SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2019), 030, 36 pages A Self-Dual Integral Form of the Moonshine Module
Scott CARNAHANUniversity of Tsukuba, Japan
E-mail: [email protected]
URL:
Received February 13, 2018, in final form April 06, 2019; Published online April 19, 2019https://doi.org/10.3842/SIGMA.2019.030
Abstract.
We construct a self-dual integral form of the moonshine vertex operator algebra,and show that it has symmetries given by the Fischer–Griess monster simple group. Theexistence of this form resolves the last remaining open assumption in the proof of the modularmoonshine conjecture by Borcherds and Ryba. As a corollary, we find that Griess’s original196884-dimensional representation of the monster admits a positive-definite self-dual integralform with monster symmetry.
Key words: moonshine; vertex operator algebra; orbifold; integral form
Contents
References 35
In this paper, we construct self-dual R -forms of the moonshine module vertex operator alge-bra [27] over various commutative rings R , culminating in the universal case where R is the ringof rational integers Z . For the Z -form, we show that the vertex operator algebra has monstersymmetry, and is self-dual with respect to an invariant bilinear form that respects the monstersymmetry. Base change then gives us self-dual monster-symmetric vertex operator algebras overany commutative ring. Our construction yields the final step in the affirmative resolution ofRyba’s modular moonshine conjecture. a r X i v : . [ m a t h . R T ] A p r S. CarnahanMonstrous moonshine began in the 1970s with McKay’s observation that the number 196884,namely the q coefficient in Klein’s modular j -invariant, is one more than the dimension of thesmallest faithful complex representation of the monster simple group M . Similar observationsby Thompson for higher-order coefficients [44] led to what is now called the McKay–Thompsonconjecture, asserting the existence of a natural graded faithful complex representation V = (cid:76) n ≥ V n of the monster with graded dimension given by (cid:88) n ≥ dim V n q n − = J ( τ ) = j ( τ ) − . The McKay–Thompson conjecture was resolved by the construction of the moonshine modu-le V (cid:92) in [26]. This construction was then refined in [27] so that V (cid:92) is not just a graded vectorspace, but a vertex operator algebra whose automorphism group is precisely the monster. Thevertex operator algebra structure on V (cid:92) turned out to be essential to Borcherds’s resolution ofthe monstrous moonshine conjecture [3] for V (cid:92) . This conjecture [13] is a substantial refinementof the McKay–Thompson conjecture, given by replacing graded dimensions with graded tracesof elements of M , and asserts that the resulting functions are Hauptmoduln for certain genuszero groups commensurable to SL ( Z ).It is natural to ask whether V (cid:92) can be defined over smaller subrings of C , and the first progresson this question was given at the end of [27], where a monster-symmetric self-dual Q -form isconstructed. In [6], the authors show that the Q -form can be defined over Z [1 / V (cid:92) with monstersymmetry was constructed in [18], but part of the construction involves passing to a monster-invariant sublattice to obtain the full symmetry, and this process generally destroys any controlover self-duality. In particular, the inner product may be quite degenerate after reductionmodulo a prime. A similar question concerns forms of the 196884-dimensional algebra V , whichis essentially what Griess used in his initial construction of the monster [29]. Conway sketchesa construction due to Norton of a monster-symmetric integral form of the 196884-dimensionalsubalgebra V in [11], and expresses hope that it may be self-dual. However, the question ofextending such a form to the rest of V (cid:92) has not been addressed. Indeed, remarks in [6] suggestthat this form would be incompatible with the form that we construct in this paper. We showin this paper that both questions have the best possible answers: there is a positive definiteself-dual integral form of V (cid:92) with monster symmetry, and its weight 2 subspace is a positivedefinite self-dual integral form of Griess’s algebra (up to a minor change in the multiplicationon the Virasoro vector, as is pointed out in [27]), with monster symmetry.The main motivation for this paper comes from the modular moonshine conjecture, whicharose out of experimental observations by Ryba in [42] that suggested the existence of a mod p version of moonshine. Modular moonshine asserted the existence of certain vertex algebrasover F p attached to elements g of M in class p A such that Brauer traces of p -regular centralizingelements h are given by certain McKay–Thompson series, i.e., graded traces on V (cid:92) . This conjec-ture, which was somewhat strengthened in [6], was essentially resolved in [6] and [4], up to someassumptions about the existence of integral versions of various constructions that were knownover R or C . Some of these assumptions, such as a description of root spaces of a Z p -form ofthe monster Lie algebra, were resolved in [5]. The remaining problem was that the conjectureasserts the existence of a self-dual Z -form of V (cid:92) with M -symmetry. Since such a form was notimmediately available, the authors of [6] noted that for the original conjecture, one only needsfor each prime p dividing the order of M , a self-dual form over the ring Z p of p -adic integers. Ifa self-dual Z p -form (or more generally, a form over any p -adic integer ring) with monster sym-metry exists, the arguments given in [6] and [4] (combined with some technical details resolvedin [5]) yield a resolution of the conjecture. For odd primes p , it therefore suffices to produce Self-Dual Integral Form of the Moonshine Module 3a self-dual Z [1 / p = 2, the problem has remained unambiguously open until now.In this paper, we give a construction of a self-dual integral form of V (cid:92) with monster symmetry,which resolves the last open assumption in modular moonshine. The construction requiresseveral very recent developments in the theory of vertex operator algebras. The most notable ofthese is the cyclic orbifold construction given in [45]. However, despite the dependence on recenttechnology, there is an effortless quality to our construction, like we are getting this object forfree. In particular, the argument requires a surprisingly small amount of group theoretic input,essentially all compressed into some information about maximal subgroups of M in the proof ofLemma 3.7. Aside from this input, the main ingredients are:1. The existence and uniqueness of self-dual R -forms of abelian intertwining algebras, fromgenerating subalgebras. This is basically given by extending some well-known skew-symmetry and associativity properties of vertex algebras to the abelian intertwining alge-bra setting.2. A very recent method for analyzing V (cid:92) by applying multiple cyclic orbifolds to the Leechlattice vertex operator algebra, due to Abe, Lam, and Yamada. Together with the previoustool, this lets us extract self-dual forms of V (cid:92) over cyclotomic S-integer rings.3. Faithfully flat descent. This technique is more than a half-century old and has broadapplications, but does not seem to get much attention in the vertex operator algebraworld, where the base ring is almost always C .While modular moonshine is now resolved, there are still many related open questions re-maining. For example, most of the questions listed at the end of [6] remain unresolved, and thequestion of what happens in the case g has composite order seems particularly natural. The main goal of this section is to show that the cyclic orbifold construction, as worked outin [45], is definable over distinguished subrings of C . For this purpose, we introduce somecommutative algebra technology.For any complex number r , we write e ( r ) to denote the normalized exponential e π i r . We willalso use the following notation conventions, some of which may be unfamiliar to non-specialistsin group schemes (see, e.g., [15]):1. For any commutative ring R and any n ∈ Z ≥ , we write µ n ( R ) to denote the group { r ∈ R | r n = 1 } . The functor µ n is represented by the affine scheme Spec Z [ x ] / (cid:0) x n − (cid:1) .2. For any commutative ring R , we write G m ( R ) to denote the group R × of invertible ele-ments. The functor G m is represented by the affine scheme Spec Z [ x, y ] / ( xy − Z (cid:2) x, x − (cid:3) of Laurent polynomials.3. Given commutative group schemes G and H , we write Hom( G, H ) for the functor thattakes a ring R to the group of natural transformations G × Spec R → H × Spec R overSpec R . In this paper, we will only consider the case G is a torus or a lattice of rank n ,and H is either µ n or G m . In particular, D ( Z n ) = Hom( Z n , G m ) ∼ = G nm , and D ( G nm ) =Hom( G nm , G m ) ∼ = Z n . Definition 2.1.
Let R be a commutative ring. We say that an R -module M is faithfully flat iffor any 3-term complex M → M → M of R -modules, the following conditions are equivalent: S. Carnahan1) M → M → M is exact,2) M ⊗ R M → M ⊗ R M → M ⊗ R M is exact.Given a homomorphism f : R → S of commutative rings, we say f is faithfully flat if S isfaithfully flat as an R -module under the induced action.See [43, Section 00H9] for a brief overview of basic properties of flat and faithfully flat modulesand ring maps. Definition 2.2.
Let f : R → S be a homomorphism of commutative rings. A descent datum formodules with respect to f is a pair ( M, φ ), where M is an S -module, and φ : M ⊗ R S → S ⊗ R M is an S ⊗ R S -module isomorphism satisfying the “cocycle condition”, i.e., that the followingdiagram commutes: M ⊗ R S ⊗ R S φ , = φ ⊗ (cid:47) (cid:47) φ , (cid:42) (cid:42) S ⊗ R M ⊗ R S φ , =1 ⊗ φ (cid:15) (cid:15) S ⊗ R S ⊗ R M, where, if we write φ ( m ⊗
1) = (cid:80) s i ⊗ m i , then the S ⊗ R S ⊗ R S -module homomorphisms φ i,j are defined by φ , ( m ⊗ ⊗
1) = (cid:88) s i ⊗ m i ⊗ ,φ , ( m ⊗ ⊗
1) = (cid:88) s i ⊗ ⊗ m i ,φ , (1 ⊗ m ⊗
1) = (cid:88) ⊗ s i ⊗ m i . A morphism of descent data from (
M, φ ) to ( M (cid:48) , φ (cid:48) ) is an S -module homomorphism ψ : M → M (cid:48) such that the following diagram commutes: M ⊗ R S φ (cid:47) (cid:47) ψ ⊗ (cid:15) (cid:15) S ⊗ R M ⊗ ψ (cid:15) (cid:15) M (cid:48) ⊗ R S φ (cid:48) (cid:47) (cid:47) S ⊗ R M (cid:48) . Theorem 2.3.
Let f : R → S be a homomorphism of commutative rings. If f is faithfully flat,then the category of R -modules is equivalent to the category of descent data for modules withrespect to f . In particular, the functor taking an R -module M to the S -module M ⊗ R S equippedwith descent datum φ (( m ⊗ s ) ⊗ s (cid:48) ) = s ⊗ ( m ⊗ s (cid:48) ) has quasi-inverse given by ( M, φ ) (cid:55)→ ˜ M = { m ∈ M | ⊗ m = φ ( m ⊗ } . Proof .
See [43, Proposition 023N] or Theorem 4.21 of [46] for freely available expositions, orTheorem 1 of [30] for the original reference. (cid:4)
We note that faithful flatness can be replaced by the weaker condition of “universally injectivefor modules”, namely that f ⊗ id M : M = R ⊗ R M → S ⊗ R M is injective for all R -modules M .The fact that this condition is necessary and sufficient for effectiveness of descent of moduleswas noted without proof in [41] and proved in [39]. Definition 2.4.
Given a homomorphism f : R → S of commutative rings, a bilinear map withdescent datum with respect to f is a quadruple ( κ : M × N → T, φ M , φ N , φ T ), where M , N , T are S -modules, κ is an S -bilinear map, and φ M : M ⊗ R S → S ⊗ R M , φ N : N ⊗ R S → S ⊗ R N , Self-Dual Integral Form of the Moonshine Module 5 φ T : T ⊗ R S → S ⊗ R T are descent data for modules with respect to f , such that the followingdiagram of S ⊗ R S -module homomorphisms commutes:( M ⊗ R S ) ⊗ S ⊗ R S ( N ⊗ R S ) φ M ⊗ φ N (cid:47) (cid:47) ¯ κ ⊗ (cid:15) (cid:15) ( S ⊗ R M ) ⊗ S ⊗ R S ( S ⊗ R N ) ⊗ ¯ κ (cid:15) (cid:15) T ⊗ R S φ T (cid:47) (cid:47) S ⊗ T. Here, ¯ κ : M ⊗ N → T is the S -module map canonically attached to κ . Lemma 2.5.
Given a faithfully flat homomorphism f : R → S of commutative rings, any bi-linear map with descent datum with respect to f is equivalent to the base change of a uniquebilinear map of R -modules. That is, given ( κ : M × N → T, φ M , φ N , φ T ) , there is a unique R -linear map ˜ κ : ˜ M ⊗ ˜ N → ˜ T such that the following diagram commutes: (cid:0) ˜ M ⊗ R S (cid:1) ⊗ S (cid:0) ˜ N ⊗ R S (cid:1) ˜ κ ⊗ (cid:47) (cid:47) ψ M ⊗ ψ N (cid:15) (cid:15) ˜ T ⊗ R S ψ T (cid:15) (cid:15) M ⊗ N ¯ κ (cid:47) (cid:47) T. Here, ψ M : ˜ M ⊗ R S → M , ψ N : ˜ N ⊗ R S → N , ψ T : ˜ T ⊗ R S → T are the S -module isomorphismsinduced by the descent data. Proof .
This amounts to applying the equivalence in Theorem 2.3 to the S -module map ¯ κ . (cid:4) Proposition 2.6.
Let f : R → S be a faithfully flat homomorphism of commutative rings, andlet M be an R -module. Then M is a finite projective ( equivalently, flat and finitely presented ) R -module if and only if M ⊗ R S is a finite projective S -module. Proof .
This is given by [31, Proposition 2.5.2]. (cid:4)
Definition 2.7.
Given a commutative ring R , we define the groupoid of finite projective R -modules with bilinear form by setting:1. Objects are given by pairs ( M, κ ), where M is a finite projective R -module, and κ : M × R M → R is an R -bilinear map.2. Morphisms ( M, κ ) → ( M (cid:48) , κ (cid:48) ) are R -module isomorphisms ψ : M → M (cid:48) such that κ (cid:48) ( ψ ( m ) , ψ ( m )) = κ ( m , m ) for all m , m ∈ M. Given a homomorphism f : R → S of commutative rings, a descent datum for finite projectivemodules with bilinear form with respect to f is a triple ( M, κ, φ ), where (
M, φ ) is a finiteprojective S -module with descent datum with respect to f , and ( κ, φ, φ, id S ) is a bilinear mapwith descent datum with respect to f . An isomorphism of descent data from ( M, κ, φ ) to( M (cid:48) , κ (cid:48) , φ (cid:48) ) is an S -module isomorphism ψ : M → M (cid:48) such that κ ( m , m ) = κ (cid:48) ( ψ ( m ) , ψ ( m ))for all m , m ∈ M , and the following diagram commutes: M ⊗ R S φ (cid:47) (cid:47) ψ ⊗ (cid:15) (cid:15) S ⊗ R M ⊗ ψ (cid:15) (cid:15) M (cid:48) ⊗ R S φ (cid:48) (cid:47) (cid:47) S ⊗ R M (cid:48) . A bilinear form κ on a finite projective R -module is called non-singular if it induces an R -moduleisomorphism M → Hom R ( M, R ) under the canonical adjunction isomorphism Hom R ( M ⊗ R M, R ) → Hom R ( M, Hom R ( M, R )). S. Carnahan
Theorem 2.8.
Let f : R → S be a homomorphism of commutative rings. If f is faithfully flat,then the groupoid of finite projective R -modules with bilinear form is equivalent to the groupoidof descent data for finite projective modules with bilinear form with respect to f . Furthermore,the subgroupoid of non-singular forms is preserved under this equivalence. Proof .
Given a descent datum (
M, κ, φ ) for finite projective modules with bilinear form withrespect to f , by Theorem 2.3 and Proposition 2.6, there exists a finite projective R -module ˜ M such that base change yields the original descent datum for modules, and by Lemma 2.5, thereis a unique R -bilinear form ˜ κ whose base-change to S yields the descent datum ( κ, φ, φ, id S ). ByTheorem 2.3, isomorphisms ψ of descent data are in natural bijection with isomorphisms ˜ ψ of R -modules that preserve ˜ κ .We now consider the condition that a form is non-singular. Given an R -module map ¯ κ : M → Hom R ( M, R ) that corresponds to a bilinear form κ , base change along f yields an S -modulemap ¯ κ ⊗ M ⊗ R S → Hom R ( M, R ) ⊗ R S . Because M is finite projective, the canonical mapcan : Hom R ( M, R ) ⊗ R S → Hom S ( M ⊗ R S, S ) is an isomorphism. We then have a commutativediagram M ⊗ R S ¯ κ ⊗ (cid:47) (cid:47) κ ⊗ (cid:40) (cid:40) Hom R ( M, R ) ⊗ R S can (cid:15) (cid:15) Hom S ( M ⊗ R S, S ) , where κ ⊗ S -module homomorphism that naturally corresponds to the S -valued bilinearform κ ⊗ M ⊗ R S . This form is non-singular if and only if ¯ κ ⊗ κ ⊗ κ is an isomorphism, i.e., if andonly if κ is non-singular. (cid:4) We consider a method of gluing that is well-suited to the objects we obtain in this paper.
Definition 2.9.
Suppose we are given a diagram R → R ← R of commutative rings. A gluingdatum for modules over this diagram is a triple ( M , M , f ), where1) M is an R -module,2) M is an R -module, and3) f : M ⊗ R R ∼ → M ⊗ R R is an R -module isomorphism.If P is a property of modules over a commutative ring, we say that a gluing datum for modulessatisfies property P if the component modules satisfy property P . A morphism of gluing datafrom ( M , M , f ) to ( M (cid:48) , M (cid:48) , f (cid:48) ) is a pair ( g : M → M (cid:48) , g : M → M (cid:48) ) of module maps suchthat the following diagram commutes: M ⊗ R R f (cid:47) (cid:47) g ⊗ id (cid:15) (cid:15) M ⊗ R R g ⊗ id (cid:15) (cid:15) M (cid:48) ⊗ R R f (cid:48) (cid:47) (cid:47) M (cid:48) ⊗ R R . We first note that gluing of quasi-coherent sheaves is effective for Zariski open covers.
Lemma 2.10.
Let R be a commutative ring, and let a, b ∈ R be coprime elements, i.e., theideal generated by a and b is all of R . Let R a and R b be the respective localizations, i.e., suchthat Spec R a and Spec R b form a Zariski open cover of Spec R . Then base change induces Self-Dual Integral Form of the Moonshine Module 7 an equivalence between the category of R -modules and the category of gluing data for modulesover the diagram R a → R ab ← R b . This restricts to an equivalence between the category offinite projective R -modules and the category of finite projective gluing data, and a correspondingequivalence for groupoids of finite projective modules with non-singular bilinear form. Proof .
The first claim amounts to faithfully flat descent for the Zariski open cover R → R a ⊕ R b .All of the components of a descent datum for this cover are given in the gluing datum, becausewe have the identifications R a ⊗ R R a = R a and R b ⊗ R R b = R b . A detailed proof can be foundat [43, Lemma 00EQ]. The remaining claims follow from the corresponding descent results inthe previous section. (cid:4) We wish to consider situations where the rings in a gluing datum are not necessarily Zariskilocalizations of R . In particular, we will consider the case where the diagram of rings has theform R → R ⊗ R R ← R , where i : R → R and i : R → R are faithfully flat. Definition 2.11.
Suppose i : R → R and i : R → R are faithfully flat. A gluing datum forthe diagram R → R ⊗ R R ← R is strict if there is some R ⊗ R R -module M such that M and M are contained in M as an R ⊗ R R -submodule and an R ⊗ R R -submodule, such thatextension of scalars gives the identifications M ⊗ R R → M and R ⊗ R M → M . Lemma 2.12.
Suppose i : R → R and i : R → R are faithfully flat. Then, any gluing datum ( M , M , f ) for the diagram R → R ⊗ R R ← R is isomorphic to a strict gluing datum (cid:0) M , f − ( R ⊗ R M ) , id M ⊗ R R (cid:1) . In particular, there is an equivalence between the category ofgluing data for the diagram R → R ⊗ R R ← R and the full subcategory of strict gluing data. Proof .
Because i is faithfully flat (in particular, universally injective for modules), R ⊗ R R is a subring of R ⊗ R R . Because f − is an R ⊗ R R -module isomorphism, f − ( R ⊗ R M ) isan R ⊗ R R -submodule of M ⊗ R R , and transporting the action of R through the canonicalisomorphism R ∼ → R ⊗ R R , we obtain an R -module isomorphism g : f − ( R ⊗ R M ) → M .It is straightforward to see that extending scalars along i yields the map f , once we make theidentification R ⊗ R f − ( R ⊗ R M ) = M ⊗ R R , by r ⊗ f − (1 ⊗ x ) = f − ( r ⊗ x ). Thus, thepair (id , g ) : (cid:0) M , f − ( R ⊗ R M ) , id M ⊗ R R (cid:1) → ( M , M , f ) induces a commutative diagram: M ⊗ R R (cid:15) (cid:15) id (cid:47) (cid:47) M ⊗ R R f (cid:15) (cid:15) M ⊗ R R f (cid:47) (cid:47) R ⊗ R M , and is therefore an isomorphism of gluing data.We therefore have a full subcategory that spans all isomorphism classes of objects, so it isequivalent to the category of gluing data. (cid:4) Lemma 2.13.
Let i : R → R and i : R → R be faithfully flat homomorphisms of commu-tative rings. Then base change induces an equivalence between the category of R -modules ( withmorphisms given by arbitrary R -module maps ) and the category of gluing data for modules overthe diagram R → R ⊗ R R ← R . This restricts to an equivalence between the category offinite projective R -modules and the category of finite projective gluing data, and a correspondingequivalence for groupoids of finite projective modules with non-singular bilinear form. Proof .
By Lemma 2.12, it suffices to consider strict gluing data instead of all gluing data.Let F be the base change functor that takes R -modules to strict gluing data. We first definea quasi-inverse functor G . Let ( M , M , id) be a strict gluing datum. Thus, we assume M is an R ⊗ R R -submodule of some R ⊗ R R -module M , and similarly for the R ⊗ R R -module M . S. CarnahanFrom M , we have a canonical descent datum on M , i.e., an ( R ⊗ R R ) ⊗ R ( R ⊗ R R )-moduleisomorphism φ : M ⊗ R ( R ⊗ R R ) → ( R ⊗ R R ) ⊗ R M , such that M is the set of elements x ∈ M satisfying φ ( x ⊗ (1 ⊗ ⊗ ⊗ x . Similarly,there is a descent datum φ : M ⊗ R ( R ⊗ R R ) → ( R ⊗ R R ) ⊗ R M , such that M is the set of elements x ∈ M satisfying φ ( x ⊗ (1 ⊗ ⊗ ⊗ x . We set G ( M , M , id) to be the R -module that is the intersection of M and M in M , or equivalently,the set of elements x ∈ M satisfying both φ ( x ⊗ (1 ⊗ ⊗ ⊗ x ∈ ( R ⊗ R R ) ⊗ R M and φ ( x ⊗ (1 ⊗ ⊗ ⊗ x ∈ ( R ⊗ R R ) ⊗ R M . Given a morphism of strict gluing data,we obtain a homomorphism of R -modules on the intersections, so G is a functor.We first check that the composite G ◦ F is isomorphic to identity on R -modules. Because i and i are faithfully flat, they are universally injective for modules, so any R -module M injectsinto both M ⊗ R R and M ⊗ R R . The intersection in M ⊗ R R ⊗ R R is then M ⊗ R R ⊗ R R ∼ = M ,and we have our isomorphism.We define a natural transformation from F ◦ G to identity by sending any object ( M , M , id)to the pair ( g , g ), where g : R ⊗ R G ( M , M , id) → M and g : R ⊗ R G ( M , M , id) → M are given by restricting the action maps R ⊗ R M → M and R ⊗ R M → M . Toshow that F is an equivalence, it remains to show that g and g are isomorphisms of R -modules and R -modules, respectively. The base change of g along i is given by the restriction g : ( R ⊗ R R ) ⊗ R G ( M , M , id) → M of the action map on M , and since i is faithfully flat,this map is injective (resp. surjective) if and only if g is also. The base change of g along i is also given by the restricted action map, so it suffices to show that at least one of these threemaps ( g , g , and g ) is injective and at least one is surjective.To prove injectivity, we consider the following diagram: R ⊗ R G ( M , M , id) ⊗ R R ⊗ ι ⊗ id (cid:47) (cid:47) g ⊗ id (cid:15) (cid:15) R ⊗ R M ⊗ R R ∼ = (cid:47) (cid:47) f (cid:15) (cid:15) ( R ⊗ R R ) ⊗ R ( M ⊗ R R ) φ − (cid:15) (cid:15) M ⊗ R R ⊗ i ⊗ id (cid:47) (cid:47) M ⊗ R R ⊗ R R ∼ = (cid:47) (cid:47) ( M ⊗ R R ) ⊗ R ( R ⊗ R R ) , where ι : G ( M , M , id) → M is the inclusion, and f is the unique R ⊗ R R ⊗ R R -moduleisomorphism that makes the square on the right commute. The horizontal arrows on the leftare injective homomorphisms, so to prove injectivity of g , it suffices to check commutativity ofthe left square. Along the lower left path, we have(id ⊗ i ⊗ id) ◦ ( g ⊗ id)( r ⊗ m ⊗ r ) = r m ⊗ ⊗ r , which is then sent to ( r m ⊗ ⊗ (1 ⊗ r ). Along the upper right path, we note that for any m ∈ G ( M , M , id), f (1 ⊗ m ⊗
1) is identified with φ − ((1 ⊗ ⊗ ( m ⊗ m ⊗ ⊗ (1 ⊗ m ⊗ ⊗
1. Thus, we have f ◦ (id ⊗ ι ⊗ id)( r ⊗ m ⊗ r ) = f ( r ⊗ m ⊗ r ) = f (( r ⊗ ⊗ r )(1 ⊗ m ⊗ r ⊗ ⊗ r ) f (1 ⊗ m ⊗
1) = ( r ⊗ ⊗ r )( m ⊗ ⊗
1) = r m ⊗ ⊗ r . This proves g ⊗ id = g is injective.We reduce the question of surjectivity to the case that R is a local ring as follows. By Claim 5of [43, Lemma 00HN], a map M → M (cid:48) of R -modules (for any commutative ring R ) is surjective Self-Dual Integral Form of the Moonshine Module 9if and only if for each prime ideal P of R , the localized map M P → M (cid:48) P of R P -modules issurjective. Each R P is a local ring, so we shall assume R is a local ring for the remainder of thisproof.Let x ∈ M , so x ⊗ ∈ M ⊗ R R , and write φ (( x ⊗ ⊗ (1 ⊗
1) = (cid:88) i ( s i ⊗ t i ) ⊗ ( x i ⊗ ∈ ( R ⊗ R R ) ⊗ R ( M ⊗ R R ) . Note that we may move multipliers in R across the middle tensor product, and this is why wemay write 1 in the rightmost factor. For each x i ∈ M , we write φ (( x i ⊗ ⊗ (1 ⊗ (cid:88) j ( s ij ⊗ t ij ) ⊗ ( y ij ⊗ . Then, because φ satisfies the cocycle condition, we have (cid:88) i ( s i ⊗ t i ) ⊗ (1 ⊗ ⊗ ( x i ⊗
1) = (cid:88) i,j ( s i ⊗ t i ) ⊗ ( s ij ⊗ t ij ) ⊗ ( y ij ⊗ (cid:88) i ( s i ⊗ t i ) ⊗ φ (( x i ⊗ ⊗ (1 ⊗ R ⊗ R R ) ⊗ R ( R ⊗ R R ) ⊗ R ( R ⊗ R R ). We now use our assumption that R isa local ring, and in particular, the property that any finitely generated submodule of an R -flatmodule is R -free. This property is not explicitly stated in Proposition 3.G of [37], which assertsthat all finitely generated R -flat modules are R -free when R is local, but Matsumura’s proof ofthe proposition yields this stronger result with no substantial change. For the case at hand, wetake the R -free submodules of R and R generated by the finite sets { s i } and { t i } , respectively.By choosing bases and performing a suitable rearrangement, we may assume the elements s i and t i in the sum (cid:80) i ( s i ⊗ t i ) ⊗ ( x i ⊗
1) are basis elements of the free submodules, and that thesummands are R -linearly independent. Then, the cocycle condition implies φ (( x i ⊗ ⊗ (1 ⊗ ⊗ ⊗ ( x i ⊗ i , and in particular, the elements x i lie in G ( M , M , id). Because φ is an ( R ⊗ R R ) ⊗ R ( R ⊗ R R )-module isomorphism, we can then write φ − (( s i ⊗ t i ) ⊗ ( x i ⊗ s i ⊗ t i ) ⊗ (1 ⊗ · φ − ((1 ⊗ ⊗ ( x i ⊗ s i ⊗ t i ) ⊗ (1 ⊗ · (( x i ⊗ ⊗ (1 ⊗ s i x i ⊗ t i ) ⊗ (1 ⊗ . Summing over i , we find that x ⊗ (cid:80) i s i x i ⊗ t i , so the image of g contains M . Because M is an R -form of M , the restricted action map ( R ⊗ R R ) ⊗ R M → M is surjective. Thecompatibility between iterated actions and ring multiplication then implies g is surjective.Summing up, we have proved that g is an isomorphism, so g and g are isomorphisms.We therefore have a natural isomorphism from F ◦ G to identity, so we conclude that F is anequivalence of categories.The claims about finite projective modules and bilinear forms then follow from essentiallythe same argument as in Theorem 2.8. (cid:4) Vertex algebras over commutative rings were first defined in [2]. Since all of the exampleswe consider will be conformal with strong finiteness conditions, we will mainly consider vertexoperator algebras, roughly following the treatment in [25].0 S. Carnahan
Definition 2.14.
A vertex algebra over a commutative ring R is an R -module V equipped witha distinguished vector ∈ V and a left multiplication map Y : V → (End V ) (cid:2)(cid:2) z, z − (cid:3)(cid:3) , written Y ( a, z ) = (cid:80) n ∈ Z a n z − n − satisfying the following conditions:1. For any a, b ∈ V , a n b = 0 for n sufficiently large. Equivalently, Y defines a multiplicationmap V ⊗ R V → V (( z )).2. Y ( , z ) = id V z and Y ( a, z ) ∈ a + zV [[ z ]].3. For any r, s, t ∈ Z , and any u, v, w ∈ V , (cid:88) i ≥ (cid:18) ri (cid:19) ( u t + i v ) r + s − i w = (cid:88) i ≥ ( − i (cid:18) ti (cid:19)(cid:0) u r + t − i ( v s + i w ) − ( − t v s + t − i ( u r + i w ) (cid:1) . Equivalently, the Jacobi identity holds: x − δ (cid:18) y − zx (cid:19) Y ( a, y ) Y ( b, z ) − x − δ (cid:18) z − y − x (cid:19) Y ( b, z ) Y ( a, y )= z − δ (cid:18) y − xz (cid:19) Y ( Y ( a, x ) b, z ) , where δ ( z ) = (cid:80) n ∈ Z z n , and δ (cid:0) y − zx (cid:1) is expanded as a formal power series with non-negativepowers of z , i.e., as (cid:80) n ∈ Z ,m ∈ Z ≥ ( − m (cid:0) nm (cid:1) x − n y n − m z m .A vertex algebra homomorphism ( V, V , Y V ) → ( W, W , Y W ) is an R -module homomorphism φ : V → W satisfying φ ( V ) = W and φ ( u n v ) = φ ( u ) n φ ( v ) for all u, v ∈ V and all n ∈ Z . Remark 2.15.
Mason notes in [36] that a vertex algebra over R is the same thing as a vertexalgebra V over Z equipped with a vertex algebra homomorphism R → V , where R is viewed asa commutative vertex algebra with product Y ( a, z ) b = a − bz = ab . Definition 2.16.
Let f : R → S be a homomorphism of commutative rings. A descent datumfor vertex algebras with respect to f is a pair ( V, φ ), where V is a vertex algebra over S , and φ : V ⊗ R S → S ⊗ R V is an isomorphism of vertex algebras over S ⊗ R S such that the cocyclecondition holds, i.e., the following diagram of isomorphisms of vertex algebras over S ⊗ R S ⊗ R S commutes: V ⊗ R S ⊗ R S φ = φ ⊗ (cid:47) (cid:47) φ (cid:42) (cid:42) S ⊗ R V ⊗ R S φ =1 ⊗ φ (cid:15) (cid:15) S ⊗ R S ⊗ R V. Here, the maps φ i,j are defined as in Definition 2.2. A morphism of descent data from ( V, φ )to ( V (cid:48) , φ (cid:48) ) is a homomorphism ψ : V → V (cid:48) of vertex algebras over S such that the followingdiagram commutes: V ⊗ R S φ (cid:47) (cid:47) ψ ⊗ (cid:15) (cid:15) S ⊗ R V ⊗ ψ (cid:15) (cid:15) V (cid:48) ⊗ R S φ (cid:48) (cid:47) (cid:47) S ⊗ R V (cid:48) . Self-Dual Integral Form of the Moonshine Module 11
Proposition 2.17.
Faithfully flat descent of vertex algebras is effective. That is, if we are givena faithfully flat ring homomorphism R → S , then there is an equivalence between the categoryof vertex algebras ˜ V over R and the category of vertex algebras V over S equipped with descentdata φ , given in one way by ˜ V (cid:55)→ ( ˜ V ⊗ R S, φ (( v ⊗ s ) ⊗ s (cid:48) ) = s ⊗ ( v ⊗ s (cid:48) )) and the other way by ( V, φ ) (cid:55)→ ˜ V = { v ∈ V | ⊗ v = φ ( v ⊗ } . Proof .
We briefly explain why this follows from the effectiveness of faithfully flat descent ofmodules. Essentially, faithfully flat descent for modules with additional structure holds as long asthat structure is defined by homomorphisms of modules, with conditions given by commutativityof diagrams. Here the additional structure is given by a distinguished map S → V inducedby the unit vector, and the “ z − n − -coefficient maps” · n : V ⊗ S V → V . The vertex algebraaxioms, such as the Jacobi identity, may be interpreted as equality of certain composites of suchmaps.Now, suppose we are given a descent datum ( V, φ ) for vertex algebras with respect to f .The isomorphism φ induces a descent datum for the underlying S -module, so faithfully flat de-scent for modules, given as Theorem 2.3, yields an R -module ˜ V and a distinguished S -moduleisomorphism ψ : ˜ V ⊗ R S ∼ = V .For each integer n , we have the “ z − n − -coefficient map” · n : V ⊗ S V → V , which we maynow rewrite as · n : (cid:0) ˜ V ⊗ R ˜ V (cid:1) ⊗ R S → ˜ V ⊗ R S using ψ . To show that this is the base change ofa map ˜ · n : ˜ V ⊗ R ˜ V → ˜ V along f , it is necessary and sufficient to show that · n is a morphism ofdescent data for modules with respect to f .By our assumption that φ is a descent datum for vertex algebras, the following diagramcommutes: (cid:0) ˜ V ⊗ R ˜ V (cid:1) ⊗ R S ⊗ R S φ ⊗ φ (cid:47) (cid:47) · n ⊗ (cid:15) (cid:15) S ⊗ R (cid:0) ˜ V ⊗ R ˜ V (cid:1) ⊗ R S ⊗· n (cid:15) (cid:15) ˜ V ⊗ R S ⊗ R S φ (cid:47) (cid:47) S ⊗ R ˜ V ⊗ R S, so we have a map of descent data. Here, the map φ ⊗ φ along the top uses an identification (cid:0) ˜ V ⊗ R ˜ V (cid:1) ⊗ R S ⊗ R S ∼ = ( V ⊗ R S ) ⊗ S ( V ⊗ R S ) induced by ψ .Essentially the same argument shows that the unit map S → V that takes 1 to is the basechange of an R -module map R → ˜ V . The remaining vertex algebra conditions then follow fromthe fact that base-change along f yields a formulas that hold for V . (cid:4) Given a vertex algebra V over R and a vertex algebra W over S , we say that V is an R -formof W with respect to a ring homomorphism φ : R → S if V ⊗ R,φ S is isomorphic to W as a vertexalgebra over S . If φ is implicitly fixed, we will simply say that V is an R -form of W . Thus, theprevious proposition amounts to the claim that the construction of an R -form with respect toa faithfully flat map is equivalent to the construction of a descent datum. Definition 2.18.
Let c ∈ R . A conformal vertex algebra over R with half central charge c isa vertex algebra over R equipped with a Z -grading V = (cid:76) n ∈ Z V n and a distinguished vector ω ∈ V such that:1. The coefficients of Y ( ω, z ) = (cid:80) k ∈ Z L k z − k − satisfy the Virasoro relations at half centralcharge c , i.e.,[ L m , L n ] = ( m − n ) L m + n + c (cid:18) m + 13 (cid:19) δ m + n, id V .
2. The product structure is homogeneous: u k v ∈ V m + n − k − for u ∈ V m , v ∈ V n .2 S. Carnahan3. L v = nv for all v ∈ V n .4. L − v is the z -term in Y ( v, z ) .A vertex operator algebra over R with half central charge c is a conformal vertex algebra suchthat all V n are finite projective R -modules, and V n = 0 for n (cid:28) Remark 2.19.
Our definition of vertex operator algebra over R differs slightly from the defi-nition in [25]. We remove the condition that the base ring R be an integral domain in which 2is invertible, and we replace the condition that each V n be a finite free R -module with thecondition that each V n be a finite projective R -module. The first change is necessary for us toconsider the case R = Z , and the second change is necessary in order for faithfully flat descent ofvertex operator algebras to be effective in general (see Proposition 2.23). That is, free modulesdon’t necessarily glue to form free modules, and standard examples come from locally trivialbut nontrivial vector bundles. This distinction doesn’t affect the strength of our final result,since Z is a principal ideal domain. Remark 2.20.
It may be reasonable someday to change the definition of “vertex operatoralgebra over R ” to require an action of the smooth integral form U + ( V ir ) of the universalenveloping algebra of Virasoro constructed in Section 5 of [5]. The self-dual integral form of V (cid:92) constructed in this paper admits such an action by virtue of the same being true of ( V Λ ) Z .Furthermore, the proof of Modular Moonshine for primes greater than 13 in [4] assumes theexistence of such an action. However, this condition is at the moment too difficult to verify ingeneral to be particularly useful. Definition 2.21.
A vertex algebra homomorphism from V to W is an R -linear map φ : V → W satisfying φ ( V ) = W and φ ( Y V ( u, z ) v ) = Y W ( φ ( u ) , z ) φ ( v ) for all u, v ∈ V . A vertex operatoralgebra homomorphism from V to W is a vertex algebra homomorphism φ : V → W thatpreserves the Z -grading, and sends ω V to ω W . Remark 2.22.
The condition that φ ( ω V ) = ω W is sometimes reserved for a special class ofhomomorphisms, using terms like “strong” or “strict” or “conformal” homomorphism in theliterature. This is because it is often fruitful to consider vertex algebra maps between vertexoperator algebras of different central charge. We will not need to consider such maps in thispaper. Proposition 2.23.
Faithfully flat descent of vertex operator algebras is effective. That is, if weare given a faithfully flat ring homomorphism R → S , then there is an equivalence between thecategory of vertex operator algebras ¯ V over R and the category of vertex operator algebras V over S equipped with descent data φ : V ⊗ R S → S ⊗ R V , given in one way by ¯ V (cid:55)→ ( ¯ V ⊗ R S,φ (( v ⊗ s ) ⊗ s (cid:48) ) = s ⊗ ( v ⊗ s (cid:48) )) and the other way by ( V, φ ) (cid:55)→ ¯ V = { v ∈ V | ⊗ v = φ ( v ⊗ } . Proof .
By Proposition 2.17, faithfully flat descent for vertex algebras is effective. It then sufficesto show that the conformal vector ω descends, and that the finite projective property of modulesdescends. The first assertion is clear from the fact that φ sends conformal vectors to conformalvectors. The second assertion is given in [43, Proposition 058S] (or [31, Proposition 2.5.2]). (cid:4) Remark 2.24.
This result allows us to consider vertex algebras and vertex operator algebrasover a broad class of geometric objects, including all schemes and all algebraic stacks.
Definition 2.25.
A M¨obius structure on a vertex algebra over R is an integer grading, togetherwith an action of the Kostant integral form U ( sl ) Z = (cid:10) L ( n ) − , (cid:0) L n (cid:1) , L ( n )1 (cid:11) n ≥ on the underlying R -module, such that the following conditions hold: Self-Dual Integral Form of the Moonshine Module 131. Y ( u, z ) = (cid:80) n ≥ L ( n ) − uz n for all u ∈ V .2. If u has weight k , then (cid:0) L n (cid:1) u = (cid:0) kn (cid:1) u .3. The subalgebra generated by { L ( n )1 } n ≥ acts locally nilpotently on V .4. For all i ∈ {− , , } , L i Y ( u, z ) v − Y ( u, z )( L i v ) = i +1 (cid:80) j =0 (cid:0) i +1 j (cid:1) z j Y ( L i − j u, v ).We note that in the embedding of U ( sl ) Z in U ( sl ), we have L ( n ) i = L ni n ! . Definition 2.26.
Let V be a vertex algebra with M¨obius structure, and let u ∈ V have weight k .For each integer n , we define the operator ( u n ) ∗ = ( − k (cid:80) i ≥ (cid:0) L ( i )1 u (cid:1) k − i − n − . An invariantbilinear form on a vertex algebra V with M¨obius structure is a symmetric bilinear form ( , ) : V × V → R such that( u n v, w ) = ( v, ( u n ) ∗ w )for all homogeneous vectors u and all vectors v and w . That is, ( u n ) ∗ is the adjoint of theoperator u n . We say that a vertex operator algebra is self-dual with respect to an invariantbilinear form if each graded piece V n is self-dual as a finite projective R -module. Remark 2.27. If R contains Z as a subring, then we may write the inner product formulaequivalently as ( Y ( u, z ) v, w ) = (cid:0) v, Y (cid:0) e zL (cid:0) − z − (cid:1) L u, z − (cid:1) w (cid:1) . Otherwise, we must replace L with an integer indicator of weight. Proposition 2.28.
Faithfully flat descent is effective for both M¨obius structure and invariantbilinear forms, on both vertex algebras and vertex operator algebras. Furthermore, the self-dualproperty for vertex operator algebras with invariant bilinear forms descends effectively.
Proof .
For M¨obius structure, the generating operators descend as module maps, and automat-ically satisfy the defining relations. For invariant bilinear forms, it suffices to use descent ofmodule maps V ⊗ R V → R as in Lemma 2.5. For the self-dual property, we may reduce to thecase of self-duality of the finite projective modules V n , and the isomorphism property of the map V n → Hom R ( V n , R ) descends effectively as in Theorem 2.8. (cid:4) We use this to describe gluing explicitly.
Definition 2.29.
Suppose we are given the following diagram of commutative ring homomor-phisms: R → R ← R . A gluing datum for vertex operator algebras over this diagram is atriple (cid:0) V , V , f (cid:1) , where1. V , V are vertex operator algebras over R and R , respectively.2. f : V ⊗ R R → V ⊗ R R is an isomorphism of vertex operator algebras over R .A gluing datum is M¨obius and self-dual if the corresponding vertex operator algebras are M¨obiusand self-dual, and the map f preserves the M¨obius structure and bilinear form. A morphism ofgluing data from (cid:0) V , V , f (cid:1) to (cid:0) V (cid:48) , , V (cid:48) , , f (cid:48) (cid:1) is a pair (cid:0) g : V → V (cid:48) , , g : V → V (cid:48) , (cid:1) of vertexalgebra homomorphisms such that the following diagram commutes: V ⊗ R R f (cid:47) (cid:47) g ⊗ id (cid:15) (cid:15) V ⊗ R R g ⊗ id (cid:15) (cid:15) V (cid:48) , ⊗ R R f (cid:48) (cid:47) (cid:47) V (cid:48) , ⊗ R R . Lemma 2.30.
Let R be a commutative ring, and let a, b ∈ R be coprime elements, i.e., theideal generated by a and b is all of R . Let R a and R b be the respective localizations, i.e., suchthat Spec R a and Spec R b form a Zariski open cover of Spec R . Then base change induces anequivalence between the category of vertex operator algebras over R and the category of gluingdata for vertex operator algebras over the diagram R a → R ab ← R b . This also yields equivalencesbetween the corresponding categories of self-dual M¨obius objects. Proof .
We obtain a quasi-inverse functor by applying Lemma 2.10 to obtain the underlyingmodules and structure maps. (cid:4)
Lemma 2.31.
Let R → R and R → R be faithfully flat homomorphisms of commutative rings.Then tensor product induces an equivalence between the category of vertex operator algebrasover R and the category of gluing data for vertex operator algebras over the diagram R → R ⊗ R R ← R . This also yields equivalences between the corresponding categories of self-dualM¨obius objects. Proof .
We first note that faithful flatness implies the tensor product functor preserves the finiteprojective property of graded pieces, so it takes vertex operator algebras over R to gluing datafor vertex operator algebras over the diagram R → R ⊗ R R ← R . Similarly, tensor pro-duct preserves the self-dual M¨obius structure. We obtain the quasi-inverse functor by applyingLemma 2.13 to obtain the underlying modules and the structure maps. It remains to check thatthe conformal vertex algebra over R that we obtain is in fact a vertex operator algebra over R ,and this follows from Proposition 2.23. For self-dual M¨obius structure, the claim follows fromProposition 2.28. (cid:4) Lemma 2.32.
Let R → R and R → R be homomorphisms of commutative rings of the formgiven in either Lemma or Lemma . Let V and V be vertex operator algebras over R and R , respectively, such that V ⊗ R R and V ⊗ R R are isomorphic as vertex operator al-gebras over R ⊗ R R . Any choice of gluing datum induces injective group homomorphisms Aut R V → Aut R ⊗ R R (cid:0) V ⊗ R R (cid:1) and Aut R V → Aut R ⊗ R R (cid:0) V ⊗ R R (cid:1) , and if the cor-responding double coset space is a singleton, then there is a unique isomorphism type of vertexoperator algebra V over R such that V ∼ = V ⊗ R R and V ∼ = V ⊗ R R . In particular, ifthe inclusions are isomorphisms, then V is unique, and the natural inclusions of Aut R V into Aut R V and Aut R V are isomorphisms. The corresponding statement also holds for M¨obiusself-dual vertex operator algebras. Proof .
There is a natural action of Aut R V × Aut R V on the set of gluing data by isomor-phisms, given by f (cid:55)→ ( g ⊗ id) ◦ f ◦ ( g ⊗ id) − = f (cid:48) , because such isomorphisms are defined by the condition that the following diagram commutes: V ⊗ R R f (cid:47) (cid:47) g ⊗ id (cid:15) (cid:15) V ⊗ R R g ⊗ id (cid:15) (cid:15) V ⊗ R R f (cid:48) (cid:47) (cid:47) V ⊗ R R . Because isomorphisms of gluing data correspond bijectively to isomorphisms of vertex opera-tor algebras over R , we get a unique vertex operator algebra over R if and only if the set oforbits is a singleton. For any two choices of gluing data f and f (cid:48) , there exists a unique h ∈ Aut R ⊗ R R (cid:0) V ⊗ R R (cid:1) such that h ◦ f = f (cid:48) . This establishes a non-canonical bijection between Self-Dual Integral Form of the Moonshine Module 15gluing data and elements of Aut R ⊗ R R (cid:0) V ⊗ R R (cid:1) . Fixing some f as a base point induces inclu-sion maps φ : Aut R V → Aut R ⊗ R R (cid:0) V ⊗ R R (cid:1) and φ : Aut R V → Aut R ⊗ R R (cid:0) V ⊗ R R (cid:1) ,and identifies the action with h (cid:55)→ φ ( g ) hφ ( g ) − . This identifies orbits with double cosets in φ (cid:0) Aut R V (cid:1) \ Aut R ⊗ R R (cid:0) V ⊗ R R (cid:1) /φ (cid:0) Aut R V (cid:1) .For the special case that φ and φ are isomorphisms, there is clearly a single double coset,hence a single isomorphism type of vertex operator algebra V over R . Furthermore, each R ⊗ R R -automorphism of (cid:0) V ⊗ R R (cid:1) descends to an R -automorphism of V , by effective descent ofmorphisms given by the equivalences of categories in Lemmas 2.30 and 2.31 (cid:4) The following fact is rather elementary, but we will use it a lot.
Lemma 2.33.
Let V be a vertex operator algebra over a ring R with M¨obius structure and aninvariant bilinear form, and suppose V is self-dual. Let g be an automorphism of order n ∈ Z ≥ that preserves the bilinear form. If n is invertible in R and µ n ( R ) ∼ = Z /n Z , then V is isomor-phic to the direct sum of its g -eigenspaces (cid:76) ζ ∈ µ n ( R ) V g = ζ , as an R -module with bilinear form.Furthermore, each eigenspace is a module for the fixed-point vertex operator subalgebra V g =1 ,and the bilinear form induces a homogeneous perfect pairing between V g = ζ and V g =1 /ζ for all ζ ∈ µ n ( R ) . Proof .
The hypotheses on R imply any action of Z /n Z on an R -module gives a unique de-composition into eigenspaces for n -th roots of unity. For V , the decomposition yields modulesfor the fixed-point vertex operator subalgebra. Self-duality of V implies the eigenspaces withreciprocal eigenvalues are paired. (cid:4) We suspect that with enough work, one may extend the definition of abelian intertwining algebra,introduced in [19], to allow definition over arbitrary commutative rings with suitable divisibilityproperties and equipped with distinguished roots of unity. However, the results in this paperonly require us to consider base rings R that lie inside C , and the formalism becomes somewhatmore manageable in this case.Recall from [35] that if we are given abelian groups ( A, +) and ( B, · ), an abelian 3-cocycleon A with coefficients in B is a pair ( F : A × A × A → B, Ω : A × A → B ) of functions satisfyingthe Eilenberg–MacLane cocycle conditions:1) F ( i, j, k ) · F ( i, j + k, (cid:96) ) · F ( j, k, (cid:96) ) = F ( i + j, k, (cid:96) ) · F ( i, j, k + (cid:96) ) for all i, j, k, (cid:96) ∈ A ,2) F ( i, j, k ) − · Ω( i, j + k ) · F ( j, k, i ) − = Ω( i, j ) · F ( j, i, k ) − · Ω( i, k ),3) F ( i, j, k ) · Ω( i + j, k ) · F ( k, i, j ) = Ω( j, k ) · F ( i, k, j ) · Ω( i, k ),and the restriction of Ω to the diagonal i (cid:55)→ Ω( i, i ) gives a bijection from abelian cohomologyclasses to B -valued quadratic functions on A . We say that an abelian 3-cocycle is normalized if F ( i, j,
0) = F ( i, , k ) = F (0 , j, k ) = 1 and Ω( i,
0) = Ω(0 , j ) = 1 for all i, j, k ∈ A , and we notethat any abelian cohomology class admits a normalized representative. Definition 2.34.
Let N ∈ Z ≥ , let R be a commutative subring of C containing N and e (1 / N ).Let A be an abelian group, let ( F, Ω) be a normalized abelian 3-cocycle on A with coefficientsin R × , and assume Ω( a, a ) N = 1 for all a ∈ A . We then let q Ω be the unique N Z / Z -valuedquadratic form on A such that Ω( a, a ) = e ( q Ω ( a )), and let b Ω : A × A → N Z be any fixed functionthat reduces to the bilinear form attached to q Ω , i.e., b Ω ( a, b ) ≡ q Ω ( a + b ) − q Ω ( a ) − q Ω ( b ) mod Z .Then an abelian intertwining algebra over R of level N and half central charge c ∈ R associatedto the data ( A, F,
Ω) is an R -module V equipped with1) a N Z × A -grading V = (cid:76) n ∈ N Z V n = (cid:76) i ∈ A V i = (cid:76) n ∈ N Z ,i ∈ A V in ,6 S. Carnahan2) a left-multiplication operation Y : V → (End V ) (cid:2)(cid:2) z /N , z − /N (cid:3)(cid:3) that we expand as Y ( a, z )= (cid:80) n ∈ N Z a n z − n − , and3) distinguished vectors ∈ V and ω ∈ V ,satisfying the following conditions for any i, j, k ∈ A , a ∈ V i , b ∈ V j , u ∈ V k , and n ∈ N Z :1) a n b ∈ V i + j ,2) a n b = 0 for n sufficiently large,3) Y ( , z ) a = a ,4) Y ( a, z ) ∈ a + zV [[ z ]],5) the coefficients of Y ( ω, z ) = (cid:80) k ∈ Z L k z − k − satisfy the Virasoro relations at half centralcharge c ,6) L v = nv for all v ∈ V n ,7) dd z Y ( a, z ) = Y ( L − a, z ) for all a ∈ V ,8) Y ( a, z ) b = (cid:80) k ∈ b Ω ( i,j )+ Z a k bz − k − ,9) the Jacobi identity: x − (cid:18) y − zx (cid:19) b Ω ( i,j ) δ (cid:18) y − zx (cid:19) Y ( a, y ) Y ( b, z ) u − B ( i, j, k ) x − (cid:18) z − y − x (cid:19) b Ω ( i,j ) δ (cid:18) z − y − x (cid:19) Y ( b, z ) Y ( a, y ) u = F ( i, j, k ) z − δ (cid:18) y − xz (cid:19) Y ( Y ( a, x ) b, z ) (cid:18) y − xz (cid:19) − b Ω ( i,k ) u, where B ( i, j, k ) = Ω( i,j ) F ( i,j,k ) F ( j,i,k ) , and ( − x ) k/N is interpreted as e ( k/ N ) x k/N .We say an abelian intertwining algebra V is well-graded if each piece V in is a finite projective R -module. A M¨obius structure on V is an A -homogeneous action of U ( sl ) Z = (cid:10) L ( n ) − , (cid:0) L n (cid:1) , L ( n )1 (cid:11) n ≥ on the underlying R -module, such that the following conditions hold:1) Y ( u, z ) = (cid:80) n ≥ L ( n ) − uz n for all u ∈ V ,2) if u has weight k , then (cid:0) L n (cid:1) u = (cid:0) kn (cid:1) u ,3) the subalgebra generated by { L ( n )1 } n ≥ acts locally nilpotently on V ,4) for all i ∈ {− , , } , L i Y ( u, z ) v − Y ( u, z )( L i v ) = i +1 (cid:80) j =0 (cid:0) i +1 j (cid:1) z j Y ( L i − j u, v ).We will need the following skew-symmetry property, which can be (more or less) found inthe proof of Proposition 2.6 of [28]. Lemma 2.35. If V is an abelian intertwining operator algebra, then V satisfies the followingskew-symmetry property: Y ( a, y ) b = Ω( i, j )e yL − Y (cid:0) b, e π i y (cid:1) a for a ∈ V i and b ∈ V j . In particular, for any n ∈ b Ω ( i, j ) + Z , a n b is an R -linear combinationof terms of the form b (cid:48) k a (cid:48) for b (cid:48) ∈ V j and a (cid:48) ∈ V i . Self-Dual Integral Form of the Moonshine Module 17
Proof .
We consider the Jacobi identity with vectors a , b , . Multiplying both sides by x b Ω ( i,j )+ N for some N ∈ Z > and taking Res x yields:( y − z ) b Ω ( i,j )+ N Y ( a, y ) Y ( b, z ) − B ( i, j, ( − π i)( b Ω ( i,j )+ N ) ( z − y ) b Ω ( i,j )+ N Y ( b, z ) Y ( a, y ) on the left side, and some power series with powers of x uniformly bounded below times Y ( Y ( a, x ) b, z ) on the right. When N is sufficiently large, the right side therefore vanishes.Because our cocycle is normalized, we have the following equality in V i + j (cid:2)(cid:2) y ± , z ± (cid:3)(cid:3) :( y − z ) b Ω ( i,j )+ N Y ( a, y )e zL − b = Ω( i, j )e ( − π i)( b Ω ( i,j )+ N ) ( z − y ) b Ω ( i,j )+ N Y ( b, z )e yL − a. This is essentially a version of the “operator-valued rational function” statement in Remark 12.31of [19]. The left side has no negative powers of z , and the right side has no negative powers of y ,so this is an equality in V i + j [[ y, z ]]. By the exponential translation formula, we find that( y − z ) b Ω ( i,j )+ N Y ( a, y )e zL − b = Ω( i, j )e ( − π i)( b Ω ( i,j )+ N ) ( z − y ) b Ω ( i,j )+ N e yL − Y ( b, z − y ) a. Because this is an equality in V i + j [[ y, z ]], we may set z = 0 to get the answer we want, aftersubstituting e π i y + z for z − y . (cid:4) We also need an associativity result, adapted from Lemma 3.12 of [33]. Li credits [20] withthis result, and I suspect he is referring to an argument leading up to Remark 3.3 in that paper.
Lemma 2.36.
Let a ∈ V i , b ∈ V j , u ∈ V k be homogeneous elements of an abelian intertwiningalgebra. Let r ∈ b Ω ( i, j + k ) + Z and s ∈ b Ω ( j, k ) + Z . Let l ∈ b Ω ( i, k ) + Z ≥ be such that z l Y ( a, z ) u ∈ V [[ z ]] , and let m ∈ Z ≥ be such that z m + s Y ( b, z ) u ∈ V [[ z ]] . Then a r b s u = F ( i, j, k ) m (cid:88) t =0 (cid:88) n ∈ Z ≥ (cid:18) r − lt (cid:19)(cid:18) ln (cid:19) ( a r − l − t + n b ) s + l + t − n u. In particular ( since this is always a finite sum ) , a r b s u is an R -linear combination of elementsof the form ( a p b ) q u . Proof .
By Proposition 2.5 of [34] (which is essentially the part of Remark 12.31 in [19] thatapplies to “generalized rationality of iterates”), the following associativity rule holds:( x + z ) l Y ( a, x + z ) Y ( b, z ) u = F ( i, j, k )( z + x ) l Y ( Y ( a, x ) b, z ) u as elements of z − m − s x − b Ω ( i,j ) − N V i + j + k [[ z, x ]] for some N ∈ Z . To extract a r b s u from the leftside, we rewrite: a r b s u = Res y Res z y r z s Y ( a, y ) Y ( b, z ) u = Res x Res y Res z x − δ (cid:18) y − zx (cid:19) y r z s Y ( a, y ) Y ( b, z ) u = Res x Res y Res z y − δ (cid:18) x + zy (cid:19) y r z s Y ( a, y ) Y ( b, z ) u = Res x Res y Res z y − δ (cid:18) x + zy (cid:19) ( x + z ) r z s Y ( a, x + z ) Y ( b, z ) u = Res x Res z ( x + z ) r z s Y ( a, x + z ) Y ( b, z ) u. Combining this with the associativity rule, we find that a r b s u = Res x Res z ( x + z ) r − l z s (cid:2) ( x + z ) l Y ( a, x + z ) Y ( b, z ) u (cid:3) = F ( i, j, k ) Res x Res z ( x + z ) r − l z s (cid:2) ( z + x ) l Y ( Y ( a, x ) b, z ) u (cid:3) . z in the last expression are bounded below, so the power series ( x + z ) r − l onlycontributes finitely many terms to the residue with respect to x . That is, we may truncate it as p ( x, z ) = m (cid:88) t =0 (cid:18) r − lt (cid:19) x r − l − t z t . We then obtain a r b s u = F ( i, j, k ) Res x Res z p ( x, z ) z s ( z + x ) l Y ( Y ( a, x ) b, z ) u, which immediately yields the answer we want. (cid:4) Definition 2.37.
Let V be a M¨obius abelian intertwining algebra over R associated to thedata ( A, F,
Ω). Then an invariant bilinear form on V is an inner product such that ( u n v, w ) =( v, e ( k/ (cid:80) i ≥ (cid:0) L ( i )1 u (cid:1) k − i − n − w ) whenever u has weight k , and ( u, v ) = 0 if u ∈ V a and v ∈ V b for a + b (cid:54) = 0. Lemma 2.38.
Let V be a well-graded M¨obius abelian intertwining algebra over R associated tothe data ( A, F, Ω) , and suppose V admits an invariant bilinear form. Then this form is self-dualif and only if for each a ∈ A the form induces V -module isomorphisms V a ∼ = ( V − a ) ∨ . Proof .
Any invariant form induces a V -module map φ a : V a → ( V − a ) ∨ by setting φ a ( v )( w ) =( v, w ), because compatibility with the V -action follows from the defining relation applied tohomogeneous vectors u ∈ V . By the orthogonality of V a and V b for a + b (cid:54) = 0, self-duality isthen equivalent to φ a being an isomorphism for all a . (cid:4) The following two results will form a primary engine behind our construction.
Proposition 2.39.
Let R be a subring of C containing /N and e π i /N . Let V be an abelianintertwining algebra over C associated to the data ( A, F, Ω) , equipped with an invariant bilinearform. Suppose the following properties hold: . V is self-dual with respect to the invariant form. . Each V a is an irreducible V -module. . V is generated by abelian intertwining subalgebras (cid:8) V A i (cid:9) i ∈ I , where A i range over a set ofsubgroups of A that generate A , and V A i = (cid:76) a ∈ A i V a . . We are given M¨obius R -forms V A i R of each V A i , which coincide on pairwise intersectionsof the subalgebras V A i .Then the abelian intertwining subalgebra V R of V over R generated by (cid:8) V A i R (cid:9) i ∈ I is a M¨obius R -form V R of V , i.e., V R ⊗ R C ∼ = V . Proof .
By induction, it suffices to consider the case where I = { , } , and A ∩ A = { } . Asit happens, this is the only case that we will use in this paper.We define V R to be the sub- R -module of V generated by products of elements in V A i R . Wethen define V aR to be the part of V R of degree a . In the course of this proof, we will showthat if a ∈ A i , then V aR is equal to the degree a part of V A i R , but before we prove this, we willwrite (cid:0) V A i R (cid:1) a for the latter space to distinguish them.To show that V R is an R -form of V , we fix a ∈ A and consider the base change map φ : V aR ⊗ R C → V a . It suffices to show that φ is an isomorphism. The map φ is surjective, because we maywrite any element of V a as a C -linear combination of products of elements of V A i , and theseelements in turn are C -linear combinations of elements of V A i R . Self-Dual Integral Form of the Moonshine Module 19To show that φ is injective, it suffices to show that any R -linearly independent set in V aR is C -linearly independent. We first show that any element of V R is an R -linear combination ofproducts u r v , where u ∈ V A R and v ∈ V A R . By skew-symmetry as in Lemma 2.35, v s u is an R -linear combination of elements of the desired form, so we may switch A and A . By inductionon word length, it suffices to show that any length 3 words of the form a r b s w and ( a r b ) s w havethis form, where a, b, w ∈ V A R ∪ V A R . Again by skew-symmetry, we do not need to consider thecase ( a r b ) s w separately. Under this reduction, if b and w are both in either V A R or V A R , thenthere is nothing to show. We therefore are reduced to considering a r b s w in the following cases:1) a, b ∈ V A R and w ∈ V A R ,2) b ∈ V A R and a, w ∈ V A R .These cases are taken to each other by applying skew-symmetry to b and w and switching A with A . It therefore suffices to consider the first case, which is handled by Lemma 2.36.Thus, V R is R -spanned by products of the form u r v , where u ∈ V A R and v ∈ V A R . It followsimmediately that V aR = (cid:0) V A i R (cid:1) a whenever a ∈ A i .To show that any R -linearly independent set in V aR is C -linearly independent, we considerthe contrapositive. Take a finite subset (cid:8) u , r u , , . . . , u ,kr k u ,k (cid:9) ⊂ V aR , and suppose c u , r u , + · · · + c k u ,kr k u ,k = 0 for some coefficients c , . . . , c k ∈ C , not all zero. We want to show that r u , r u , + · · · + r k u ,kr k u ,k = 0 for some r , . . . , r k ∈ R , not all zero. We may assume thatthis set of vectors is minimal with respect to C -linear dependence, and in particular, that (cid:8) u , r u , , . . . , u ,k − r k − u ,k − (cid:9) is a C -linearly independent set. By rescaling, we may assume u ,kr k u ,k = c u , r u , + · · · + c k − u ,k − r k − u ,k − for uniquely defined c , . . . , c k − ∈ C .Proposition 11.9 of [19] asserts that if w and w are nonzero elements of irreducible mo-dules M and M of a generalized vertex algebra, and Y is an intertwining operator, then Y (cid:0) w , z (cid:1) w = 0 implies Y is identically zero. Here, our hypothesis is that V a and V − a areirreducible V -modules, and the self-duality of the invariant bilinear form on V implies themultiplication operation is nonzero. Thus, for any nonzero v ∈ V − a R , Y (cid:0) v , z (cid:1) is an injectionto z s V a (( z )) for some s ∈ Q . Thus, (cid:8) Y (cid:0) v , z (cid:1) u , r u , , . . . , Y (cid:0) v , z (cid:1) u ,k − r k − u ,k − (cid:9) is a C -linearly independent set in z s V a (( z )), so Y (cid:0) v , z (cid:1) u ,kr k u ,k = b Y (cid:0) v , z (cid:1) u , r u , + · · · + b k − Y (cid:0) v , z (cid:1) u ,k − r k − u ,k − holds if and only if b i = c i for all i ∈ { , . . . , k − } . However, both Y (cid:0) v , z (cid:1) u ,kr k u ,k and eachsummand Y (cid:0) v , z (cid:1) u ,ir i u ,i lie in z s V a R (( z )), and by hypothesis, this space is an R -form of somesubspace of z s V a (( z )). We conclude that all c i lie in the fraction field of R . By clearingdenominators, we see that there is some nonzero x ∈ R such that xc u , r u , + · · · + xc k − u ,k − r k − u ,k − − xu ,kr k u ,k = 0and in particular, the set (cid:8) u , r u , , . . . , u ,kr k u ,k (cid:9) is R -linearly dependent. Thus, φ is injective,and V R ⊗ R C ∼ = V as abelian intertwining algebras.The M¨obius structure on V R follows from the formula describing the action of L ( n )1 on productsof generators. (cid:4) We now refine the previous result, showing that V R is self-dual and well-graded.0 S. Carnahan Proposition 2.40.
Let R be a subring of C containing /N and e π i /N . Let V be an abelianintertwining algebra over C associated to the data ( A, F, Ω) , equipped with an invariant bilinearform. Suppose the following properties hold: . V is self-dual with respect to the invariant form. . Each V a is an irreducible V -module. . V is generated by abelian intertwining subalgebras (cid:8) V A i (cid:9) i ∈ I , where A i range over a set ofsubgroups of A that generate A , and V A i = (cid:76) a ∈ A i V a . . We are given M¨obius R -forms V A i R of each V A i such that the restriction of the invariantbilinear form takes values in R , and V A i R is self-dual with respect to this bilinear form. . The R -forms and invariant bilinear forms coincide on pairwise intersections of the subal-gebras V A i .Then, the following holds: . The invariant bilinear form on V restricts to an R -valued invariant bilinear form on the R -form V R of V given in Proposition . . V R is self-dual with respect to this bilinear form. . V R is the unique R -form of V whose intersection with V A i is V A i R , such that the invariantbilinear form on V restricts to an R -valued invariant bilinear form. . V R is well-graded as an R -module, i.e., the graded components ( V R ) an are finite projective R -modules. Proof .
We begin by considering the invariant inner product. For u r u ∈ V aR and v s v ∈ V − aR ,we apply Lemma 2.36 to see that the inner product is given by (cid:0) v s v , u r u (cid:1) = (cid:18) v , e ( k/ (cid:88) i ≥ (cid:0) L ( i )1 v (cid:1) k − i − s − u r u (cid:19) = e ( k/ (cid:88) i ≥ m (cid:88) t =0 (cid:88) n ∈ Z ≥ (cid:18) k − i − s − − lt (cid:19) ×× (cid:18) ln (cid:19)(cid:0) v , (cid:0)(cid:0) L ( i )1 v (cid:1) k − i − s − − l − t + n u (cid:1) r + l + t − n u (cid:1) , which is an R -linear combination of inner products of vectors in V A R . By hypothesis, this is anelement of R , so the inner product on V R is R -valued. We therefore find that for each a ∈ A and r ∈ Q , there is a canonical map (cid:0) V aR (cid:1) r → Hom R (cid:0)(cid:0) V − aR (cid:1) r , R (cid:1) of R -modules induced by the innerproduct, and by self-duality of the base change to C , this map is injective. To show that V R isself-dual, it suffices to show that this map is surjective.Let f : (cid:0) V − aR (cid:1) r → R be an R -module map. Because V is self-dual, f is induced by taking theinner product with a unique element u ∈ V ar . We shall show that u ∈ (cid:0) V aR (cid:1) r .By precomposing f with any (cid:0) v t (cid:1) ∗ for homogeneous v ∈ V − a R , we obtain an R -linear mapfrom some homogeneous piece (cid:0) V − a R (cid:1) r +wt( v ) − t − to R . By self-duality of V A R , this is necessarilygiven by the inner product with a homogeneous vector v (cid:48) ∈ (cid:0) V a R (cid:1) r +wt( v ) − t − . That is, (cid:0) v t u, − (cid:1) = (cid:0) u, (cid:0) v t (cid:1) ∗ − (cid:1) = f (cid:0)(cid:0) v t (cid:1) ∗ − (cid:1) = ( v (cid:48) , − ) . In other words, all v t operators take u to elements of V a R . This implies that for any v ∈ V − a R , Y (cid:0) v , z (cid:1) u ∈ z s V a R (( z )) for some s ∈ Q . By the skew-symmetry Lemma 2.35, Y ( u, z ) takeselements of V − a R to elements of z s V a R (( z )). Self-Dual Integral Form of the Moonshine Module 21Because V A R is self-dual, we may present as a finite R -linear combination of products ofthe form v k u , where u ∈ V a R and v ∈ V − a R . Then u − = u is a finite R -linear combinationof products of the form u − v k u . By Lemma 2.36, this is an R -linear combination of productsof the form (cid:0) u s v (cid:1) t u . Since each u s v ∈ V a R , we find that u ∈ V aR . Thus, the inner producton V R is self-dual.Self-duality implies each of the graded pieces ( V aR ) r are finite projective R -modules, becausefinite projective R -modules are precisely the dualizable R -modules. That is, V R is well-gradedas an R -module. Furthermore (as an anonymous referee has pointed out), self-duality of V R implies uniqueness of R -forms containing the subalgebras V A i R with R -valued inner product.This is because V R is generated by those subalgebras, so any other such form must strictlycontain V R , contradicting self-duality of V R . (cid:4) Corollary 2.41.
Let A be a finite abelian group, and let A , A be subgroups that generate A .Let n be the exponent of A , and let R be a subring of C containing /n and e (1 / n ) . Let V bea simple vertex operator algebra over C , and suppose V admits an A -grading ( i.e., given by anaction of the Pontryagin dual group A ∗ ) . If we are given self-dual R -forms V R and V R of thevertex operator subalgebras of V given by the parts graded by A and A , such that the A ∩ A -graded subalgebras of V and V are isomorphic, then there exists a unique R -form of V whose A -graded subalgebra is V and whose A -graded subalgebra is V . Proof .
By Theorem 3 of [23] our assumption that V is simple implies the graded pieces of V are simple V -modules. Then the hypotheses of Proposition 2.40 are satisfied, where ( F, Ω) istrivial. (cid:4)
A description of an integral form for a lattice vertex algebra is given in the original paper [2]where vertex algebras are defined, and more properties are established in [5]. We will follow thetreatments in [18] and [38], because the proofs are somewhat more detailed.Let L be an even integral lattice, i.e., a finite rank free abelian group equipped with a Z -valuedbilinear form that is even on the diagonal. Then there is a double cover ˆ L , written as a nontrivialcentral extension by (cid:104) κ (cid:105) ∼ = {± } , that is unique up to non-unique isomorphism. Choosinglifts { e a } a ∈ L of lattice vectors, the cocycle defining the central extension is determined up toequivalence by the signs relating e a e b to e b e a , and in our case it is by e a e b = ( − ( a,b ) e b e a . Thetwisted group ring C { L } is then the quotient of C (cid:2) ˆ L (cid:3) by the ideal ( κ + 1), and we write ι ( e a ) forthe image of e a . The vertex algebra V L is given by the tensor product of C { L } with a Heisenbergvertex algebra M (1) which we will not describe further.If we choose a basis { γ , . . . , γ d } of L , then for any α ∈ L ∨ , we set E − ( − α, z ) = exp (cid:32)(cid:88) n> α ( − n ) n z n (cid:33) = (cid:88) n ≥ s α,n z n . Note that E − ( − α, z ) E − ( α, z ) = 1 for all α ∈ L , i.e., n (cid:80) i =0 s α,i s − α,n − i = δ n, for all n ≥
0. Define( V L ) Z to be the Z -span of s α ,n · · · s α k ,n k e α for α i ∈ { γ , . . . , γ d } , n ≥ · · · ≥ n k , k ≥
0, and α ∈ L . Here, e α denotes the image of ι ( e α ) for some lift e α – this choice of notation hasan ambiguous sign, but in this paper we will not do calculations where the choice of sign isimportant. By Proposition 3.6 of [18], ( V L ) Z is an integral form of V L , generated by e ± γ i , andif L is positive definite and unimodular, then ( V L ) Z is a direct sum of positive definite unimodularlattices under its usual invariant bilinear form. By Proposition 5.8 of [38], there is a conformalelement ω in an integral form for V L if and only if L is unimodular, and the central charge isequal to the rank.2 S. Carnahan Definition 2.42.
Let L be a positive definite even unimodular lattice, and let R be a com-mutative ring. We call the base change ( V L ) Z ⊗ Z R the standard R -form of V L , and denote itby ( V L ) R . Lemma 2.43.
Let L be a positive definite even unimodular lattice, and let R be a commutativering. The standard R -form ( V L ) R is a M¨obius vertex operator algebra over R , and admits aninvariant bilinear form for which it is self-dual. Proof .
The M¨obius claim is a special case of Lemma 5.6 of [5], and in fact Borcherds proves theclaim for an action of an integral form of the universal enveloping algebra of the Virasoro algebra.The existence of the bilinear form is asserted in [2], and self-duality is proved as Proposition 3.6of [18]. (cid:4)
We describe some finer details of automorphisms.
Proposition 2.44.
Let L be a positive definite even unimodular lattice of rank d with no roots ( i.e., no vectors of norm . The contravariant functor that sends an affine scheme Spec R tothe automorphism group of ( V L ) R is represented by a finite type affine group scheme Aut( V L ) Z over Z . This group scheme has the form T O (cid:0) ˆ L (cid:1) , where T is a normal subgroup scheme that isa split torus over Z , and O (cid:0) ˆ L (cid:1) is the finite flat group scheme of isometries of the double coverof L , and lies in a canonical exact sequence → Hom(
L, µ ) → O (cid:0) ˆ L (cid:1) → O ( L ) → . The torus T is given as the diagonalizable group D ( L ) = Spec Z [ L ] , isomorphic to G dm . Theintersection between T and O (cid:0) ˆ L (cid:1) is Hom(
L, µ ) , which is isomorphic to the constant groupscheme {± } d over any field of characteristic not equal to . Proof .
In the beginning of Section 2 of [17], we have a description of the automorphism groupof any finitely generated vertex operator algebra V over C as a closed subgroup of GL( U ) for U = (cid:76) km =0 V m a generating subspace. The arguments given there extend straightforwardly toany finitely generated vertex operator algebra over a commutative ring in a way that commuteswith base change, so the automorphism functor is represented by a finite type affine groupscheme.Let t R denote the weight 1 subspace of ( V L ) R . By our assumption that L has no roots, t isisomorphic to L ⊗ R as an R -module with inner product given by ( a, b ) = a b . By [2], t R hasa canonical Lie algebra structure given by [ a, b ] = a b , and in this case it is abelian. Consideran automorphism σ of ( V L ) R . Because σ respects the weight grading, it acts as an R -linearisometry on t R that induces an isometry on the L -grading of ( V L ) R . Thus, the isometry on t R is defined over Z , i.e., the restriction is given by some ¯ σ ∈ O ( L ).The group O (cid:0) ˆ L (cid:1) acts by permutations on the set { (cid:15) e α } (cid:15) ∈ µ ( R ) ,α ∈ L , and this action inducesvertex operator algebra automorphisms on ( V L ) R that act by O ( L ) on the L -grading. Thereexists a lift τ of ¯ σ to O (cid:0) ˆ L (cid:1) , so ψ = στ − is an automorphism of ( V L ) R that fixes t R pointwise andhence fixes the L -grading. Because the generators e ± γ i are minimal weight in their respective L -graded components, ψ must act on these generators by scalars. Furthermore, Lemma 2.5 in [24](which is proved for the case R = C but extends without change) implies this ψ is necessarily anelement of the torus Hom( L, R × ), which is the group of R -points of D ( L ). The L -grading givenby assigning degree α to e α endows ( V L ) Z with a Z [ L ]-comodule structure, so the automorphismgroup contains the rank d split torus D ( L ). We conclude that Aut( V L ) Z = T O (cid:0) ˆ L (cid:1) .Because T acts trivially on t R , it is clear that the intersection of T ( R ) with O (cid:0) ˆ L (cid:1) is thepreimage of the identity element of O ( L ), and this is precisely Hom( L, µ ( R )). (cid:4) Self-Dual Integral Form of the Moonshine Module 23
Lemma 2.45.
Let L be a positive definite even unimodular lattice of rank d with no roots, andlet ¯ g and ¯ h be commuting automorphisms of L . Then for any lifts ˆ g of ¯ g and ˆ h of ¯ h to Aut( V L ) R ,we have ˆ g ˆ h = c ˆ g, ˆ h ˆ h ˆ g for some c ˆ g, ˆ h ∈ T ( R ) . Furthermore, if ˜ g = γ ˆ g and ˜ h = δ ˆ h are differentlifts, where γ, δ ∈ T ( R ) , then c ˜ g, ˜ h = ¯ g · δδ γ ¯ h · γ c ˆ g, ˆ h , where ¯ g · δ denotes the image of δ under the canonical action of Aut L on T ( R ) = Hom( L, R × ) . Proof .
The first claim is straightforward from the description of the automorphism groupof ( V L ) R given in Proposition 2.44. The second claim follows from a short calculation, essentiallyusing the fact that ˆ gδ = (¯ g · δ )ˆ g . (cid:4) Lemma 2.46.
Let L be a positive definite even unimodular lattice of rank d with no roots, let ¯ g be a fixed-point free automorphism of L such that all nontrivial powers are also fixed-point free,and let n be its order. Then the set of automorphisms g of ( V L ) R that map to ¯ g in O ( L ) isa torsor under T ( R ) , and each such lift has order n . Given any pair ˆ g and ˜ g of lifts of ¯ g to Aut( V L ) R , there exists an extension R (cid:48) of R given by adjoining finitely many roots of units, suchthat ˜ g and ˆ g are conjugate in Aut( V L ) R (cid:48) , in fact by an element of T ( R (cid:48) ) . In particular, if R × is n -divisible ( i.e., each unit has an n th root ) , then all lifts of ¯ g are conjugate in Aut( V L ) R ,and if R is a subring of C containing e (1 / n ) , then all lifts of ¯ g in Aut( V L ) Z are conjugate in Aut( V L ) R . Finally, if R is a subring of C containing n and e (1 /n ) , then for any fixed lift g of ¯ g , ( V L ) R splits into a direct sum of irreducible ( V L ) gR -modules (cid:8) ( V L ) g = e ( k/n ) R (cid:9) n − k =0 , and the invariantbilinear form induces a homogeneous perfect pairing between ( V L ) g = e ( k/n ) R and ( V L ) g = e ( − k/n ) R . Proof .
The parametrization of lifts of ¯ g by a T ( R )-torsor is given in Proposition 2.44. To showthat each lift has order n , we first show that there is a lift to O (cid:0) ˆ L (cid:1) with order n . The obstructionto the existence of an order n lift of any order n automorphism is described in [3, Lemma 12.1]in the case of the Leech lattice, but the argument applies in general. Namely, if n is odd, thereis no obstruction, and if n is even, there is an obstruction if and only if (cid:0) ¯ g n/ v, v (cid:1) is an oddinteger for some v ∈ L . Since we assume ¯ g n/ is a fixed-point free element of order 2, all of itseigenvalues are −
1, so (cid:0) ¯ g n/ v, v (cid:1) = ( − v, v ) ∈ Z , and the obstruction vanishes.For the general problem of conjugation, we consider lifts ˆ g and ˜ g of ¯ g , and note that theysatisfy γ ˆ g = ˜ g for a unique γ ∈ T ( R ). If there is some δ ∈ T ( R (cid:48) ) such that δ − ˆ gδ = ˜ g forsome extension R (cid:48) of R , then by Lemma 2.45, we have δ − (¯ g · δ ) = γ . We note that by theidentification T ( R ) = Hom( L, R × ), we may define γ by the values in R × it takes on a basis of L .Then, if we extend R to R (cid:48) by adjoining n -th roots of those values, we find that γ has an n -throot γ (cid:48) ∈ T ( R (cid:48) ). Then by setting δ = γ (cid:48) (¯ g · γ (cid:48) ) · · · (cid:0) ¯ g n − · γ (cid:48) (cid:1) n ∈ T ( R (cid:48) ) , we find that δ − (¯ g · δ ) = ( γ (cid:48) ) n = γ . Thus, δ − ˆ gδ = δ − (¯ g · δ )ˆ g = γ ˆ g = ˜ g, so ˆ g and ˜ g are conjugate by an element of T ( R (cid:48) ).For the claim about n -divisible R × , the identification T ( R ) = Hom( L, R × ) implies T ( R ) isalso n -divisible. For the claim about subrings of C , we note that Aut( V L ) Z = O (cid:0) ˆ L (cid:1) , so anydiscrepancy γ of lifts of ¯ g necessarily lies in T ( Z ) = Hom( L, ± R contains e (1 / n ),then T ( R ) contains all n -th roots of elements of T ( Z ).The last claim follows immediately from Lemma 2.33. (cid:4) Proposition 2.47.
Let Λ be the Leech lattice, i.e., the unique positive definite even unimo-dular lattice of rank with no roots, and let ¯ g be a fixed-point free automorphism of primeorder p ( therefore, an element in one of the classes a , a , a , a , a according to the notationof [7]) . Let R be a subring of C that contains e (1 /p ) , and let g be a lift of ¯ g to ( V Λ ) R . Then C Aut( V Λ ) R ( g ) ∼ = C Aut V Λ ( g ) , and in particular, has the form p / ( p − .C Co (¯ g ) . Proof .
It suffices to show that any automorphism of ( V Λ ) C that commutes with g also preservesthe standard R -form, viewed as an R -submodule. From the description of the automorphismgroup given in Proposition 2.44, it is clear that both the C -form and the R -form have the O (cid:0) ˆΛ (cid:1) part of the automorphism group in common, so it suffices to show that all centralizing complexelements in the torus T = D (Λ) are defined over R .We claim that the split torus T equivariantly decomposes under ¯ g as a direct sum of p − copies of the torus G p − m with ¯ g acting on points as( a , . . . , a p − ) (cid:55)→ (cid:18) a · · · a p − , a , . . . , a p − (cid:19) . To show this, we may use the fact that D gives an involutive anti-equivalence between thecategory of split tori and the category of free abelian groups of finite rank (see, e.g., [15, Ex-pos´e VIII, Section 1]). The claim then follows from the classification of indecomposable Z -free Z [ Z /p Z ]-modules given in [16] (see also [14, Theorem 74.3]), and in particular, the fact that onlyone isomorphism type is fixed-point free.Any centralizing C -point ( a , . . . , a p − ) in the torus G p − m is fixed by this action of ¯ g , so thecoordinates satisfy a = · · · = a p − = 1 a · · · a p − ∈ µ p ( C ) . However, R contains a full set of p -th roots of unity by our hypothesis, so all of the centralizingcomplex points in T are defined over R . (cid:4) We recall that if V is a simple, C -cofinite, holomorphic vertex operator algebra V , and g isan automorphism of finite order n , then by Theorem 10.3 of [21], there is a unique g -twisted V -module, up to isomorphism (which we will call V ( g )), and its L (0)-spectrum lies in some cosetof n Z in Q . We say that g is anomaly-free if this coset is n Z , and we say that g is anomalousotherwise.The construction that makes this paper possible is the following: By Theorem 5.15 of [45],if V is a simple, C -cofinite, holomorphic vertex operator algebra V of CFT type, and g isan automorphism of finite order n , such that the nontrivial irreducible twisted modules V ( g i )have strictly positive L (0)-spectrum, then there is some t ∈ Z /n Z (uniquely determined by theproperty that the L (0)-spectrum of V ( g ) lies in tn + n Z ) and an abelian intertwining algebrastructure on g V = (cid:76) n − i =0 V ( g i ), graded by an abelian group D that lies in an exact sequence0 → Z /n Z → D → Z /n Z →
0, with addition law determined by the “add with carry” 2-cocycle c t ( i, k ) = (cid:40) , i n + k n < n, t, i n + k n ≥ n, where the notation i n denotes the unique representative of i ∈ Z /n Z in { , . . . , n − } . By loc.cit. Proposition 5.13, the quadratic form q ∆ on D given by conformal weights is isomorphic tothe discriminant form on the even lattice with Gram matrix (cid:0) − t n nn (cid:1) . Self-Dual Integral Form of the Moonshine Module 25Furthermore, by Theorem 5.16, if t = 0 (i.e., g is anomaly-free), then the abelian intertwiningalgebra g V is naturally graded by D = Z /n Z × Z /n Z , such that V is the sum of the degree (0 , i )pieces, and that there is a simple C -cofinite, holomorphic vertex operator algebra V /g of CFTtype given by the sum of the degree ( j,
0) pieces, for 0 ≤ j < n . The natural Z /n Z -grading fromthis decomposition endows V /g with a canonical automorphism g ∗ whose order is equal to | g | ,such that ( V /g ) /g ∗ ∼ = V and g ∗∗ = g . More generally, they showed that if H is any order n subgroup of Z /n Z × Z /n Z that is isotropic with respect to q Ω , then (cid:76) a ∈ H ( g V ) a is a holomorphic C -cofinite vertex operator algebra of CFT type. Proposition 2.48.
Let V be a holomorphic C -cofinite vertex operator algebra of CFT type, andlet g be an anomaly-free automorphism of order n . Suppose both V and V /g admit R -forms V R and ( V /g ) R for some subring R ⊂ C containing /n and e (1 / n ) , such that the R -forms coincidein V ∩ V /g in g V , and both g and g ∗ are automorphisms of the respective R -forms. Supposefurther that both V R and ( V /g ) R admit R -invariant bilinear forms for which they are self-dual,and that coincide on their intersection in g V , and assume that g and g ∗ preserve the bilinearform. Let G be the automorphism group of V R and let G ∗ be the automorphism group of ( V /g ) R .Then: . The abelian intertwining algebra g V has a unique R -form ( g V ) R with a bilinear formextending those on V R and ( V /g ) R , and it is self-dual with respect to this form. . The group of homogeneous automorphisms of ( g V ) R is equal to a central extension of C G ( g ) by (cid:104) g ∗ (cid:105) , and also a central extension of C G ∗ ( g ∗ ) by (cid:104) g (cid:105) . . For any divisor d of n , (cid:76) n/d − j =0 (cid:76) d − i =0 ( g V ) dj, ( n/d ) iR is an R -form of V /g d that is self-dualwith respect to the induced invariant inner product. Proof .
The first claim follows immediately from Proposition 2.40.For the second claim, we note that restriction yields the following commutative diagram:Aut( g V ) R (cid:47) (cid:47) (cid:15) (cid:15) C Aut V R ( g ) (cid:15) (cid:15) C Aut(
V/g ) R ( g ∗ ) (cid:47) (cid:47) Aut (cid:0) V gR (cid:1) , where the centralizers of g (resp. g ∗ ) are precisely the groups of automorphisms that are com-patible with the grading by eigenspaces for g (resp. g ∗ ). It suffices to show that the maps outof Aut( g V ) R are surjective with cyclic central kernel of order | g | , and the argument given in theproof of Proposition 2.5.2 of [8] works here with minimal change.For the third claim, the fact that (cid:76) n/d − j =0 (cid:76) d − i =0 g V dj, ( n/d ) i is a holomorphic vertex operatoralgebra follows from the fact that the group of degrees in question is isotropic of order n , and theidentification with V /g d is straightforward. The fact that we have an R -form that is self-dualfollows from the corresponding claims for ( g V ) R . (cid:4) We now use the tools from the previous section to construct R -forms of V (cid:92) , as R ranges oversome cyclotomic S -integer rings. In [1], several constructions of V (cid:92) were given by cyclic orbifolds of odd prime order p on V Λ , andanalyzed using cyclic orbifolds of order 2 p in order to produce a comparison with the original6 S. Carnahanorder 2 orbifold construction of [27]. We will apply a similar method to produce actions of themonster on orbifolds over various rings.To be specific, we consider cyclic orbifolds of V Λ with respect to lifts of fixed-point freeisometries of Λ, such that those whose order is prime yield V (cid:92) , and those whose order is a productof two primes yield V Λ . The key is that we may use Proposition 2.40 to produce self-dual R -forms of abelian intertwining algebras from those orbifolds yielding V Λ , and this automaticallyyields self-dual R -forms for V (cid:92) by restriction.We summarize the information about cyclic orbifolds over C that we need. Lemma 3.1.
Let P = { , , , , } . Then: . P is the set of primes p such that there exists a fixed-point free automorphism of the Leechlattice of order p . . For each p ∈ P , there is a unique conjugacy class [¯ g p ] of fixed-point free automorphismsin Co of order p , and there exists a unique conjugacy class [ g p ] of automorphisms of V Λ lifting [¯ g p ] . For any representative element g p , the order of g p is p , and we have anisomorphism V Λ /g p ∼ = V (cid:92) of vertex operator algebras over C . . Given a pair p , p of distinct elements of P , if there exists an automorphism of Λ of order p p , then there is a unique algebraic conjugacy class [¯ g p p ] of automorphismsof Λ such that ¯ g p p p ∈ [¯ g p ] and ¯ g p p p ∈ [¯ g p ] . When such an automorphism exists, itis fixed-point free, and there exists a unique algebraic conjugacy class [ g p p ] of automor-phisms of V Λ lifting [¯ g p p ] . For any representative element g p p , we have an isomorphism V Λ /g p p ∼ = V Λ of vertex operator algebras over C . Proof .
All of the claims about automorphisms of Λ can be checked by examination of char-acters and power maps of Co in [7]. As it happens, each [¯ g p ] is labeled p a with frame shape1 − /p − p /p − , and each [¯ g p p ] is labeled p p a with frame shape 1 k p − k p − k ( p p ) k , with theexception of order 39, where (39a, 39b) is an algebraically conjugate pair satisfying our criteria.We note that the inadmissible pairs are those satisfying p p ∈ { , } .The claims about existence and uniqueness of lifts of automorphisms to V Λ follow fromSection 4.2 of [32], and a short version of the argument is Proposition 2.1 in [9]. The identificationof prime order cyclic orbifolds with V (cid:92) follows from the main construction of [27] for p = 2,Theorem 1.1 of [10] for p = 3, and Theorem 4.4 in [1] for p = 5 , ,
13. The identification oforder p p cyclic orbifolds with V Λ follows from Theorem 4.1 in [1] for p p ∈ { , , , } , andfor the others, the result follows from essentially the same argument: it suffices to show that theweight 1 subspace of the irreducible twisted module V Λ ( g p p ) has dimension p − p − , andone can do this by manipulating the frame shape. (cid:4) We note that for p ∈ { , , , } , these orbifold constructions were conjectured in [27], andpartially worked out in [22] and [40].We now consider forms over rings. Note that in this section, we are not claiming that anyparticular R -form of V (cid:92) necessarily carries monster symmetry. Lemma 3.2.
Let P = { , , , , } , let p and q be distinct elements of P such that pq (cid:54)∈{ , } , and let R = Z [1 /pq, e (1 / pq )] . Let g be an automorphism of V Λ in the class [ g pq ] described in Lemma namely the fixed-point free class pq a in [7]) such that g preserves ( V Λ ) Z ( such g exists by Lemma . Then g V Λ has a unique self-dual R -form that restricts to thestandard R -forms on V Λ and ( V Λ ) /g ∼ = V Λ . Proof .
Consider the decomposition (cid:76) pq − i =0 V iR of the standard R -form of V Λ into eigen- R -modules for g . This decomposition preserves self-duality by Lemma 2.33. By Lemma 3.1,the abelian intertwining subalgebras (cid:76) pq − i =0 ( g V Λ ) i, and (cid:76) pq − j =0 ( g V Λ ) ,j are isomorphic to V Λ Self-Dual Integral Form of the Moonshine Module 27under its decomposition (cid:76) pq − i =0 V i into eigenspaces for g . Fix embeddings φ, ψ : V Λ → g V Λ suchthat φ | V i is an isomorphism to V i, and ψ | V j is an isomorphism V ,j , and φ | V = ψ | V . Theseembeddings are unique up to composition with automorphisms of V Λ that commute with g .Then restriction to ( V Λ ) R yields embeddings that satisfy the hypotheses of Proposition 2.40.Thus, there is a unique R -form ( g V Λ ) R for g V Λ that extends the R -forms given by φ and ψ , andby the uniqueness of φ and ψ , it is the unique R -form that restricts to the standard R -formon the two copies of V Λ . Furthermore, the proposition asserts that ( g V Λ ) R admits a uniqueinvariant bilinear form that extends the form on each copy of V Λ , and ( g V Λ ) R is self-dual underthis form. (cid:4) Proposition 3.3.
With notation as in Lemma , the abelian intertwining subalgebras q − (cid:77) i =0 p − (cid:77) j =0 ( g V Λ ) pi,qjR and p − (cid:77) i =0 q − (cid:77) j =0 ( g V Λ ) qi,pjR of ( g V Λ ) R are isomorphic self-dual R -forms of V (cid:92) . Proof .
By Lemma 3.1, the two abelian intertwining subalgebras (cid:76) q − i =0 (cid:76) p − j =0 ( g V Λ ) pi,qj and (cid:76) p − i =0 (cid:76) q − j =0 ( g V Λ ) qi,pj are both isomorphic to V (cid:92) . Thus, the abelian intertwining subalgebras (cid:76) q − i =0 (cid:76) p − j =0 ( g V Λ ) pi,qjR and (cid:76) p − i =0 (cid:76) q − j =0 ( g V Λ ) qi,pjR are R -forms for V (cid:92) that are self-dual under theinduced invariant bilinear form.We now consider the automorphism of g V Λ given by switching coordinates, i.e., sending( g V Λ ) i,j (cid:55)→ ( g V Λ ) j,i . Explicitly, the map is defined by taking the composite isomorphisms( g V Λ ) i, φ ← V i ψ → ( g V Λ ) ,i and their inverses, and extending uniquely to g V Λ by the fact that g V Λ is generated by theimages of φ and ψ . This automorphism restricts to an automorphism of the R -form ( g V Λ ) R ,and it transports the two R -forms of V (cid:92) to each other. Thus, the two R -forms of V (cid:92) areisomorphic. (cid:4) Definition 3.4.
We write V (cid:92) [1 /pq, e (1 / pq )] to denote the R -form of V (cid:92) given in Proposition 3.3. Remark 3.5.
Because R is a subring of C containing e (1 / pq ), Lemma 2.46 implies g is uniqueup to conjugation. Thus, the formation of V (cid:92) [1 /pq, e (1 / pq )] does not depend on our choice oflift g .The following lemma implies the automorphism g ∗ is compatible with the cyclic orbifold dualsof V Λ arising from g p and g q , in the sense that ( g p ) ∗ = ( g ∗ ) p and ( g q ) ∗ = ( g ∗ ) q . Lemma 3.6.
With notation as in Lemma , there are unique self-dual R -forms of g p V Λ and g q V Λ such that: g p V i, = V (cid:92) [1 /pq, e (1 / pq )] ( g ∗ ) p = e ( i/q ) and g q V i, = V (cid:92) [1 /pq, e (1 / pq )] ( g ∗ ) q = e ( i/p ) for all i ∈ Z , g p V ,j Λ = ( V Λ ) g p = e ( j/q ) R and g q V ,j Λ = ( V Λ ) g q = e ( j/p ) R for all j ∈ Z .These R -forms naturally embed into ( g V Λ ) R by decomposing into g -eigenspaces. Proof .
Existence follows from applying Proposition 2.40 to the embeddings of ( V Λ ) R and V (cid:92) [1 /pq, e (1 / pq )] into g p V Λ and g q V Λ . Uniqueness follows from the first claim of Proposi-tion 2.48. The abelian intertwining subalgebras (cid:76) q − i =0 (cid:76) pq − j =0 ( g V Λ ) pi,jR and (cid:76) p − i =0 (cid:76) pq − j =0 ( g V Λ ) qi,jR of ( g V Λ ) R are self-dual R -forms of g p V Λ and g q V Λ , equipped with decompositions into g -eigen-spaces. (cid:4) We now consider symmetries of these R -forms of V (cid:92) , where once again R = Z [1 /pq, e (1 / pq )].The next lemma is where we use recent developments in finite group theory. I suspect lesspowerful results can yield the same answer, and I welcome any insights from specialists in finitegroup theory. Lemma 3.7.
Let P = { , , , , } , let p and q be distinct elements of P such that pq (cid:54)∈{ , } . Let g p and g q be elements of M in classes pB and qB , respectively ( i.e., the uniquenon-Fricke classes of those orders ) . Then any subgroup of the monster simple group M thatcontains C M ( g p ) and C M ( g q ) is M itself. Proof .
This follows from known constraints on the maximal subgroups of M , e.g., given in [48].The important point is that for p ∈ { , , } , C M ( g p ) is contained in only one isomorphism typeof maximal subgroup of M . However, for each prime q under consideration, C M ( g q ) containsthe Sylow q -subgroup of M , so it suffices to check that the order of the maximal subgroupcontaining C M ( g p ) has insufficient q -valuation. (cid:4) Lemma 3.8.
The R -form (cid:0) g p V Λ (cid:1) R given in Lemma has automorphism group given bya central extension of C Aut V Λ ( g p ) by (cid:104) ( g ∗ ) p (cid:105) . Furthermore, the abelian intertwining subalge-bra V (cid:92) [1 /pq, e (1 / pq )] admits a faithful action of C M (( g ∗ ) p ) . The same claims also hold with p and q switched. Proof .
The second claim of Proposition 2.48 gives the description of the automorphism groupof (cid:0) g p V Λ (cid:1) R as a central extension of C Aut( V Λ ) R ( g p ) by (cid:104) ( g ∗ ) p (cid:105) , but in Proposition 2.47, we identifythis centralizer with C Aut V Λ ( g p ). By the cyclic orbifold correspondence for prime-order orbifoldsgiven in [1], the restriction of this action to V (cid:92) [1 /pq, e (1 / pq )] induces a surjection to C M (( g ∗ ) p ),with kernel generated by g p . (cid:4) Theorem 3.9.
The embedded R -form V (cid:92) [1 /pq, e (1 / pq )] of V (cid:92) is preserved by the action of M given in [27] . In particular, the automorphism group of V (cid:92) [1 /pq, e (1 / pq )] is the monster simplegroup M . Proof .
By Lemma 3.8, V (cid:92) [1 /pq, e (1 / pq )] admits faithful actions of C M ( g p ) and C M ( g q ). Basechange to C yields V (cid:92) , whose automorphism group is M , so we obtain embeddings of thesegroups in M . However, by Lemma 3.7, these subgroups generate M . We conclude that theaction of M on V (cid:92) preserves the R -form. (cid:4) Corollary 3.10 (weak modular moonshine) . The modular moonshine conjecture holds in ( a slight weakening of ) its original form given in [42] . That is, for each prime p dividing theorder of the monster, and each element g in class p A in the monster, there is a vertex algebra V p over F p n for some n ( where n = 1 is asserted in the original statement ) equipped with an actionof C M ( g ) such that the graded Brauer character (cid:80) Tr( h | V p ) of a p -regular element h ∈ C M ( g ) is equal to the McKay–Thompson series T gh ( τ ) = (cid:80) Tr( gh | V (cid:92) ) . Proof .
Following [6], we may take V p to be the Tate cohomology group ˆ H ( g, V ) for V a formof V (cid:92) defined over a p -adic integer ring. Then this is a special case of the modular moonshineconjecture proved in [6] and [4], under some assumptions that were not known to be true atthe time. The assumption about homogeneous pieces of the Lie algebra m ⊗ Z p was provedas Theorem 7.1 of [5] by applying an integral enhancement of the no-ghost theorem. The lastremaining open assumption is the existence of a suitable Z [1 / V (cid:92) . However, for ourweakened version of Theorem 5.2 of [6], it suffices to have a form of V (cid:92) with monster symmetrydefined over a 2-adic integer ring, i.e., a construction that does not involve division by 2, that Self-Dual Integral Form of the Moonshine Module 29decomposes into 2 A -modules that are submodules of a corresponding form of V Λ . Theorem 3.9gives 4 separate constructions, by setting pq ∈ { , , , } , and the decomposition conditionfollows from the discussion in Section 5 of [6]. In particular, the existence of a normalizingSL ( F ) follows from the computation of normalizers of elementary abelian subgroups of themonster in [47]. The resulting 2-adic forms are defined over unramified extensions of Z ofdegree 4, 6, 12, and 12, respectively. Reduction mod 2 then implies the existence of a suitablevertex algebra V defined over F n for some n ≤ (cid:4) Remark 3.11.
It is possible that the method given in Section 5 of [6] to reduce the conjectural Z [1 / , e (1 / Z [1 / Now that we have R -forms of V (cid:92) with monster-symmetric self-dual invariant bilinear forms forvarious R , we will glue them. In this section, we will describe the glue. Lemma 3.12.
Let p , q , r be distinct elements in P , and let g ∈ p B be an anomaly-free non-Fricke element of order p . Assume pq, pr (cid:54)∈ { , } . Let R = Z [1 /pqr, e (1 / pqr )] . Then (cid:0) V (cid:92) [1 /pq, e (1 / pq )] (cid:1) gR ∼ = (cid:0) V (cid:92) [1 /pr, e (1 / pr )] (cid:1) gR . Proof .
By Lemma 3.6, both sides are isomorphic to ( V Λ ) g ∗ R . (cid:4) Lemma 3.13.
Let p ∈ { , } . Then there exists a subgroup H p ⊂ M such that . H p is isomorphic to ( Z /p Z ) . . All non-identity elements in H p lie in the conjugacy class p B, i.e., they are non-Fricke.
Proof .
The existence of p B-pure non-cyclic elementary abelian subgroups can be extractedfrom the table “maximal p -local subgroups” in [12]. (cid:4) Proposition 3.14.
Let p ∈ { , } , and let q, r be distinct elements of P \ { p } . Let R = Z [1 /pqr, e (1 / pqr )] . Then V (cid:92) [1 /pq, e (1 / pq )] ⊗ Z [1 /pq,e (1 / pq )] R ∼ = V (cid:92) [1 /pr, e (1 / pr )] ⊗ Z [1 /pr,e (1 / pr )] R. Proof .
Let H p be a group of the form given in Lemma 3.13, and let g ∈ H p \ { } be a nontrivialelement.Decomposing V pq = (cid:0) V (cid:92) [1 /pq, e (1 / pq )] (cid:1) R and V pr = (cid:0) V (cid:92) [1 /pr, e (1 / pr )] (cid:1) R into eigenmodulesfor H p , we obtain V i,jpq and V i,jpr , for i, j ∈ { , . . . , p − } . Because H p is generated by p B-elements,we have isomorphisms: p − (cid:77) i =0 V ,ipq ∼ = V gpq ∼ = p − (cid:77) i =0 V i, pq , p − (cid:77) i =0 V ,ipr ∼ = V gpr ∼ = p − (cid:77) i =0 V i, pr . Lemma 3.12 yields V gpq ∼ = V gpr , so all of the vertex operator subalgebras are isomorphic. Both V (cid:92) [1 /pq, e (1 / pq )] ⊗ Z [1 /pq,e (1 / pq )] R and V (cid:92) [1 /pr, e (1 / pr )] ⊗ Z [1 /pr,e (1 / pr )] R are generated bythese vertex operator subalgebras, so by the uniqueness claim of Corollary 2.41, the two R -forms are isomorphic. (cid:4) Proposition 3.15.
Let p , q , r be distinct elements of P , such that pq and pr are not elementsof { , } . Let R = Z [1 /pqr, e (1 / pqr )] . Then V (cid:92) [1 /pq, e (1 / pq )] ⊗ Z [1 /pq,e (1 / pq )] R ∼ = V (cid:92) [1 /pr, e (1 / pr )] ⊗ Z [1 /pr,e (1 / pr )] R. Proof .
We first note that if p ∈ { , } , then this follows from Proposition 3.14. Otherwise, wemay assume q is the smallest prime among p , q , r , and therefore, that q ∈ { , } . Let g, h ∈ M lie in classes pB and rB , respectively. By Lemma 3.12, we have isomorphisms (cid:0) V (cid:92) [1 /pq, e (1 / pq )] (cid:1) gR ∼ = (cid:0) V (cid:92) [1 /pr, e (1 / pr )] (cid:1) gR , and (cid:0) V (cid:92) [1 /qr, e (1 / qr )] (cid:1) hR ∼ = (cid:0) V (cid:92) [1 /pr, e (1 / pr )] (cid:1) hR . However, by Proposition 3.14, we have an isomorphism (cid:0) V (cid:92) [1 /pq, e (1 / pq )] (cid:1) R ∼ = (cid:0) V (cid:92) [1 /qr, e (1 / qr )] (cid:1) R , hence an isomorphism (cid:0) V (cid:92) [1 /pq, e (1 / pq )] (cid:1) hR ∼ = (cid:0) V (cid:92) [1 /pr, e (1 / pr )] (cid:1) hR . Both (cid:0) V (cid:92) [1 /pq, e (1 / pq )] (cid:1) R and (cid:0) V (cid:92) [1 /pr, e (1 / pr )] (cid:1) R are R -forms of abelian intertwining sub-algebras of gh V Λ generated by isomorphic R -forms of (cid:0) V (cid:92) (cid:1) g and (cid:0) V (cid:92) (cid:1) h , so by the uniquenessclaim of Corollary 2.41, they are therefore isomorphic. (cid:4) Remark 3.16.
An alternative proof of the previous proposition can be given by using thesame technique as in Proposition 3.14, because there are p B-pure non-cyclic elementary abeliansubgroups of M for all p ∈ P (see [47]). Before we start gluing forms of V (cid:92) , we show that the output of gluing is unique and has monstersymmetry. Lemma 3.17.
Let i : R → R , i : R → R be homomorphisms of subrings of C , such that R ⊗ R R is also a subring of C , and suppose either: R and R are Zariski localizations of R with respect to a coprime pair of elements, or i and i are faithfully flat.Suppose we have a gluing datum (cid:0) V , V , f (cid:1) of self-dual forms of V (cid:92) over the diagram R → R ⊗ R R ← R , and suppose both forms have monster symmetry. Then there is a uniqueform V of V (cid:92) over R such that both V and V are base-changes of V . Furthermore, this formhas monster symmetry. Proof .
Because the double quotient M \ M / M is a singleton, Lemma 2.32 asserts that we geta unique isomorphism type of R -form, and the same result also shows that this form has monstersymmetry. (cid:4) Lemma 3.18.
Let n ∈ Z ≥ . Suppose we have a commutative diagram R (cid:33) (cid:33) (cid:125) (cid:125) (cid:15) (cid:15) R (cid:33) (cid:33) · · · (cid:15) (cid:15) R n (cid:125) (cid:125) T of inclusions of commutative subrings of C , where all maps are either Self-Dual Integral Form of the Moonshine Module 311) faithfully flat, and R ∩ · · · ∩ R n = R , or Zariski localizations, forming a Zariski open cover of
Spec R .Suppose we are given a self-dual T -form V (cid:92)T of V (cid:92) with M -symmetry, and for each i ∈ { , . . . , n } ,we are given a self-dual R i -form V (cid:92)R i of V (cid:92) with M -symmetry, together with an isomorphism V (cid:92)R i ⊗ R i T ∼ → V (cid:92)T . Then there exists a self-dual R -form V (cid:92)R of V (cid:92) , unique up to isomorphism,such that base change along the diagram of inclusions yields the original diagram of forms. Inparticular, for each i ∈ { , . . . , n } , we have V (cid:92)R i ∼ = V (cid:92)R ⊗ R R i . Proof .
The case n = 1 is trivial, and the case n = 2 is covered in Lemma 3.17. When n ≥
3, weapply induction, assuming existence and uniqueness for all smaller collections of rings. Then,we may reduce the question to the case n = 3 by partitioning { , . . . , n } into three nonemptysets X , Y , Z , and introducing subrings R X = (cid:84) i ∈ X R i , R Y = (cid:84) i ∈ Y R i , and R Z = (cid:84) i ∈ Z R i of T .By our induction assumption, we have unique self-dual forms V (cid:92)X , V (cid:92)Y , V (cid:92)Z of V (cid:92) over R X , R Y , R Z satisfying the expected tensor product compatibility. It suffices to show that gluing to makean R X ∩ R Y -form V (cid:92)XY followed by gluing with V (cid:92)Z yields an R -form V (cid:92)XY,Z that is isomorphicto the R -form V (cid:92)X,Y Z we get by forming an R Y ∩ R Z -form V (cid:92)Y Z followed by gluing with V (cid:92)X .By uniqueness of pairwise gluing, to obtain the isomorphism V (cid:92)XY,Z ∼ = V (cid:92)X,Y Z it suffices toshow that V (cid:92)XY,Z ⊗ R R X ∼ = V (cid:92)X and V (cid:92)XY,Z ⊗ R ( R Y ∩ R Z ) ∼ = V (cid:92)Y Z . The first isomorphism followsfrom the fact that the formation of V (cid:92)XY,Z yields isomorphisms V (cid:92)X ∼ = V (cid:92)XY ⊗ R X ∩ R Y R X ∼ = V (cid:92)XY,Z ⊗ R ( R X ∩ R Y ) ⊗ R X ∩ R Y R X ∼ = V (cid:92)XY,Z ⊗ R R X . A similar argument then yields analogous expansions of V (cid:92)Y and V (cid:92)Z . The second isomorphismthen follows from the uniqueness statement of Lemma 3.17. To elaborate, both V (cid:92)XY,Z ⊗ R ( R Y ∩ R Z ) and V (cid:92)Y Z are self-dual R Y ∩ R Z -forms satisfying the property that tensor product with R Y and R Z yield V (cid:92)Y and V (cid:92)Z , respectively. They are therefore isomorphic over R Y ∩ R Z . (cid:4) Lemma 3.19.
Let p , q , r be distinct elements of P , such that pq and pr are not in { , } .Then, there exists a unique self-dual Z [1 /pqr, e (1 / p )] -form V (cid:92) [1 /pqr, e (1 / p )] of V (cid:92) such that: V (cid:92) [1 /pqr, e (1 / p )] ⊗ Z [1 /pqr,e (1 / p )] Z [1 /pqr, e (1 / pq )] ∼ = V (cid:92) [1 /pq, e (1 / pq )] ⊗ Z [1 /pq,e (1 / pq )] Z [1 /pqr, e (1 / pq )] ,V (cid:92) [1 /pqr, e (1 / p )] ⊗ Z [1 /pqr,e (1 / p )] Z [1 /pqr, e (1 / pr )] ∼ = V (cid:92) [1 /pr, e (1 / pr )] ⊗ Z [1 /pr,e (1 / pr )] Z [1 /pqr, e (1 / pr )] . Furthermore, this form has monster symmetry.
Proof .
By Proposition 3.15, we have a gluing datum for the diagram Z [1 /pqr, e (1 / pq )] → Z [1 /pqr, e (1 / pqr )] ← Z [1 /pqr, e (1 / pr )] in self-dual vertex operator algebras. By Lemma 3.18we obtain a unique self-dual vertex operator algebra over Z [1 /pqr, e (1 / p )] such that tensorproduct yields the input vertex operator algebras, and furthermore, this vertex operator algebrahas monster symmetry. (cid:4) Lemma 3.20.
Let p , q , r , (cid:96) be distinct elements of P such that pq, pr, p(cid:96) (cid:54)∈ { , } . Thenthere exists a unique self-dual Z [1 /p, e (1 / p )] -form V (cid:92) [1 /p, e (1 / p )] of V (cid:92) such that: V (cid:92) [1 /p, e (1 / p )] ⊗ Z [1 /p,e (1 / p )] Z [1 /pq, e (1 / pq )] ∼ = V (cid:92) [1 /pq, e (1 / pq )] ,V (cid:92) [1 /p, e (1 / p )] ⊗ Z [1 /p,e (1 / p )] Z [1 /pr, e (1 / pr )] ∼ = V (cid:92) [1 /pr, e (1 / pr )] ,V (cid:92) [1 /p, e (1 / p )] ⊗ Z [1 /p,e (1 / p )] Z [1 /p(cid:96), e (1 / p(cid:96) )] ∼ = V (cid:92) [1 /p(cid:96), e (1 / p(cid:96) )] . Furthermore, this form has monster symmetry.
Proof .
We apply Lemma 3.19 for the triples ( p, q, r ), ( p, q, (cid:96) ) and ( p, r, (cid:96) ) to obtain unique self-dual forms over Z [1 /pqr, e (1 / p )], Z [1 /pq(cid:96), e (1 / p )], and Z [1 /pr(cid:96), e (1 / p )] with monster sym-metry, and they are all isomorphic to each other when base-changed to Z [1 /pqr(cid:96), e (1 / p )]. Foreach pair of these forms, we apply Zariski descent (following Lemma 2.30) to obtain uniqueself-dual forms V (cid:92) [1 /pq, e (1 / p )] over Z [1 /pq, e (1 / p )], V (cid:92) [1 /pr, e (1 / p )] over Z [1 /pr, e (1 / p )],and V (cid:92) [1 /p(cid:96), e (1 / p )] over Z [1 /p(cid:96), e (1 / p )], satisfying V (cid:92) [1 /pq, e (1 / p )] ⊗ Z [1 /pq,e (1 / p )] Z [1 /pq, e (1 / pq )] ∼ = V (cid:92) [1 /pq, e (1 / pq )] ,V (cid:92) [1 /pr, e (1 / p )] ⊗ Z [1 /pr,e (1 / p )] Z [1 /pr, e (1 / pr )] ∼ = V (cid:92) [1 /pr, e (1 / pr )] ,V (cid:92) [1 /p(cid:96), e (1 / p )] ⊗ Z [1 /p(cid:96),e (1 / p )] Z [1 /p(cid:96), e (1 / p(cid:96) )] ∼ = V (cid:92) [1 /p(cid:96), e (1 / p(cid:96) )] . Applying Zariski descent to any pair of these forms yields a self-dual Z [1 /p, e (1 / p )]-form withmonster symmetry, and the uniqueness claim in Lemma 3.18 implies any pair yields an isomor-phic object. (cid:4) Lemma 3.21.
Let p , q , r be distinct elements of P , such that pq , qr , and pr are not in { , } .Then, there exists a unique self-dual Z [1 /pqr ] -form V (cid:92) [1 /pqr ] of V (cid:92) such that V (cid:92) [1 /pqr ] ⊗ Z [ e (1 / pq )] ∼ = V (cid:92) [1 /pq, e (1 / pq )] ⊗ Z [1 /pq,e (1 / pq )] Z [1 /pqr, e (1 / pq )] ,V (cid:92) [1 /pqr ] ⊗ Z [ e (1 / pr )] ∼ = V (cid:92) [1 /pr, e (1 / pr )] ⊗ Z [1 /pr,e (1 / pr )] Z [1 /pqr, e (1 / pr )] ,V (cid:92) [1 /pqr ] ⊗ Z [ e (1 / qr )] ∼ = V (cid:92) [1 /qr, e (1 / qr )] ⊗ Z [1 /qr,e (1 / qr )] Z [1 /pqr, e (1 / qr )] . Furthermore, V (cid:92) [1 /pqr ] has monster symmetry. Proof .
We apply Lemma 3.19 for the triples ( p, q, r ), ( q, r, p ), and ( r, p, q ) to obtain unique self-dual forms over Z [1 /pqr, e (1 / p )], Z [1 /pqr, e (1 / q )], and Z [1 /pqr, e (1 / r )]. For any pair of theseforms, applying faithfully flat gluing as in Lemma 2.31 yields a self-dual Z [1 /pqr ]-form satisfyingthe expected conditions, and uniqueness and monster symmetry follow from Lemma 3.18. (cid:4) We now come to the main theorem.
Theorem 3.22.
There exists a unique self-dual Z -form V (cid:92) Z of the vertex operator algebra V (cid:92) such that for any distinct p, q ∈ P satisfying pq (cid:54)∈ { , } , we have V (cid:92) Z ⊗ Z Z [1 /pq, e (1 / pq )] ∼ = V (cid:92) [1 /pq, e (1 / pq )] . Furthermore, this integral form has monster symmetry.
Proof .
If we let ( p, q, r ) range over triples in { , , , } , Lemma 3.21 yields Z [1 /pqr ]-forms V (cid:92) [1 /pqr ]. Since the four rings Z [1 /pqr ] form a Zariski cover of Spec Z , we obtain a self-dual Z -form V (cid:92) Z by pairwise gluing, and Lemma 3.18 implies monster symmetry and uniqueness withrespect to isomorphisms V (cid:92) Z ⊗ Z Z [1 /pq, e (1 / pq )] ∼ = V (cid:92) [1 /pq, e (1 / pq )]as p , q range over distinct elements of { , , , } .We now consider the remaining cases, both of which involve the prime 13. Let p ∈ { , } .Recall from Lemma 3.20 that for any distinct q , r , (cid:96) in P \ { p } , including the case q = 13, wehave a Z [1 /p, e (1 / p )]-form V (cid:92) [1 /p, e (1 / p )] satisfying V (cid:92) [1 /p, e (1 / p )] ⊗ Z [1 /p,e (1 / p )] Z [1 /pq, e (1 / pq )] ∼ = V (cid:92) [1 /pq, e (1 / pq )] . Self-Dual Integral Form of the Moonshine Module 33We claim that this form is independent of the choice of the primes q , r , (cid:96) . Indeed, applyingLemma 3.19 to all possible triples ( p, q, r ) yields a collection of forms over the various rings Z [1 /pqr, e (1 / p )]. By the tensor product compatibilities, these forms are isomorphic on Zariskiintersections, so all of the Spec Z [1 /p, e (1 / p )]-forms obtained by gluing are isomorphic byuniqueness.Thus, it suffices to show that V (cid:92) Z ⊗ Z [1 /p, e (1 / p )] ∼ = V (cid:92) [1 /p, e (1 / p )]. We have establishedthat the isomorphism type of V (cid:92) [1 /p, e (1 / p )] is independent of the choice of q , r , (cid:96) , so we maychoose q, r, (cid:96) ∈ { , , , } \ { p } , and then the isomorphism follows from the uniqueness of ourconstruction of V (cid:92) Z in the first paragraph. (cid:4) Corollary 3.23.
The inner product on V (cid:92) Z is positive definite. Proof .
The Z [1 / V (cid:92) that was constructed in [6] is positive definite, because base-change to Q yields the positive-definite Q -form constructed in [27].It therefore suffices to show that V (cid:92) Z ⊗ Z [1 /
2] is isomorphic to the Z [1 / V (cid:92) constructedin [6]. This follows from the same method as Proposition 3.14: we decompose both forms underthe action of a 2B-pure 4-group H , and obtain identifications with the 2a-fixed point vertexoperator subalgebra of ( V Λ ) Z [1 / . In fact, we don’t need the full isomorphism to show positive-definiteness, since it suffices to show that the corresponding H -eigenspaces are isomorphic,hence positive-definite. (cid:4) R. Griess has informed me that the following result is new, even without the positive definitecondition.
Corollary 3.24.
There exists a -dimensional positive-definite unimodular lattice witha faithful monster action by orthogonal transformations.
Proof .
The weight 2 subspace of V (cid:92) Z satisfies the required properties. (cid:4) Corollary 3.25 (newer modular moonshine) . The stronger version of the modular moonshineconjecture, as stated in [6] and [4] , is unconditionally true. That is, there exists a self-dualintegral form V of V (cid:92) with M symmetry, such that for each element g of prime order p , thegraded Brauer character of any p -regular element h ∈ C M ( g ) on the Tate cohomology ˆ H ∗ ( g, V ) is given by Tr( h | ˆ H ( g, V )) = T gh ( τ ) , g ∈ p A , C ,T gh ( τ ) + T gh ( τ + 1 / , g ∈ B ,T gh ( τ ) + T ghσ ( τ )2 , g ∈ p B , | ( p − , Tr( h | ˆ H ( g, V )) = , g ∈ p A , C ,T gh ( τ ) − T gh ( τ + 1 / , g ∈ B ,T gh ( τ ) − T ghσ ( τ )2 , g ∈ p B , | ( p − , where the element σ is the unique involution in C M ( g ) /O p ( C M ( g )) that acts as on ˆ H ( g, V ) and − on ˆ H ( g, V ) , when p ∈ { , , , } . Proof .
The conditional proof given in [6] and [4] only requires the assumptions that V (cid:92) Z exist(with a small technical condition at p = 2), and that a statement about Z p -forms of the monsterLie algebra hold. The existence of V (cid:92) Z is given in Theorem 3.22, and the assumption about4 S. Carnahan Z p -forms was shown to hold in [5]. The technical condition at p = 2 appears in Section 5of [6], where Assumption 5.1 asserts that V (cid:92) Z ⊗ Z [1 /
3] decomposes into graded 2 .M . V Λ ) Z [1 / . We will not prove this assumption, but wenote that it is only used to transfer the vanishing properties of Tate cohomology from theform of V Λ to the form of V (cid:92) . The proofs of vanishing only use the existence of involutions inclasses 2A and 2B instead of a 2 .M . q ∈ { , , , } , Lemma 3.6 gives an isomorphism from the 3B-fixed vectors in V (cid:92) Z ⊗ Z [1 / q, e (1 / q )] to a submodule of ( V Λ ) Z [1 / q,e (1 / q )] , and the order 9 3B-pure group H given in Lemma 3.13 gives a decomposition of V (cid:92) Z ⊗ Z [1 / q, e (1 / q )] into pieces that embedequivariantly (with respect to C M (3 B ) ∩ N M ( H )) into the 3B-fixed vectors. When q is odd, thisis sufficient to transfer the Tate cohomology vanishing properties that we need from V Λ . (cid:4) We showed in Lemma 3.6 that some prime order orbifold constructions of V (cid:92) can be definedover rather small rings. In particular, if p, q ∈ { , , , , } and pq (cid:54)∈ { , } , then we haveisomorphisms( V Λ ) σ Z [1 /pq,e (1 /pq )] ∼ → ( V (cid:92) Z ⊗ Z [1 /pq, e (1 /pq )]) g for σ an order p lift of a fixed-point free automorphism ¯ σ of Λ, and g ∈ M in class pB . Here, weshow that these isomorphisms can be defined over Z [1 /p, e (1 /p )]. For p = 3, this refinement wasconjectured in [6], in the hope that such a construction could be used to construct a self-dualintegral form. Thus, we are approaching this question in a somewhat backward way. Corollary 3.26.
Let p ∈ { , , , } . Then there exists: an order p automorphism σ of ( V Λ ) Z [1 /p,e (1 /p )] lifting a fixed-point free automorphism ¯ σ of Λ of order p , an integral form V of V (cid:92) with invariant bilinear form and monster symmetry, admittingan order p automorphism g in class pB , an isomorphism ( V Λ ) σ Z [1 /p,e (1 /p )] ∼ → ( V ⊗ Z [1 /p, e (1 /p )]) g of Z [1 /p, e (1 /p )] -vertex algebraspreserving inner products. Proof .
By Lemma 3.6, we have isomorphisms( V Λ ) σ Z [1 /pq,e (1 / pq )] ∼ → ( V (cid:92) Z ⊗ Z [1 /pq, e (1 / pq )]) g for all q ∈ { , , , } \ { p } . We apply a descent argument following the same lines as in Theo-rem 3.22: Faithfully flat gluing for Z [1 /pqr, e (1 / pq )] → Z [1 /pqr, e (1 / pqr )] ← Z [1 /pqr, e (1 / pr )]yields isomorphic Z [1 /pqr, e (1 / p )]-forms, as q and r vary over the set { , , , } \ { p } . Zariskigluing then yields isomorphic Z [1 /p, e (1 / p )]-forms. When p is odd, we have Z [1 /p, e (1 / p )] = Z [1 /p, e (1 /p )], and when p = 2, the isomorphism of Z [1 / (cid:4) It seems reasonable to expect that one can improve this result by removing the roots of unity.
Conjecture 3.27.
For any p ∈ { , , , , } , there is an isomorphism ( V Λ ) σ Z [1 /p ] ∼ → (cid:0) V (cid:92) Z ⊗ Z [1 /p ] (cid:1) g of vertex operator algebras over Z [1 /p ] , where σ is an order p lift of a fixed-point free automor-phism of Λ , and g is an element of the monster in conjugacy class p B. Self-Dual Integral Form of the Moonshine Module 35Section 5 of [6] describes how to produce a Z [1 / V (cid:92) using a Z [1 / , e (1 / ( F )-action on a 3B-pure elementarysubgroup. This method works for p = 5 and p = 7, as well, but I do not know how to showthat the Galois actions on (cid:0) V (cid:92) Z ⊗ Z [1 /p, e (1 /p )] (cid:1) g and ( V Λ ) σ Z [1 /p,e (1 /p )] have matching fixed-pointsubmodules. For p = 13, one may need to do more explicit work. Acknowledgements
I would like to thank Toshiyuki Abe for describing the constructions in [1] in detail at the “VOAand related topics” workshop at Osaka University in March 2017. I would also like to thank theanonymous referees for many helpful comments, and one referee in particular for their help withthe proof of Lemma 2.13. This research was partly funded by JSPS Kakenhi Grant-in-Aid forYoung Scientists (B) 17K14152.
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