KK3 spectra
Markus SzymikJanuary 2009
The notion of a K3 spectrum is introduced in analogy with that of an ellipticspectrum and it is shown that there are “enough” K3 spectra in the sensethat for all K3 surfaces X in a suitable moduli stack of K3 surfaces there isa K3 spectrum whose underlying ring is isomorphic to the local ring of themoduli stack in X with respect to the etale topology, and similarly for thering of formal functions on the formal deformation space. Stable homotopy theory may be defined as the theory of the sphere spectrum S and its homotopy groups π ∗ S , the stable homotopy groups of spheres. But, just asnumber theory may be defined as the theory of the integers, in practice that entailsthe study of other structures such as fields, both global and local, Galois groups,their representations, deformations, and many other things which – typically – areharder to construct, but easier to understand.In stable homotopy theory, some of the auxiliary spectra encountered are theEilenberg-MacLane spectra such as H Q , leading to rational homotopy theory,topological K-theory spectra KU and KO, elliptic spectra and TMF, and many a r X i v : . [ m a t h . A T ] F e b ore. The “chromatic” hierarchy of these is organised by the complex bordismspectrum MU, which is relevant for the fact that π ∗ MU represents graded formalgroup laws rather than for its relationship with manifolds. For example, H Q cor-responds to the additive formal group law, KU to the multiplicative formal grouplaw, and elliptic spectra to formal group laws coming from elliptic curves.There have been efforts to extend the connection between elliptic curves and spec-tra to other geometric objects such as curves of higher genus, see [13] and [29],and abelian varieties of higher dimension, see [5]. The aim of this report is topresent some other part of arithmetic geometry which may be lurking behind thenext layers of the chromatic hierarchy: K3 surfaces and their corresponding K3spectra.It is not a new idea to use K3 surfaces, and more generally Calabi-Yau varieties, asgeneralisations of elliptic curves, not even in topology. I first read about the ideaof cohomology theories related to K3 surfaces in Thomas’ writings [35] and [34],where he refers to Morava. The latter was so kind to share his notes [25] basedthen again on a lecture of Hopkins. See also [8] for a recent contribution to theproblem of finding a differential geometric (rather than homotopical) descriptionof K3 cohomology.In recent years a lot of work has been done, on the side of algebraic topology justas well as on the side of arithmetic geometry, and suggests to have a fresh lookat this connection. For example, the papers [10] and [28] give detailed accountsof the height stratification on the moduli stacks of K3 surfaces in prime charac-teristic. And there are now various different methods to produce new species inthe “brave new world” of highly structured ring spectra and localisations thereof,see [30], [12], and [22] to name a few. How far these apply to K3 spectra is theauthor’s work in progress and will be addressed elsewhere [33].The purpose here is to show that there are “enough” K3 spectra in the sense thatfor all K3 surfaces X in a suitable moduli stack of K3 surfaces there is a K3 spec-trum whose underlying ring is isomorphic to the local ring of the moduli stack in X The aim of this section is to review the geometry of K3 surfaces as far as neededin the rest of the text. All fields will be assumed perfect from now on.
An elliptic curve over a field k is a smooth proper curve X such that the canonicalbundle ω X = (cid:86) ( Ω X ) is trivial, with a chosen base point. The base point can beused for various purposes. It can be used to impose an abelian group structure onthe curve. And it can be used to define a Weierstrass embedding into the projectiveplane.One dimension higher, there are two kinds of smooth proper surfaces X such thatthe canonical bundle ω X = (cid:86) ( Ω X ) is trivial: abelian surfaces and another class ofsurfaces which satisfy the additional condition H ( X , O X ) =
0: the
K3 surfaces .See the textbooks [4], Chapter VIII, and [3], Chapter 10, for the general theoryof K3 surfaces.One generalisation of this to even higher dimensions would be Calabi-Yau n -folds,which are defined by the triviality of the canonical bundle ω X = (cid:86) n ( Ω X ) and the3anishing of H i ( X , O X ) for all 1 (cid:54) i (cid:54) n −
1. See [15] and [11], for example. Apartfrom a few side remarks, these will play no rˆole in the following.The Euler characteristic of K3 surfaces is 24, so that these will never be abeliangroups. However, every K3 surface X can be embedded in some projective spaceby means of an ample line bundle L on X ; the choice of such an L is called a polarisation of X . This corresponds to the choice of the base point for ellipticcurves. The self-intersection of L is called the degree of the polarisation; this willalways be an even integer 2 d for some d (cid:62) It will be useful to bear the following two classes of examples in mind. For these,the base field k needs to be of characteristic char ( k ) (cid:54) = Example.
The most famous example for a K3 surface is the
Fermat quartic defined by the equation T + T + T + T = P . More generally, all smooth quartics in P define K3surfaces. (See [4], Example VIII.8, and 10.4 in [3].) This is analogous to the factthat smooth cubics in P define elliptic curves. However, not every K3 surfacecan be embedded into P . Example. If A is an abelian surface, the inversion [ − ] : A → A is an involution,and it extends to the blow-up of A along the sixteen 2-torsion points. The quotientof the resulting free action on the blow-up by this involution turns out to be a K3surface, called the Kummer surface of A . See [4], Example VIII.10, and 10.5in [3]. 4 .3 Moduli of K3 surfaces Given the two classes of examples above, one might want to get an overviewof all K3 surfaces and how they vary in families, leading to the question oftheir moduli. While the moduli stack of elliptic curves is only 1-dimensional,the moduli stack of Kummer surfaces is 3-dimensional, and that of quarticsin projective space is 19-dimensional. In general, one may consider the mod-uli stack M d of polarised K3 surfaces of fixed degree 2 d . See [31]. That isa separated Deligne-Mumford stack of finite type, which is smooth of dimen-sion 19 over Z [ / d ] . The objects of M d should be the pairs ( X , L ) , where X is a K3 surface over some Z [ / d ] -scheme S , and L is a polarisation for X ; mor-phisms ( X (cid:48) , L (cid:48) ) → ( X , L ) are pullback diagrams X (cid:48) ξ (cid:47) (cid:47) (cid:15) (cid:15) X (cid:15) (cid:15) S (cid:48) σ (cid:47) (cid:47) S such that ξ ∗ L = L (cid:48) . But, in order to circumvent difficulties with etale descent, onebuilds this into the definition of M d by allowing for objects which become K3surfaces only after an etale base change. Again, see [31] for details. The differenceis clearly irrelevant over the spectrum of a strictly henselian local ring. Example.
The case d = P . (See [26] for othersmall d .) The space of quartics in P is a projective space of dimensiondim H ( P ; O P ( )) − = (cid:18) (cid:19) − = . The discriminant locus ∆ ⊂ P corresponding to the singular quartics is a hyper-surface. The Veronese embedding shows that the open complement P \ ∆ isaffine. The universal K3 surface over it has a canonical polarisation given by therestriction of O P ( ) . There results a morphism P \ ∆ −→ M . P may acquire ordinary double points. However,the map factors over the quotient stack (cid:0) P \ ∆ (cid:1) // PGL ( ) of the affine scheme P \ ∆ by the action of the affine group PGL ( ) which actsby change of co-ordinates, and links M to a stack associated to a Hopf algebroid.Finally, it should be pointed out that the moduli problem of (polarised) Calabi-Yau n -folds is much more complicated if n (cid:62)
3. There are 3-folds with differentHodge numbers, so that – globally – there will be many components. The Hodgenumbers also show that – locally – the obstruction space for deformations neednot be zero (as it is for K3 surfaces). And in fact there are examples of Calabi-Yau 3-folds which do not lift to characteristic zero, see [15].
The aim of this section is to review the arithmetic of K3 surfaces as far as neededin the rest of the text.
Let R be a local ring with residue field k . A functor Γ from the category of artinianlocal R -algebras with residue field k to the category of abelian groups is a (1-dimensional, commutative) formal group if the underlying functor to the categoryof sets is (pro-)representable, formally smooth, and 1-dimensional. However, theactual representation is not part of the data. The choice of a representation (nec-essarily by a power series ring R [[ T ]] ) yields a co-ordinate for Γ . With respect6o this co-ordinate the formal group is described by the formal group law , whichis a power series in R [[ T , T ]] . See the lecture notes [9] and [21] for backgroundinformation. There are ways to globalise the notion of a formal group in order towork over arbitrary rings or even base schemes. In that case, co-ordinates neednot exist globally, only locally. This generality will not be needed here. Let X be an elliptic curve over a field k . Its Picard group Pic X ( k ) is the groupof isomorphism classes of line bundles on X . As it stands, this is just a set withan abelian group structure, but with some work it can be made into an algebraicgroup Pic X , whose component of the identity is isomorphic to X by a map whichsends the base point to the unit. The formal completion (cid:98) Pic X is a 1-dimensionalformal group. It has been those formal groups that arose the interest of algebraictopologists in elliptic curves, see [18] for example.For a K3 surface X , the Picard group is 0-dimensional, so that its formal Picardgroup is trivial, but the isomorphismPic X ( k ) ∼ = H ( X ; G m ) suggests that one should consider H ( X ; G m ) instead. As for a geometric inter-pretation, there is an isomorphismH ( X ; G m ) ∼ = Br X ( k ) with the Brauer group Br X ( k ) of X . Although the notation may suggest that, itturns out that these are not the k -valued points of an algebraic group, let aloneone whose component of the identity is isomorphic to X . However, Artin andMazur [2] have shown that the functor A (cid:55)−→ Ker (cid:0) H ( X × Spec ( A ) ; G m ) → H ( X ; G m ) (cid:1)
7n artinian k -algebras A with residue field k is (pro-)representable and formallysmooth. This is referred to as the formal Brauer group (cid:98) Br X of X . Its dimensionis 1; in fact Lie ( (cid:98) Br X ) ∼ = H ( X ; G a ) ∼ = H ( X , O X ) (3.1)gives the Lie algebra. These formal groups are the reason for the topologists’interest in K3 surfaces.As a remark, the results of Artin and Mazur are general enough to show that everyCalabi-Yau n -fold gives rise to a 1-dimensional formal group, using the sameconstruction based on H n et ( X ; G m ) . Over rings of prime characteristic, formal groups have an associated height, whichcan be a positive integer or infinite. It counts the maximal number of Frobeniusmorphisms over which multiplication by p factors. It is the most basic invariantof formal groups in prime characteristic, and over separably closed fields it is infact their only invariant. See [20] and [9], III.2. Over local rings, the height isdominated by the height over the residue field.The multiplicative formal group ˆ G m has height 1, whereas the additive formalgroup ˆ G a has infinite height. The height of the formal Picard group of an ellip-tic curve is either 1 or 2, in which case the elliptic curve is called ordinary orsupersingular, respectively. Example.
It is known that the height of the formal Brauer group of the Fermatsurface is 1 in the case when p ≡ p ≡ (cid:48) ( T ) = ∞ ∑ n = ( n ) ! ( n ! ) T n Example.
The height of formal Brauer groups of Kummer surfaces are 1, 2 orinfinite, see [11]. Specifically, to get height 2, take the Kummer surface of aproduct of an ordinary and a supersingular elliptic curve.In general, the height of the formal Brauer group of a K3 surface will be at most 10or infinite, and all these possibilities actually occur. There are explicit examplesknown for all heights except for 7, see [38] and [14]. A K3 surface is called ordinary or supersingular (in the sense of Artin, see [1]) if the height is 1 orinfinite, respectively.The pattern continues: there is a bound on the height of the Artin-Mazur formalgroup associated to a Calabi-Yau n -fold, see [11] again, but note that the boundgiven there is not sharp even in the case of K3 surfaces.Now let p be a prime that does not divide 2 d , and let M d , p be the base change of M d from Z [ / d ] to Z / p . For all h (cid:62)
1, there is a closedsubstack M d , p , h of M d , p defined by those K3 surfaces which have a formal Brauer group of heightat least h . These define the height stratification M d , p = M d , p , ⊇ M d , p , ⊇ M d , p , ⊇ . . . (3.2)of M d , p , see [10] and [28]. By what has been said above, the chain stabilisesat h =
11, with M d , p , being the supersingular locus, at least set-theoretically.Its open complement M fin2 d , p , h M d , p , h is the moduli stack of polarised K3 surfaces of finite height atleast h in characteristic p . These are known to be smooth of dimension 20 − h for h = , . . . ,
10, and empty for h =
11. The substack M fin2 d , p , h + of M fin2 d , p , h isdefined by the vanishing of a section of an invertible sheaf of ideals, see again [10]and [28]. It follows that these sections define a regular sequence locally on themoduli stack. The aim of this section is to define the notion of a K3 spectrum in analogy withelliptic spectra.
Let E be a ring spectrum in the weak “up to homotopy” sense. In other words E is just a monoid with respect to the smash product in the homotopy category ofspectra. Nowadays, there are various models for the category of spectra whichcome with a well-behaved smash product before passage to the homotopy cate-gory, so that it also makes sense to study monoids in a category of spectra itself.See [23] for a comparison of some of the most common models. In this writing,the stronger notion will not be discussed.The ring spectrum E is called even if its homotopy groups are trivial in odddegrees, and periodic if the multiplication induces isomorphisms π m E ⊗ π E π n E ∼ = −→ π ( m + n ) E for all integers m and n . An even periodic ring spectrum has an associated formalgroup Γ E , represented by the ring π E B T = E B T . This ring will be interpretedas the ring of functions on the formal group over π E .10he projection from B T to a point induces the structure map π E → π E B T of the π E -algebra. The unit ∗ = B1 → B T of the group T induces a co-unit π E B T → π E . This is to be interpreted as the evaluation map at the ori-gin. Its kernel I is the ideal of functions vanishing at the origin. The cotangentspace I / I at the origin can then be identified with π E , so that its dual π − E isthe Lie algebra Lie ( Γ E ) ∼ = π − E (4.1)of Γ E . Recall, or see [16] for example, that an elliptic spectrum is a triple ( E , X , ϕ ) con-sisting of an even periodic ring spectrum E , an elliptic curve X over π E , andan isomorphism ϕ of the formal Picard group of X with the formal group Γ E associated to E over π E . There may be reasons to allow X to be some sort ofgeneralised elliptic curve, but these will not matter here. Definition. A K3 spectrum is a triple ( E , X , ϕ ) consisting of an even periodicring spectrum E , a K3 surface X over π E , and an isomorphism ϕ of the formalBrauer group of X with the formal group Γ E of E .The purpose of the rest of this text is to show how one obtains examples of K3spectra. Let X be a K3 surface over a field of characteristic 0, so that the formal Brauergroup of the K3 surface X is automatically additive. By (3.1), its Lie algebra11s Lie ( (cid:98) Br X ) ∼ = H ( X ; G m ) , and the usual logarithm G m ∼ = G a induces an isomor-phism H ( X ; G m ) ∼ = H ( X ; G a ) . The latter group is just H ( X ; O X ) which calcu-lates the Lie algebra of (cid:98) Br X : Lie ( (cid:98) Br X ) ∼ = H ( X ; O X ) . Let E be the Eilenberg-MacLane spectrum for the (graded) canonical ring of X ,so that π n E = H ( X ; ω ⊗ nX ) . As has been noted above, see (4.1), the Lie algebra of Γ E is π − E , which is thedual of π E : Lie ( Γ E ) ∼ = H ( X ; ω X ) ∨ . Let ϕ be the unique isomorphism between (cid:98) Br X and Γ E such that the inducedisomorphism on Lie algebras is Serre dualityH ( X ; O X ) ∼ = H ( X ; ω X ) ∨ . Then ( E , X , ϕ ) is a K3 spectrum.The next section will explain how to obtain examples in non-trivial characteristic. The aim of this section is to show that there are “enough” K3 spectra. Theirexistence will be a consequence of the regularity of the height stratification andLandweber’s exact functor theorem, as suggested in [25]. However, in view oflater applications to highly structured K3 spectra, extra care will be taken to workover torsion-free rings instead of rings of characteristic p .12 .1 The exact functor theorem Let us first recall Landweber’s theorem from [17]. See also [24].Let Γ be a formal group over a local ring R . Choose a co-ordinate T and let [ p ]( T ) = a T + a T + · · · + a p − T p + a p T p + + . . . be the p -series of Γ with respect to that co-ordinate. For an integer n (cid:62) I p , n be the ideal generated by the first p n − coefficients. It is known that this does notdepend on the co-ordinate. There results an increasing sequence0 = I p , ⊆ I p , ⊆ I p , ⊆ . . . (5.1)of ideals. Note that I p , is the ideal ( p ) generated by a = p . For n (cid:62)
1, the sur-jection R / p → R / I p , n corresponds to the closed subscheme of Spec ( R / p ) wherethe height of Γ is at least n . It is also known that there is a sequence ( v n | n (cid:62) ) of elements in R such that I p , n + is generated by I n , p and v n . The formal group Γ is called regular at p if ( v n | n (cid:62) ) is a regular sequence in R . This does notdepend on the choice of the v n . For example, if p is invertible in R , then Γ isautomatically p -regular.The graded formal group law over the graded ring R [ u ± ] which is defined by Γ is classified by a graded morphism MU ∗ → R [ u ± ] . Landweber’s theorem statesthat the functor X (cid:55)→ MU ∗ X ⊗ Γ MU ∗ R [ u ± ] is a homology theory if Γ is p -regular for all primes p and the sequence (5.1)eventually stabilises. This homology theory is representable by an even periodicring spectrum E which has π E ∼ = R and Γ E ∼ = Γ .13 .2 The statement As before, let us fix an integer d (cid:62)
1, and consider the moduli stack M d ofpolarized K3 surfaces of degree 2 d over Z [ / d ] . Suppose that X is a K3 surfaceover an affine scheme Spec ( R ) on which 2 d is invertible. Then X is classified bya map X : Spec ( R ) −→ M d . The following theorem will assert that – under certain conditions – the associatedformal Brauer group is Landweber exact. Before giving the precise statement, letme motivate the choice of hypotheses.First of all, flatness of the map classifying X will be indispensable for the argu-ment. This will imply that R is flat over Z [ / d ] , so that R is torsion-free, asrequired by Landweber’s theorem. The primes which divide 2 d are automaticallyunits in R . If R is a Q -algebra, there is no need to invoke Landweber’s theorem toget examples, see the previous Section 4.3. On the other hand, the height stratifi-cation does not vary nicely from prime to prime, see Section 3.3. Therefore, wewill concentrate on one prime and assume that the ring R is a local Z ( p ) -algebrafor some prime p which does not divide 2 d . Recall that “local” means that R has aunique maximal ideal m , and that this maximal ideal contains p , so that the residuecharacteristic of R is p . There may be other settings which make the followingargument work, but this one has its merits, as will hopefully become clear in duecourse. Theorem 1.
Let R be a noetherian local Z ( p ) -algebra for some prime p whichdoes not divide d. Let X be a polarised K3 surface of degree d over R such thatthe height of the closed fibre is finite. If the mapX : Spec ( R ) −→ M d (5.2) classifying X is flat, then the formal Brauer group (cid:98) Br X is Landweber exact, so thatthere is an even periodic ring spectrum E with π E ∼ = R and Γ E ∼ = (cid:98) Br X . By assumption on the ring R , all primes different from p are invertible, so that theformal group will automatically be q -regular for all q (cid:54) = p . It remains to show thatit is p -regular as well. Lemma 2.
The prime p is a nonzerodivisor on R.Proof.
As has already been remarked before, this follows from the flatness of R as an algebra over Z [ / d ] .Let us now reduce everything modulo p . The reduction X / p of X modulo p isclassified by a morphism X / p : Spec ( R / p ) −→ M d , p . (5.3) Lemma 3.
If the map (5.2) classifying X is flat, so is the map (5.3) classify-ing X / p.Proof. Consider the following diagram.Spec ( R / p ) X / p (cid:15) (cid:15) (cid:47) (cid:47) Spec ( R ) X (cid:15) (cid:15) M d , p (cid:15) (cid:15) (cid:47) (cid:47) M d (cid:15) (cid:15) Spec ( Z / p ) (cid:47) (cid:47) Spec ( Z [ / d ]) M d , p . The outer rectangle isa pullback by elementary algebra. It follows that the upper square is a pullback aswell. The result follows by base change for flat morphisms.Recall from 3.3 that the finite height part M fin2 d , p = M fin2 d , p , denotes the complementof the supersingular locus M d , p , in the moduli stack M d , p = M d , p , . Lemma 4.
The map (5.3) classifying X / p factors over the open substack M fin2 d , p .Proof. By assumption, the height of the closed fibre X / m of X / p is finite, andthis height bounds the height of X / p .Let us now see how the height stratification (3.2) from Section 3.3 manifests in thiscontext. The pullbacks of the ideal sheaves which cut out the height strata M d , p , h in M d , p = M d , p , along the morphism (5.3) give rise to a sequence of ideals0 = J ⊆ J ⊆ J ⊆ · · · ⊆ R / p , such that I p , h is the pre-image of J h along R → R / p . Lemma 5.
If the height of the closed fibre is h, one has J h (cid:54) = R / p and J h + = R / p,as well as I p , h (cid:54) = R and I p , h + = R.Proof.
The results for the J imply those for the I , so we only need to prove thefirst ones.The ideal J h cuts out the locus in Spec ( R / p ) where the height is at least h , and wehave to show that this locus is not empty. But it contains the closed point.Similarly, the ideal J h + cuts out the locus in Spec ( R / p ) where the height is atleast h +
1, and we have to show that this locus is empty. But it is closed and doesnot contain the closed point Spec ( R / m ) by assumption. A geometric interpreta-tion of Nakayama’s Lemma gives the result: a closed subset of a spectrum of anoetherian local ring which does not contain the closed point is empty.16he preceding lemma implies that v h is a unit in R , and, unless the closed fibre isordinary, we are left to deal with the v , . . . , v h − . Lemma 6.
The sequence ( v , . . . , v h ) is regular on R / p.Proof. This requires the more delicate results about the height stratificationdescribed in Section 3.3: the height stratification on M d , p is defined locallyby a regular sequence of sections of line bundles on the moduli stack. As flatmorphisms preserve regularity, these sections pull back to a regular sequenceon R / p .As p is a nonzerodivisor on R by Lemma 2, it follows that the sequence ( p , v , . . . , v h ) is regular on R , with v h a unit. This finishes the proof of Theorem 1.There are weaker versions of Theorem 1 based on corresponding versions ofLandweber’s result modulo p . (See [36] and [37] for the latter.) However, inorder to impose “brave new rings” structures on the resulting spectra, it is desir-able to work with torsion-free coefficient rings. The extra effort it took to achievethis will pay off in the extra examples encompassed, to which we turn now. The rest of this section serves the purpose to show that all geometric points of M d of finite height can be thickened to give rise to K3 spectra by means of the preced-ing Theorem 1. The problem does not lie so much in finding a lifting to character-istic 0, as there always is a lifting to the Witt ring, for example, see [27], [6]. It liesin finding such a lift with a flat map to the moduli stack. However, the algebraicityof the stack provides such, as will now be explained.17or every degree 2 d , there is a smooth surjection H −→ M d where H is a suitable piece of a Hilbert scheme, see [31] for example. Let R be one of the local rings of H . As M d has finite type over Z [ / d ] , this willbe noetherian. And if the residue field k of R has characteristic prime to 2 d , thering R will be local over Z ( p ) . The compositionSpec ( R ) −→ H −→ M d is flat as a composition of flat maps, and classifies a K3 surface X over R : the pull-back of the universal family over the Hilbert scheme. If the closed fibre of X over k has finite height, Theorem 1 applies to give an even periodic ring spectrum E suchthat π E ∼ = R and Γ E ∼ = (cid:98) Br X . Proposition 7.
Let R be one of the local rings of the Hilbert scheme covering M d with residue field of characteristic prime to d. Then there is an even periodic ringspectrum E such that π E ∼ = R and Γ E is isomorphic to the formal Brauer groupof the germ of the universal family over R. The reader may wonder why a smooth cover has been used in the preceding dis-cussion, while an etale cover is available for the Deligne-Mumford stack M d .The reason is that the smooth cover can be made fairly concrete using the Hilbertschemes above, whereas the existence of an etale cover is only guaranteed bymeans of an unramified diagonal, which implies for abstract reasons the existenceof etale slices for the smooth cover, see (4.21) in [7] or (8.1) in [19]. This isin contrast to the case of elliptic curves, where etale covers can be written downexplicitly. Proposition 8.
If X : Spec ( k ) → M d is a geometric point of finite height andcharacteristic prime to d, there is an even periodic ring spectrum E such that π Eis isomorphic to the local ring of M d in X (with respect to the etale topology) andthe reduction of Γ E to k is the formal Brauer group of X . roof. The local ring (cid:101) O M d , X (with a tilde to indicate the etale topology) is thecolimit of the local rings O U , u , where ( U , u ) runs over the etale neighbourhoodsSpec ( k ) u (cid:35) (cid:35) X (cid:47) (cid:47) M d U etale (cid:61) (cid:61) of X in M d . As in the proof of the previous proposition, the hypotheses of Theo-rem 1 are satisfied for the local rings O U , u . As (cid:101) O M d , X is the (strict) henselisationof the O U , u , it is flat over them. Therefore, the hypotheses of Theorem 1 aresatisfied for (cid:101) O M d , X as well.There is a weaker variant of the preceding proposition which replaces thehenselian rings by complete rings. To state it, let (cid:98) O M d , X be the completion ofthe local ring (cid:101) O M d , X . Its (formal) spectrum is, by definition, the formal deforma-tion space of M d at X , so that, conversely, (cid:98) O M d , X is the ring of formal functionson it. Proposition 9.
If X : Spec ( k ) → M d is a geometric point of finite height andcharacteristic prime to d, there is an even periodic ring spectrum E such that π Eis isomorphic to the ring of formal functions on the formal deformation spaceof M d in X and such that the reduction of Γ E to k is the formal Brauer groupof X .Proof. As all rings involved are noetherian, the completion (cid:101) O M d , X → (cid:98) O M d , X isflat and one may argue as before. 19 cknowledgements I would like to thank those who have discussed this material with me before,especially Mike Hopkins and Jack Morava; my debts to them are clear.
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