aa r X i v : . [ m a t h . A T ] D ec Structured ring spectra and displays
Tyler LawsonSeptember 25, 2018
Abstract
We combine Lurie’s generalization of the Hopkins-Miller theoremwith work of Zink-Lau on displays to give a functorial construction ofeven-periodic E ∞ ring spectra E , concentrated in chromatic layers 2and above, associated to certain n × n invertible matrices with coeffi-cients in the Witt ring of π ( E ). This is applied to examples relatedto Lubin-Tate and Johnson-Wilson spectra. We also give a Hopf alge-broid presentation of the moduli of p -divisible groups of height greaterthan or equal to 2. One of the most successful methods for understanding stable homotopy the-ory is its connection to formal groups. By work of Quillen, a homotopy com-mutative and associative ring spectrum R has M U -homology
M U ∗ R an alge-bra over the the Lazard ring L , which carries a universal 1-dimensional formalgroup law, and the M U -homology cooperations precisely provide
M U ∗ R withrules for change-of-coordinates on the formal group law.In recent years much study has been devoted to the study of the converseproblem. Given a ring R with a formal group on R , can we reconstruct a ringspectrum E whose associated formal group lifts that on R ? In addition, canmore rigid structure (such as the structure of an E ∞ algebra) be imposed on E ? Can these constructions be made functorial?1n 2005 Lurie announced a theorem that lifts formal groups to E ∞ ringstructures, generalizing the Goerss-Hopkins-Miller theorem [GH04]. The ap-plication of this theorem requires extra data: an extension of the formalgroup to a p -divisible group. In addition, the p -divisible group on R is re-quired to satisfy a universality condition at each point of Spec( R ). Thisspecifically can be applied to produce elliptic cohomology theories and thetheory of topological modular forms, and served as the basis for previousjoint work with Behrens generalizing topological modular forms to moduli ofhigher-dimensional abelian varieties that reach higher chromatic levels [BL].From a geometric point of view these are some of the most natural fami-lies of 1-dimensional p -divisible groups, as the study of p -divisible groupsoriginated with their connection to abelian varieties. However, one of themajor obstructions to understanding the associated cohomology theories isthat such an understanding rests on an understanding of certain moduli ofhigher-dimensional abelian varieties, and in particular their global geometry.This presents barriers both because of the necessary background and becausethese moduli seem to be intrinsically difficult. Moreover, from the point ofview of homotopy theory one does not have as many of the “designer” toolsof the subject [Hop08], which construct spectra with the explicit goal ofcapturing certain homotopy-theoretic phenomena.The aim of this paper is to provide a method for constructing E ∞ ring spec-tra from purely algebraic data. Specifically, theorem 5.2 allows the functorialconstruction of even-periodic E ∞ ring spectra with π = R from the dataof certain n × n invertible matrices with coefficients in the Witt ring of R . This is obtained using Zink’s displays on R [Zin02], which correspondto certain p -divisible groups over Spec( R ). This generalizes the Dieudonn´ecorrespondence between p -divisible groups over a perfect field and their asso-ciated Dieudonn´e modules, and in particular restricts to this structure at anyresidue field k . Due to restrictions on the p -divisible groups constructible bythese displays, the associated spectra are concentrated in chromatic heightgreater than or equal to 2.The layout of this paper is as follows. In section 2 we recall the definitionsof displays and nilpotent displays over a ring R from [Zin02], specificallyconcentrating on those in matrix form. We state the equivalence of categoriesbetween nilpotent displays and formal p -divisible groups on Spec( R ) due toZink and Lau. In section 3 we apply Serre duality to obtain a classification2f p -divisible groups of dimension 1 and formal height ≥ R ),and give a presentation of the moduli of such p -divisible groups by a largeHopf algebroid. In section 4 we study the deformation theory of nilpotentdisplays in matrix form over a ring R and use this to give a criterion forthese to satisfy the universal deformation criterion. Specifically, a display inmatrix form determines a map from Spec( R ) to projective space P h − thatis ´etale if and only if the associated p -divisible group is locally a universaldeformation. In section 5 we recall the statement of Lurie’s theorem andapply it to functorially obtain even-periodic ring spectra associated to certaindisplays. This is applied to construct “almost-global” objects which arerelated to Lubin-Tate and Johnson-Wilson spectra. Finally, in section 6 werelate this to the work of Gross and Hopkins on the rigid analytic period map[HG94]. In the specific case of a Lubin-Tate formal group over R = W ( k ) J u i K ,there is a choice of coordinates such that the above map Spec( R ) → P h − induces a rigid-analytic map that agrees with the Gross-Hopkins period mapmodulo the ideal ( u i ) p .We mention that Zink’s theory of Dieudonn´e displays provides objects equiv-alent to general p -divisible groups over certain complete local rings. Thesecould be applied to produce spectra associated to universal deformations of p -divisible groups of dimension 1 over a field, analogous to Lubin-Tate spec-tra, which are worth study in their own right. However, our goal in thispaper is to allow more global rather than local constructions. Notation . For a ring R , we write W ( R ) for the ring of p -typical Witt vectorsover R . This carries Frobenius and Verschiebung maps f, v : W ( R ) → W ( R ),the Teichm¨uller lift [ − ] : R → W ( R ), and ghost maps w k : W ( R ) → R . Wewrite I R for the ideal of definition v ( W ( R )). We will first briefly recall the classical Dieudonn´e correspondence. Let k be aperfect field. The Dieudonn´e module functor D is a contravariant equivalenceof categories between p -divisible groups over Spec( k ) and finitely generatedfree modules M over the Witt ring W ( k ) that are equipped with operators F and V satisfying the following properties.3 F is semilinear: for x ∈ W ( k ), m ∈ M , F ( xm ) = f ( x ) F ( m ). • V is anti-semilinear: for x ∈ W ( k ), m ∈ M , xV ( m ) = V ( f ( x ) m ). • F V = V F = p .The inverse equivalence can be made explicit. The group schemes Spec( W n )representing Witt vectors assemble into an inductive system using the Ver-schiebung maps, and the colimit (the “unipotent” Witt covectors) is a formalgroup scheme with self-maps F and V and an action of the Witt vectors W ( k ).For p -divisible groups which accept no nontrivial maps from the multiplica-tive group scheme G m , D ( G ) is the set of maps from G to this direct limit.The generators of the Dieudonn´e module provide an embedding of G into aproduct of copies of the Witt covectors. The theory for general G is harder,and the unipotent Witt covectors must be replaced by a suitable completionviewed as a sheaf on commutative k -algebras [Fon77].Classical Dieudonn´e theory also incorporates duality. Each p -divisible group G has a Serre dual G ∨ , whose associated Dieudonn´e module D ( G ∨ ) is iso-morphic to the dual module D ( G ) t = Hom W ( k ) ( D ( G ) , W ( k ))equipped with Frobenius operator V t and Verschiebung operator F t . The covariant Dieudonn´e module of G is the Dieudonn´e module of G ∨ , and thisprovides a covariant equivalence of categories between Dieudonn´e modulesand p -divisible groups.Zink’s theory of displays is a generalization of the Dieudonn´e correspondence.However, over a non-perfect ring R defining the map V is no longer sufficient.This is instead replaced by a choice of “image” of V , together with an inversefunction V − . Definition 2.1 ([Zin02], Definition 1) . A display over a ring R consists ofa tuple ( P, Q, F, V − ), where P is a finitely generated locally free W ( R )-module, Q is a submodule of P , and F : P → P and V − : Q → P areFrobenius-semilinear maps. These are required to satisfy the following: • I R P ⊂ Q ⊂ P , 4 the map P/I R P → P/Q splits, • P is generated as a W ( R )-module by the image of V − , and • V − ( v ( x ) y ) = xF ( y ) for all x ∈ W ( R ), y ∈ P .If p is nilpotent in R , then W ( R ) → R is a nilpotent thickening, and P beinglocally free on W ( R ) implies this is also true locally on Spec( R ). In addition, P/Q and
Q/I R P are locally free R -modules.Therefore, locally on such R we may choose a basis { e i | ≤ i ≤ h } of P suchthat Q = I R P + h e d +1 , · · · , e h i . We refer to h as the height and d as thedimension of the display, and these are locally constant on Spec( R ). As in[Zin02], in such a basis we may define an h × h matrix ( b ij ) as follows: F e j = X b ij e i for 1 ≤ j ≤ d,V − e j = X b ij e i for ( d + 1) ≤ j ≤ h. These determine all values of F and V − : F e j = V − ( v (1) · e j ) = X ( pb ij ) e i for ( d + 1) ≤ j ≤ h,V − ( v ( x ) · e j ) = X ( xb ij ) e i for 1 ≤ j ≤ d, x ∈ W ( R ) . (We remark that the reason this last statement includes an indeterminate x is that Q is often not free as a W ( R )-module.)The image of V − generates all of P if and only if this matrix is invertible,or equivalently that its image under the projection M h W ( R ) → M h R isinvertible.In block form, we may write ( b ij ) as (cid:2) u u (cid:3) , and find that in this basis the functions F and V − are given by blockmultiplication as follows: F (cid:20) xy (cid:21) = (cid:2) u pu (cid:3) (cid:20) f xf y (cid:21) ,V − (cid:20) vxy (cid:21) = (cid:2) u u (cid:3) (cid:20) xf y (cid:21) .
5o aid calculation in section 4 and further, we will refer to the inverse matrix B = ( b ij ) − as a matrix form for the given display. Specifying the matrixform is equivalent to specifying the inverse matrix.If R is a perfect field of characteristic p , the operator V − has a genuineinverse defining an anti-semilinear map V : P → Q ⊂ P , and the mapssatisfy V F = F V = p . Under these circumstances, if B − = (cid:2) u u (cid:3) , B = (cid:20) w w (cid:21) are block forms, then the operators reduce to the classical operators F and V on a Dieudonn´e module which is free over W ( k ) with basis { e i } , and theseoperators have the matrix expression F (cid:20) xy (cid:21) = (cid:2) u pu (cid:3) (cid:20) f xf y (cid:21) , V x = (cid:20) vw f − w (cid:21) ( f − x ) . A map between two displays is a W ( R )-module map P → P ′ preservingsubmodules and commuting with F and V − . If P and P ′ have bases { e i } and { e ′ i } as above, a map φ : P → P ′ takes Q to Q ′ if and only if, whenwe write φ ( e j ) = P φ ij e ′ i , we have φ ij ≡ W ( R ) /I R when i ≥ ( d + 1), j ≤ d . Given an isomorphism φ from ( P, Q, F, V − ) to ( P ′ , Q ′ , F ′ , ( V ′ ) − ),the operators F ′ and ( V ′ ) − are determined uniquely by F ′ = φF φ − andsimilarly for V ′ . If these displays have matrix forms B and B ′ respectively,and we write φ in the block form φ = (cid:20) a vbc d (cid:21) , then we find by direct calculation that the matrix form for the display( P ′ , Q ′ , φF φ − , φV − φ − ) is given by the change-of-coordinates formula B ′ = (cid:20) f a bp · f c f d (cid:21) · B · (cid:20) a vbc d (cid:21) − . (1)The associated map of displays induces the map of modules Q/I R P → Q ′ /I R P ′ given in matrix form by w ( d ).Given a matrix form B , let B be the ( h − d ) × ( h − d ) matrix in R/ ( p ) whichis the image of lower-right corner of B under the projection W ( R ) → R/ ( p ).6e say that the display is nilpotent if the product f n B · · · f B · f B · B is zero for some n ≥
0. (This is independent of the choice of basis, as it isequivalent to the semilinear Frobenius map acting nilpotently on the quotientof Q by ( p ) + I R P .) Here f is the Frobenius on R/ ( p ) applied to each entry ofthe matrix. A general display over R is nilpotent if it is locally nilpotent inthe Zariski topology. We refer to a display on a formal Z p -algebra R = lim R i as nilpotent if its restrictions to the R i are nilpotent. Theorem 2.2 ([Zin02], [Lau08]) . If R is a formal Z p -algebra, there is a(covariant) equivalence of categories between nilpotent displays over R andformal p -divisible groups on Spec( R ) .Under this correspondence, the Lie algebra of the p -divisible group associatedto a display ( P, Q, F, V − ) is the locally free R -module P/Q . -dimensional p -divisible groups Beginning in this section we specialize to the case of topological interest: thetheory of 1-dimensional p -divisible groups. Unfortunately, there are very few1-dimensional p -divisible groups to which Theorem 2.2 applies. The onlyones satisfying the conditions of Lurie’s theorem (see 5.1) are analogues ofthe Lubin-Tate formal groups.However, the category of p -divisible groups has a notion of duality, compati-ble with a corresponding duality on the display. Serre duality is a contravari-ant self-equivalence of the category of p -divisible groups over a general base X that associates to a p -divisible group G of constant height h and dimension d a new p -divisible group G ∨ = Hom( G , G m ) of height h and dimension h − d .This equivalence takes formal p -divisible groups to p -divisible groups with nosubobjects of height 1 and dimension 1. There is a compatible notion of du-ality for displays [Zin02, 1.13, 1.14], sending a display ( P, Q, F, V − ) to a newdisplay ( P t , Q t , F t , V − t ) where P t = Hom( P, W ( R )) and Q t is the submoduleof maps sending Q into I R . The operators F t and V − t are determined by theformula v (( V − t f )( V − x )) = f ( x ) for f ∈ Q t , x ∈ Q .Composing this duality equivalence with Theorem 2.2, we find the following.7 orollary 3.1. If R is a formal Z p -algebra, there is a contravariant equiv-alence of categories between nilpotent displays of height h and dimension ( h − over R and p -divisible groups of dimension and formal height ≥ on Spec( R ) .Under this correspondence, the Lie algebra of the p -divisible group associatedto a display ( P, Q, F, V − ) is the locally free R -module Hom R ( Q/I R P, R ) , andthe module of invariant -forms is isomorphic to Q/I R P .Remark . In particular, the p -divisible group associated to a display inmatrix form B , with basis e . . . e h , has a canonical nowhere-vanishing invari-ant 1-form u which is the image of e h in Q/I R P , and a change-of-coordinatesas in equation (1) acts on u by multiplication by d .We can use this data to given a presentation for the moduli of p -divisiblegroups of height ≥
2. We recall that the Witt ring functor W is representedby the ring W = Z [ a , a , a , . . . ]. Proposition 3.3.
Displays in matrix form of height h and dimension ( h − are represented by the ring A = Z [( β n ) ij , det ( β ) − ] ∼ = W ⊗ h [ det ( β ) − ] The indices range over n ∈ N , ≤ i, j ≤ h . The element det ( β ) is thedeterminant of the matrix (( β ) ij ) .Isomorphisms between displays are represented by the ring Γ = A [( φ n ) ij , det ( φ ) − ] ∼ = A ⊗ W ⊗ h [ det ( φ ) − ] The indices range over n ∈ N , ≤ i, j ≤ h , with the convention that ( φ ) ij is zero if ≤ i ≤ ( h − , j = h . The element det ( φ ) is the determinant ofthe matrix (( φ ) ij ) .The ideal J = ( p, ( β ) hh ) of A is invariant. A display represented by A → R for R a formal Z p -algebra is nilpotent if and only if it factors through thecompletion of A at this ideal.Proof. The ring W ⊗ h represents the functor R
7→ { h × h matrices with entries in W ( R ) } , A represents h × h invertible matrices ( β ij ) = B with entriesin the Witt ring. Similarly, the ring Γ represents pairs of a display in matrixform and an isomorphism to a second display in matrix form, according tothe change-of-coordinates formula (1).The change-of-coordinates formula, mod I R , takes β hh to a unit times itself,so the ideal ( p, ( β ) hh ) is invariant.Suppose R is a Z /p k -algebra and A → R represents a matrix B , which weview as the matrix form of the display. The display is nilpotent as in section 2if and only if ( β ) p n hh · · · ( β ) phh · ( β ) hh is zero in R/ ( p ) for some n . This is equivalent to ( β ) hh being nilpotent in R , and so the display is then nilpotent over R if and only if the map A → R factors through a continuous map A ∧ → R . The corresponding statementfor formal Z p -algebras follows. Corollary 3.4.
The pair ( A, Γ) forms a Hopf algebroid, and the completion ( A ∧ , Γ ∧ ) at the invariant ideal J has an associated stack isomorphic to themoduli of p -divisible groups of height h , dimension , and formal height ≥ .Proof. The existence of a Hopf algebroid structure on ( A, Γ) is a formalconsequence of the fact that this pair represents a functor from rings togroupoids.Let M p ( h ) be the moduli functor of p -divisible groups of height h and di-mension 1. The universal nilpotent display on A ∧ gives rise to a naturaltransformation of functors Spf( A ∧ ) → M p ( h ), and Theorem 3.1 implies thatthe 2-categorical pullback Spf( A ∧ ) × M Spf( A ∧ ) is Spf(Γ ∧ ).The resulting natural transformation of groupoid valued functors from thepair (Spf( A ∧ ) , Spf(Γ ∧ )) to M p ( h ) is fully faithful by Theorem 3.1; it remainsto show that the map from the associated stack is essentially surjective.Given a p -divisible group G on Z /p k -scheme X of height h , dimension 1, andformal height ≥
2, there exists an open cover of X by affine coordinate chartsSpec( R i ) → X and factorizations Spec( R i ) → Spf( A ∧ ) → M p ( h ). It followsthat ( A ∧ , Γ ∧ ) gives a presentation of the moduli as desired. Remark . The Hopf algebroid described is unlikely to be the best possible.Zink’s equivalence of categories shows that locally in the Zariski topology,9 general p -divisible group can be described in this general matrix form; itis possible that there are more canonical matrix forms locally in the flattopology.For example, if h = 2 then a general matrix form (cid:20) α βγ δ (cid:21) (with δ nilpotent) can be canonically reduced to the form (cid:20) γ ′ δ ′ (cid:21) , and by adjoining elements to R to obtain a solution of f t = tγ ′ we obtaina faithfully flat extension in which the matrix can canonically be reduced tothe form (cid:20) δ ′′ (cid:21) . The existence of canonical forms at higher heights, as well as more explicitdetermination of the associated Hopf algebroids, merits further study.
In this section we briefly study the deformations of displays in matrix form.We note that [Zin02] has already fully interpreted the deformation theoryof displays in terms of the deformations of the Hodge structure
Q/I R P .The approach there is specifically in terms of fixing deformations of F and V − and classifying possible deformations of the “Hodge structure” Q ; forcalculational reasons we will instead fix the deformation of Q and studypossible deformations of the operators.Let GL h ⊂ A h Z p be the group scheme of h × h invertible matrices. Thereis a projection map Spec( A ) → GL h classifying the map that sends a dis-play represented by a matrix B ∈ GL h ( W ( R )) to the matrix w ( B ). Let p : GL h → P h − be the projection map sending a matrix ( β ij ) to the pointwith homogeneous coordinates [ β h : β h : . . . : β hh ].10 heorem 4.1. Let k be a field, and Spec( k ) → Spec( A ) ⊂ Spec( W ) h be apoint that defines a nilpotent display over a field k with matrix form B . Thenthe composite map Spec( A ) → GL h → P h − identifies the set of isomorphismclasses of lifts of this display to k [ ǫ ] / ( ǫ ) with the tangent space of P h − k at Spec( k ) .Proof. Suppose we are given the matrix form B of a display over k , and writein block form B = (cid:20) α βγ vδ (cid:21) ∈ GL h ( W ( k )) . (Nilpotence of the display forces the final entry to reduce to zero in k .) Givenany lift of the display on k to a display on k [ ǫ ] /ǫ , Nakayama’s lemma impliesthat any lift of the basis of the display gives a basis of the lift, whose matrixform is a lift of the matrix form over k .Lifts of the matrix form B to k [ ǫ ] /ǫ are precisely of the form B + s where s isa matrix in W ( ǫk ), as any such element is automatically invertible. Applyingthe change-of-basis formula (1), we find that the lifts isomorphic to this oneare of the form (cid:18) I + (cid:20) f a bp · f c f d (cid:21)(cid:19) ( B + s ) (cid:18) I + (cid:20) a vbc d (cid:21)(cid:19) − for a , b , c , d matrices in W ( ǫk ). The ideal W ( ǫk ) is square-zero and annihi-lated by f , so this reduces to B + (cid:18) s + (cid:20) b (cid:21) B − B (cid:20) a vbc d (cid:21)(cid:19) . The space of all isomorphism classes of lifts is therefore the quotient of M h ( W ( ǫk )) by the subspace generated by elements of the form B (cid:18) − (cid:20) a c d (cid:21) − (cid:20) vb (cid:21) + B − (cid:20) bγ
00 0 (cid:21)(cid:19) . As B is invertible, this subspace consists of the h × h matrices in W ( ǫk )whose final column is congruent to a multiple of the final column of B (mod I R ).However, this coincides with the kernel of the (surjective) map on relativetangent spaces Spec( A ) → P h − over Z p at Spec( k ).11 orollary 4.2. Suppose that we are given a display ( P, Q, F, V − ) of height h and dimension ( h − over a formal Z p -algebra R in matrix form ( β ij ) . Then R gives a universal deformation of the associated -dimensional p -divisiblegroup at all points if and only if the map Spec( R ) → P h − Z p given by [ w ( β h ) : w ( β h ) : · · · : w ( β hh )] is ´etale.Proof. The ring R gives a universal deformation of the associated p -divisiblegroup at a residue field Spec( k ) = x if and only if the completed local ring R ∧ x is mapped isomorphically to the universal deformation ring of the p -divisiblegroup, which is a power series ring W ( k ) J u , . . . , u h − K . In particular, R givesa universal deformation at x if and only if: • R is smooth over Z p at x , and • the relative tangent space of R over Z p at x , which is the set of liftsSpec( k [ ǫ ] /ǫ ) → Spec( R ), maps isomorphically to the set of lifts of thedisplay to k [ ǫ ] /ǫ .However, because P h − Z p is smooth over Spf( Z p ), the map Spf( R ) → P h − Z p is´etale at Spec( k ) if and only if: • R is smooth over Z p at x , and • the relative tangent space of R over Z p at x maps isomorphically to thetangent space of P h − k at x .By the previous theorem, these conditions coincide. We recall a statement of Lurie’s theorem (as yet unpublished) from [Goe09].We write M p ( h ) for the moduli of p -divisible groups of height h and dimen-sion 1, M F G for the moduli of 1-dimensional formal groups, and M p ( h ) →M F G for the canonical map representing completion at the identity.12 heorem 5.1 (Lurie) . Let X be an algebraic stack, formal over Z p , equippedwith a morphism X → M p ( h ) classifying a p -divisible group G . Suppose that at any point x ∈ X , thecomplete local ring of X at x is mapped isomorphically to the universal de-formation ring of the p -divisible group at x . Then the composite realizationproblem X → M p ( n ) → M F G has a canonical solution; that is, there is a sheaf of E ∞ even weakly periodic E on the ´etale site of X with E locally isomorphic to the structure sheaf andthe associated formal group isomorphic to the formal group G for . The spaceof all solutions is connected and has a preferred basepoint. We may then combine this result with Corollary 4.2 to produce E ∞ -ringspectra associated to schemes or stacks equipped with an appropriate coverby coordinate charts carrying displays. Rather than stating in maximal gen-erality, we have the following immediate consequence. Theorem 5.2.
Suppose R is a formal Z p -algebra and B is the matrix formof a nilpotent display over R , with associated p -divisible group G . If theassociated map Spf( R ) → P h − Z p is ´etale, then there is an E ∞ even-periodic E = E ( R, B ) with E ∼ = R , E ∼ = Q/I R P , and formal group isomorphic tothe formal group G for .Given matrix forms B , B ′ of such displays over R and R ′ respectively, g : R → R ′ a ring map, and φ a change-of-coordinates from g ∗ B to B ′ asin equation (1), there exists a map of E ∞ ring spectra E ( R, B ) → E ( R ′ , B ′ ) inducing g on π and lifting the associated map G for → ( G ′ ) for . This con-struction is functorial in B ′ as an object over B .Proof. The existence of E ( R, B ) follows from Theorem 5.1 and Corollary 4.2.Given any such map Spf( R ′ ) → Spf( R ), the maps Spf( R ′ ) → P h − andSpf( R ) → P h − both being ´etale forces Spf( R ′ ) to be ´etale over Spf( R ), andhence represents an element of the ´etale site. Lurie’s theorem then impliesthat the map lifts.We recall from remark 3.2 that in matrix form there is a nowhere vanishing1-form u on the cotangent space Q/I R P of the formal group G for , implying13hat the tensor powers of the cotangent bundle are all free. This implies thestrictly even-periodic structure on E .Associated to a display not in matrix form, we would instead obtain a weaklyeven-periodic ring spectrum whose 2 k ’th homotopy group is the k ’th tensorpower of the locally free module Q/I R P of invariant 1-forms. Example . Let h ≥ R = ( Z [ u , · · · , u h − ]) ∧ ( p,u ) . Then we have thefollowing display over R : · · · u h − ]0 1 0 0 [ u h − ]... ...0 0 0 0 [ u ]0 0 0 · · · u ] (2)Here [ x ] denotes the Teichm¨uller lift of the element x . The associated mapSpf( R ) → P h − is the map [1 : u h − : · · · : u ] , which is the completion of a coordinate chart of P h − at the ideal ( p, u ) andis therefore ´etale. Because this display is given in matrix form, there is acanonical non-vanishing invariant 1-form u and the resulting spectrum hashomotopy groups ( Z [ u , . . . , u h − ]) ∧ ( p,u ) [ u ± ] . Here | u | = 2. This represents a “partial” thickening of a Lubin-Tate spectrumof height h to a global object. Example . Let S = W ( F p h )[ u , · · · , u h − ] ∧ ( p,u ) , with the same display asgiven in the previous example. Let ζ be a primitive ( p h − F p h with Teichm¨uller lift [ ζ ]. Then the group F × p h = h ζ i acts on S withgenerator ζ acting by ζ · ( u , . . . , u h − ) = ( ζ − p u , ζ − p u , . . . , ζ − p h − u h − ) , ζ lifts to an action on the display via the change-of-coordinates matrix [ ζ p h − ] 0 0 00 [ ζ p h − ] · · · · · · [ ζ p ] 00 0 0 [ ζ ] . This acts on the invariant 1-form u by multiplication by ζ . This gives awell-defined action on the p -divisible group associated to the display.Similarly, there is a Galois automorphism σ of S which acts by the Frobe-nius on W ( F p h ) and acts trivially on the generators u i . This automorphismpreserves the display, and satisfies the relations σ h = id , σζ = ζ p σ . Togetherthese give an action of G = ( F × p h ⋊ Gal F ph / F p ) on Spf( S ) which lifts to anaction on the associated p -divisible group.Theorem 5.2 implies that this lifts to an associated spectrum with an actionof G . (More generally, there is an associated sheaf of E ∞ -ring spectra onthe quotient stack [Spf( S ) //G ].) The canonical invariant 1-form u is actedon by ζ by left multiplication, and acted on trivially by σ . The homotopyfixed point object (which is the global section object of the quotient stack)has homotopy groups (cid:0) W ( F p h )[ u , . . . , u h − ] ∧ ( p,u ) [ u ± ] (cid:1) G ∼ = Z [ v , . . . , v h − , v ± h ] ∧ ( p,u ) . Here v i = u p i − u i has degree 2 p i −
2. This has the homotopy type of aJohnson-Wilson spectrum completed at the height ≥ In section 4 we associated to each display in matrix form over R a mapSpec( R ) → P h − . In this section we will briefly relate this, in a specificchoice of coordinates, to the rigid analytic period map constructed by Grossand Hopkins [HG94]. We first recall the construction of this period map.Let k be a perfect field of characteristic p carrying a formal group law of finiteheight h , and K = W ( k ) ⊗ Q . Associated to this formal group law there is15 universal deformation to a formal group law over the Lubin-Tate ring R ∼ = W ( k ) J u , . . . , u h − K . The formal group law gives rise to a Dieudonn´e crystalon R of rank h , and for a sufficiently large ring R ⊂ S ⊂ K J u , . . . , u h − K the horizontal sections of this crystal on Spec( S ) form a vector space V over K of rank h , containing a family of rank ( h − S ) → P ( V )sends points of this rigid-analytic extension of Lubin-Tate space to the Hodgestructure at that point.Let R = W ( k ) J u , . . . , u h − K , equipped with the ring homomorphism σ : R → R which acts by the Frobenius on W ( k ) and sends u i to u pi ; this is a lift of theFrobenius map on R/p , and provides a splitting R → W ( R ) commuting withthe Frobenius. We write J for the ideal ( u , . . . , u h − ). This ring R carries thedisplay of equation 2. This display is a universal deformation of a p -divisiblegroup of height h over the residue field k , and so the associated p -divisiblegroup on Spf( R ) is a universal deformation of the formal group on Spec( k ).The map Spec( R ) → P h − of section 4 is the map [1 : u h − : · · · : u ].To translate this into the (covariant) language of the Gross-Hopkins map,we first convert the display into the dual, covariant, display, which is a free W ( R )-module P t with dual basis e , . . . , e h and Hodge structure Q t ⊂ P t generated by e , . . . , e h − . (This Hodge structure is determined by the linearfunctional P a i e i a h .) A straightforward calculation finds that the matrixof F t with respect to this basis is p p · · · p
00 0 0 0 1 p p [ u h − ] p [ u h − ] · · · p [ u ] [ u ] . (3)As in [Zin02], there is a Dieudonn´e crystal associated to this display. Thedata of such a crystal produces: a module M = R ⊗ W ( R ) P t , a Hodge struc-ture Q t /I R P t ⊂ M , a σ -semilinear Frobenius map F : M → M , and a σ -antisemilinear map V : M → M satisfying F V = V F = p . Associated tothis data there is a unique connection ∇ : M → M ⊗ Ω R/ W ( k ) for which F and V are horizontal. 16et Ψ be the matrix of F in this basis (the reduction mod I R of equation3) and Ψ the image given by sending u i to 0. There exists a deformation ofthe basis { e i } to a basis of horizontal sections for this connection; i.e., thereis a matrix A ∈ GL h ( K J u , . . . , u h − K ) whose columns are annihilated by ∇ satisfying A ≡ I mod J . The expression in this basis for the linear functionalcutting out the Hodge structure is given by the last row of A .As F is horizontal, it takes horizontal sections to horizontal sections, andhence applying the Frobenius to the columns of A gives a linear combinationof the combinations of A . This implies that Ψ A σ = AB for some matrix B with coefficients in K . Reducing mod J we find that B = Ψ. Thus, such amatrix must satisfy A = Ψ A σ Ψ − . (Taking a limit of iterative substitutionsrecovers A itself.)As A ≡ I mod J , A σ ≡ I mod J p . Therefore, we find A ≡ ΨΨ − mod J p .Applying this to equation 3, we find A ≡ · · · u h − u h − · · · u mod J p . As a result, the Gross-Hopkins map classifying the Hodge structure is congru-ent to [ u h − : · · · : u : 1] mod J p . With an appropriate choice of coordinateswe can then regard the map defined in section 4 as an approximation of theGross-Hopkins map. References [BL] Mark Behrens and Tyler Lawson,
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