8K + 4) v 4 2u j, (4 j 8K + 4) xα k i 16K+9, xv 4 2α k i 16K+9, (k 0) RP 2(m+α(m) 1) does not immerse in R 2(2m α(m))+1. This shall give us new non-immersions that are often new and different from those of [15] and [16]. Using Davis s table [6] the first new result is RP does not immerse in R

3 2.2 Tate Cohomology of ER(n) Let G be a compact Lie group and k G a G-equivariant spectrum/space. One of the most standard ways of understanding the equivariant homotopy groups of k G is by splitting k G into G-free and G-acyclic parts, EG + k G and t(k G ) the Tate spectrum associated to k G. These fit into a homotopy cofiber sequence EG + k G F (EG +, k G ) t(k G ) The homotopy groups of these can be computed using spectral sequences whose E 2 pages begin with group homology, group cohomology and Tate cohomology of the action of G on the non-equivariant coefficient ring of k G. Theorem 2. The Tate spectrum t(er(n)) is trivial. The triviality of the Tate spectrum for Atiyah s K-theory with reality and subsequently the spectrum KR was proved by Fajstrup. Results in similar directions can be found in [11] and [12]. In [11] Hu shows the fixed points of the Tate spectrum for BP R is HZ/2 and also computes the coefficient ring of t(bp R < n >). The triviality of the Tate spectrum also means that we can define ER(n) as the homotopy orbit spectrum ER(n) hz/ Moduli of Real formal groups and MU R The Morava stabilizer groups at prime 2 contains additionl families of automorphism coming from the formal inverse [ 1] F. We will define a real formal group over a ring R as an ordinary 1-dimensional formal group over R along with an involution coming from the formal inverse. Let µ be the constant group scheme Z/2 acting on Spec(MU ) by F ( F ) where ( F )(x, y) = F ( x, y). This lifts to an action of the the constant group scheme (Z) by the strict isomprphism i F (x). However one needs to invoke equivariance to obtain a covering by an algebraic space. Let µ act on SpecMU G m by (F, σ) ( F, σ). Define the quotient stack FGL G m //µ to be the moduli stack of real formal group laws FGL real. The isomorphisms of real formal group laws are represented by the stack FGL real MF G FGL real. Consider the non-equivariant spectrum MU R = (MU[σ ±1 ]) hz/2 where σ is the periodicity element in degree 1 α in the Z/2 equivariant MU R coming from the equivalence S 1 Z/2 + S α Z/2 +. (Notation: 1 + α is the regular representation of Z/2.) The general yoga of formal groups in stable homotopy theory lets us associate to any complex oriented ring spectrum an affine scheme over M F G and more generally, to any homotopy commutative ring spectrum X a stack M X = Spec (MU +1 X) over M F G. Let X denote the coarse moduli functor associated to a moduli functor X. Theorem 3. M MUR = FGL real M MUR MU R = FGL real MF G FGL real Futhermore, these are algebraic spaces and if S R = Tot MU +1 R is a pro-smooth atlas. M SR = colim (FGL real MF G FGL real FGL real ) 3

4 3 Current Research 3.1 A modular description of ER(2) In this paper we study the chromatic layers of ER(2) and how they fit together using the chromatic fracture square. Fix p = 2. Let E n be the n-th Morava E-theory. This is a Landweber exact theory with π E n = W (F 2 n)[[u 1,..., u n 1 ]][u ±1 ] which classifies deformations of the height n Honda formal group law. The n-th Morava stabilizer group S n = O D 1n,Q 2 (4) (the group of units in the maximal order of the division algebra over Q 2 with Hasse invariant 1 n ) acts on the spectrum E n through E -ring maps. This action extends to an action of the extended stabilizer group Gal(F 2 n/f 2 ) S n. Consider the subgroup K = Gal(F 2 n/f 2 ) F 2n. Then the homotopy fixed points can be related to the K(n)-localization, L K(n) E(n) En hk. The group Z/2 acts on E(n) by complex conjugation via the Z/2 equivariant MU R -module ER(n) and on E n by the action of the formal inverse. It is known that previous equivalence is also a Z/2-equivariant equivalence; L K(n) ER(n) ((E n ) hk ) hz/2 (5) For n = 1, the stabilizer group S 1 = Z 2 has the maximal finite subgroup Z/2 = {±1} and L K(1) ER(2) = E hz/2 1. For n = 2, the stabilizer group S 2 = O D 12,Q 2 has the maximal finite subgroup Â4 = Q 8 C 3 and L K(2) ER(2) = E h(c2 C3 Z/2) 2. Let KO denote the spectrum of orthogonal K-theory and T MF denote the spectrum of topological modular forms. The spectrum T M F arises from the moduli stack of smooth elliptic curves Ell in the same way KO arises from the moduli stack Forms(G) of forms of the multiplicative group scheme. The localizations of KO and T MF at the chromatic primes can be described; for prime 2, L K(1) KO = x Forms(G) p(f p) E h Aut(x) 1 hg(f p/f p) Since, by definition, there is only one isomorphism class of x over F p and the group Aut(x) = Z/2 is the maximal finite subgroup of S 1, the right hand side is the Hopkins-Miller higher real K-theory EO hg(fp/fp) 1. For primes 2 and 3, L K(2) T MF = x Ell ss (F p) E h Aut(x) 2 hg(f p/f p) where Ell ss is the locus of supersingular curves at prime p. Since there is a unique isomorphism class of elliptic curves x at these prime and Aut(x) is the maximal finite subgroup of S 2 the right hand side is the higher real K-theory EO hg(fp/fp) 2. We wish to identify the localizations of ER(2) at the chromatic primes using appropriate étale extensions of KO and T MF. This raises the questions (6) (7) 4

5 (i) Given p = 2 and chromatic height 1 does there exist x Forms(G) 2 for which Aut(x) is the group Z/2 = S 1? This is trivial. (ii) Given p = 2 and chromatic height 2 does there exist an étale cover of X of Ell ss so that there exists x X for which Aut(x) is the subgroup C 2 C 3 of S 2? The purpose of this paper is to answer these questions in the affirmative and as a consequence show that the stack associated to the spectrum ER(2) can be covered with certain modular curves and quadratic curves. If M ER(2) denotes the moduli stack over M F G associated to ER(2), we give a precise description of its 2-completion. Theorem 4. (M ER(2) ) 2 Forms(G) 2 Ell 0 (3) ss 2 (8) (Ell 0(3) ss 2 )ord Notation: Ell 0 (3) ss 2 is the completion of Ell 0 (3), the moduli stack of smooth elliptic curves with Γ 0 (3) level structures, at the supersingular locus at the prime 2 and (Ell 0 (3) ss 2 ) ord is its ordinary locus. Equivalently, there is a homotopy pullback square in the stable homotopy category. ER(2) 2 L K(2) T MF 0 (3) L K(1) KO L K(1) L K(2) T MF 0 (3) 3.2 T AF and modular descriptions of ER(n) Behrens and Lawson [4] have constructed p-complete spectra T AF GU (K) of topological automorphic forms associated to unitary similitude groups over Q of signature (1, n 1) and compact open subgroups K GU(A p, ). The spectrum arises from the Shimura stack Sh(K) in the same way T MF arises from the moduli stack of elliptic curves Ell. The spectra T AF GU (K) detect E(n)-local phenomenon the same way T MFp detects E(2)-local phenomenon. The following equivalence makes this precise. L K(n) T AF GU (K) = x Sh [n] (K)(F p) E h Aut(x) n h G(F p/f p) where Sh [n] (K) is the non-empty finite 0-dimensional substack of Sh(K) where the associated formal group has height n and the automophism groups Aut(x) are finite subgroups of S n. Hopkins and Behrens in [3] answers the question For a given prime p and chromatic level n does there exist a pair (GU, K) such that there exists a x Sh [n] (K) for which Aut(x) is the maximal finite subgroup of S n? Since we are interested only in the case p = 2, we quote from Behrens-Hopkins [3] the part relevant to us; Theorem 5. (Behrens-Hopkins) If p = 2 and n = 2 r 1, r > 2, the maximal subgroup G r 1 of S n, as defined by Hewett in [10], can be realized as an automorphism group. (9) From equation (5) we have L K(n) ER(n) = ( ) h G E h F 2 n Z/2 n We want to identify the fibers of M ER(n) over the geometric points of M F G of height > 2 with Shimura varieties in the same way we identified the geometric fibers of M ER(2) over M F G with 5

6 quadrics and supersingular modular curves. In other words we want to identify the chromatic layers of ER(n) with T AF GU(K) for some pair (GU, K) or an appropriate étale extension. This raises the question (i) For the prime 2 and chromatic level n does there exist a pair (GU, K) such that there exists a x Sh [n] (K) for which Aut(x) is the finite subgroup F 2 n Z/2 of S n? (ii) and further, For the prime 2 and chromatic levels k < n does there exist a pair (GU, K) such that there exists a x Sh [k] (K) for which Aut(x) is the finite subgroup F 2 k Z/2 of S k? 3.3 E ring structures on ER(n) An H ring structure is the algebraic structure on the stable homotopy of X implied by the existence of an E ring structure. Such a structure is expressed by Dyer-Lashoff operations on the homotopy of X. The question of whether an H structure on π X comes from an E ring structure on X can be answered by solving an obstruction theory problem formulated by Goerss-Hopkins. Ando-Hopkins-Strickland [1] further relates H ring structures to descent data for level structures on formal groups. Let X be a K(n)-local homotopy commutative ring spectrum. As before, this lets us construct a stack M X along with a natural map to M n, the formal neighborhood of the geometric point in M F G classifying a heigth n formal group. AHS then says that X has an H structure iff there is a lift M n (l) M X M n where M n (l) is the moduli of formal groups of height n with descent data for level structures [20]. The description of the 2-completion of M ER(2) in Theorem 4 enables us to lift the K(1) and K(2) localizations of the structure sheaf to sheaves of H and even E ring spectra (Hopkins-Miller). In order to extend the H structure sheaf from the chromatic localizations to the entire moduli stack we have to show the trans-chromatic map is a map of stacks over M 1 (l). Claim 1. ER(2) is an H -ring spectrum (Ell 0 (3) ss 2 ) ord Forms(G) 2 The existence of E ring structures on ER(2) then rests on Goerss-Hopkins obstruction theory calculations. For ER(n), the proposed decomposition of M ER(n) at the chromatic primes k < n utilizing the Shimura stacks mentioned in section 3.2 should also give us means to lift the transchromatic maps to maps of H and subsequently E -rings. Part II Derived Algebraic Geometry P.May et.al. constructed the category of S-modules as a model for the category of spectra. The commutative ring objects in S-modules, or commutative S-algebras, are simply E rings and the smash product S defined in S-modules makes it into a symmetric monoidal stable model category whose homotopy category is the classical stable homotopy category. This makes way for a version of derived commutative algebra with E ring spectra analogous to the way one does commutative algebra with ordinary commutative rings. 6

7 In derived algebraic geometry the affine schemes are representable moduli functors from the category of E rings to the category of spaces similar to way affine schemes in ordinary algebraic geometry are representable functors from commutative rings to sets. More complex spaces like derived schemes and derived stacks are obtained by considering more general moduli functors satisfying appropriate homotopical geometricity properties. J.Lurie lays the foundations of algebraic geometry with E rings in [17] and [18]. 4 Current Research and Future Plans 4.1 Categories of Modules and their deformations The purpose of this program is to understand the stable -category of modules over a (derived) algebraic stack and the extent to which it is compactly generated. We develop an obstruction theory for lifting compact objects to the stable category of quasicoherent modules over a derived geometric stack X from the category of modules over it s underlying classical stack X cl. The obstructions live in Andre-Quillen cohomology. An explicit description of the space of realizations of a given module over X as a colimit of perfect modules can be given in terms of the k-invariants of a postnikov tower of X and the cotangent complex of the moduli functor of perfect modules. The central question is the following Given a (derived) algebraic stack, when is the derived category of quasi-coherent sheaves compactly generated? When is the derived category of comodules over a Hopf algebroid compactly generated? Neeman in [19] shows the unbounded derived category of modules of a quasi-compact separated affine scheme is compactly generated. Zvi-Francis-Nadler in [5] proves this for quasi-compact derived affine schemes. Neeman s key idea: Let U X be a quasi-compact affine subscheme of X. The induced map D qc (X) D qc (U) is a part of a Verdier fibration of triangulated categories D qc X (U) D qc(x) D qc (U) Let x be an arbitrary object D qc (X), and let u be a perfect complex in D qc (U). Suppose that there is a map u x in D qc (U). Then there exits a perfect complex u in D qc (U) so that the map u u u x lifts to D qc (X). There exists a perfect complex ũ D qc (X) restricting to u u on U and a map ũ x in D qc (X) restricting to u u x on U. We want to extend this idea to derived stacks. Let X be a derived -stack. Let X cl be it s associated classical (non-derived stack). There is a natural map i : X cl X. The induced map on derived categories: D qc (X) D qc (X cl ) Given arbitrary x in D qc (X) and a perfect u in D qc (X cl ), with a map u x. In this paper we find cohomological obstructions for lifting u to a perfect module ũ over X and a map ũ x over X which restricts to u x over X cl In the following theorem QC is the moduli stack of quasi-coherent modules considered as a functor from the -category of connective E ring spectra to the category of stable -categories and QC ω is the moduli of perfect modules. Theorem 6. Let X be a perfect derived algebraic n-stack for some n and let X be a square-zero extension of X. Let x : X QC be a complex of quasi-coherent modules over X and let u : X QC ω be a complex of perfect modules over X, along with a map u x in QC(X). 7

8 There exists an obstruction theory for deforming u to a ũ : X QC ω. The space of deformations is isomorphic to ΩHom OX (α L QC ω, N) with loops based at the trivial derivation. If this space in non-empty and ũ is a deformation of u, then there exists a perfect module y β : X QC ω along with maps β : u y β and y β x in QC(X) such that the triangle commutes in QC(X) u β y β x There is an obstrucion theory for lifting β to β : ũ ỹ β such that ũ ỹ β x is a deformation of α : u x. More precisely, there exists a moduli functor G : Ω u,yβ QC /X X and an cocycle in the Andre- Quillen cohomology α(u, y β ) Hom OX (β L G, N) such that, if α(u, y β ) = 0 there exists a lift β. The space of all such deformations is isomorphic to ΩHom OX (β L G, N) where the loops are based at the trivial derivation. We want to use our obstruction theory to construct examples of derived algebraic stacks whose categories of modules are compactly generated. In order to make full use of our obstruction theory we ll need to understand the cotangent complex of the stack of perfect modules L QC ω. 4.2 Grothendieck duality and Tate spectra The purpose of this program is to formulate a Grothedieck duality for derived schemes and relate the Tate spectrum associated to any G-equivariant spectrum k G to dualizing complexes of derived schemes. A toy example. Let Forms(G) denote the stack of forms of the multiplicative group scheme. There is an equivalence of stacks Forms(G) BAut (G). The structure sheaf of Forms(G) lifts to a sheaf of E rings whose ring of global sections is KO. Call this derived stack Forms(G) top. There is an equivalence Forms(G) top BAut(Spec KU) It s known that the Tate spectrum associated to the Z/2 action on KU is trivial. One might ask, what geometric property of the underlying derived stack Forms(G m ) top is responsible for this? The answer can be expressed in terms of the dualizing complex of this derived stack. More precisely, Proposition 1. The norm map N : KU hz/2 KU hz/2 can be viewed as the composition of the derived Grothendieck Trace and the counit Rf f! KU KU Rf f KU Here f is the map to the final object in derived stacks, f : Forms(G m ) top Spec S and S is the sphere spectrum. 8

9 More generally, for every derived Deligne-Mumford stack X, there is an ordinary Deligne- Mumford stack X cl. The Grothendieck dualizing complex for X cl is obtained from the right adjoint of the canonical map to the final object in algebraic stacks f : X cl Spec Z, whereas the dualizing complex for X is obtained from the right adjoint to canonical map to the final object in derived algebraic stacks f : X Spec S. We study how these two dualizing complexes are related by means of the diagram of topoi X Spec S X cl Spec Z We also define a notion of local cohomology and local homology on -topoi with derived structure sheaf of rings. We show that under certain conditions these -local cohomology and homology functors are representable by certain stable -Koszul complexes. This results in a generalization of the work of Greenlees and May [9] on local cohomology duality in the category of spectra to derived stacks. References [1] M.Ando, M.J.Hopkins, N.Strickland, The sigma orientation is an H map [2] M.F.Atiyah, K-theory and Reality, Quar. J. Math., 17(1966), [3] M.Behrens, M.J.Hopkins, Higher Real K-theories and Topological Automorphic Forms [4] M.Behrens, T.Lawson, Topological Automorphic Forms, Memoirs of the American Mathematical Society. [5] David Ben-Zvi, John Francis, David Nadler, Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry [6] D.Davis, Table of immersions and embeddings of real projective spaces, [7] D.Davis, A strong immersion theorem for real projective spaces, Ann. of Math., 120: , [8] J.P.C.Greenlees, J.P.May, Generalized Tate Cohomology, Mem. Amer. Math. Soc., 113. [9] J.P.C Greenlees, J.P.May, Completions in algebra and topology [10] T.Hewett, Finite subgroups of division algebras over local fields, J.Algebra, vol 173 (1995), no.3, [11] P.Hu, The Ext 0 -term of the Real-oriented Adams-Novikov spectral sequence, Homotopy Methods in algebraic topology, 271, Contemporary Mathematics, , AMS. [12] P.Hu, I.Kriz, Real-oriented Homotopy Theory and an analogue of the Adams-Novikov Spectral Sequence, Topology, 40(2): , [13] I.M.James, On the immersion problem of real projective spaces, Bull. Amer. Math., Soc. 69: , [14] N.Kitchloo, W.S.Wilson, On fibrations related to real spectra. Proceedings of the Nishida fest (Kinosaki 2003), volume 10, Geometry and Topology Monographs, pages , [15] N.Kitchloo, W.S.Wilson, The second real Johnson-Wilson theory and non-immersions of RP n, Homology, Homotopy Appl., 10(3), 2008, [16] N.Kitchloo, W.S.Wilson The second real Johnson-Wilson theory and non-immersions of RP n, Part 2, Homology, Homotopy Appl., 10(3), 2008, [17] Jacob Lurie, Higher Topos Theory, Annals of Mathematics Studies. [18] Jacob Lurie, Higher Algebra [19] A. Neeman The Grothendieck duality via Bousefield s technique s and Brown representability. [20] N.Strickland Formal groups and their subgroups 9