The Differential Brauer Group Raymond Hoobler City College of New York and Graduate Center, CUNY Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 1 / 13 Outline 1 Review of Brauer groups of fields, rings, and D-rings 2 Cohomology 3 Cohomological interpretation of D Brauer groups (with connections to Hodge theory) Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 2 / 13 Brauer Groups of Fields Finite dimensional division algebras L over a field K are classified by Br (K ) , the Brauer group of K. If L is a central, simple algebra over K, then it is isomorphic to Mn (D) for some division algebra D. Given two such algebras L and G, L K G is again a central simple K algebra. They are said to be Brauer equivalent if there are vector spaces V , W and a K algebra isomorphism L K End (V ) u G K End (W ). This is an equivalence relation, and Br (K ) is then defined to be the group formed from the equivalence classes with as product. Br (K ) classifies division algebras over K. For any such algebra L over K, there is a Galois extension L/K such that L K L u EndL (V ). Galois cohomology is then used to classify all such equivalence classes 2 using an isomorphism Br (K ) u H GK /K , K . Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 3 / 13 Brauer Groups of Rings Azumaya algebras over a commutative ring R A finitely generated central R algebra L is an Azumaya algebra algebra if L R L is a central simple algebra over L for any homomorphism from R to a field L. An Azumaya algebra L is a central, finitely generated R algebra which is a projective L Lop algebra. R Two such Azumaya algebras L and G are Brauer equivalent if there are faithful, projective R modules P, Q and an R algebra isomorphism L R End (P) u G R End (Q) . If R is a local ring, there is an etale extension S/R such that L R S u EndS (P) for some projective S module P. Etale cohomology is then used to classify all such equivalence classes 2 using an isomorphism ¶ : Br (R) H (Ret , Gm ) . !u Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 4 / 13 Brauer Groups of D-rings Let D = d1, ..., dn be a set of n commuting derivations on R, a ring containingf Q. g A D Azumaya algebra over R is an Azumaya algebra L over R equipped with derivations extending the action of D on R. Two such D Azumaya algebras L and G are D Brauer equivalent if there are faithful, projective D R modules P, Q and a D R algebra isomorphism L R End (P) u G R End (Q). This is an equivalence relation, and BrD(R) is the resulting group on the set of equivalence classes with R as the product. If R is local, there is an etale extension S and a D S isomorphism L R S u EndS (P) for some D S projective module P. Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 5 / 13 Cohomology Let C be a category with fibred products. A pretopology on C consists of specifying for all X ob(C ), a set Cov(X ) whose members are collections 2 f : U X a A Cov(X ) satisfying f a a ! j 2 g 2 1 If f : X X is an isomorphism, f Cov (X ) . ! f ga 2 a 2 If f : U X Cov (X ) and g : V U Cov (Ui ) for all f a a ! g 2 f i i ! ag 2 i , then f g a : V a X Cov (X ) . f a i i ! g 2 3 If f : U X Cov (X ) and Y X , then f a a ! g 2 ! 2 C f X Y : U X Y Y Cov (Y ) . f a a ! g 2 A presheaf F : op ((Sets)) is a sheaf if for all X and U X CovC (!X ) , 2 C f a ! g 2 F (X ) , F (U ) F U X U ! ∏ a ∏ a b is exact. Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 6 / 13 Cohomology Example 1 Xet has fa : Va U fa is an etale map and U = fa (Va) = Cov (Uff) . ! j gg et S 2 XD fl has g : Va U g is a flat D map of finite type and U = g (Va) = ff a ! j a a gg CovD fl (U) . S If G is a scheme, then its functor of points defines a sheaf in either of these topologies. Moreover there is a map of sites t : XD fl Xet since 1 ! any etale map is a flat D map. Thus t ( fa ) CovD fl (U) . Moreover f g 2 H (Xet , G ) H (XD fl, G ) for sheaves G defined by smooth, !u quasi-projective group schemes over X like the sheaf of units, Gm. Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 7 / 13 Cohomological Interpretation In particular on XD fl we have the exact sequence D d ln 1 0 G Gm Z 0 ! m ! ! X ! whose cohomology sequence contains 0 1 1 D c1 1 1 H XD fl, ZX H XD fl, Gm Pic (X ) H XD fl, ZX ! ! ! 2 D H XD fl, Gm Br (X ) 0 ! ! ! 2 if X is smooth since then H (Xet , Gm ) is torsion unlike the vector space 2 1 H XD fl, ZX ! How do we interpret this?? Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 8 / 13 Cohomological Interpretation Theorem Let X be a quasi-projective variety of finite type over a field K of 2 characteristic 0. If x H (X , mN ) , then there is an Azumaya algebra L equipped with an integrable2 connection constructed from x such that 2 ¶ ([L]) = iN (x) H (Xet , Gm ) where iN : m Gm is inclusion. 2 N ! For simplicity, let’sconsider the case where X = Spec (R) is a local ring and R contains a primitive Nth root of unity. Then there is an etale extension R S Cov (Spec (R)) , i.e. U = Spec (S) Spec (R) , and ! 2 3 ˇ 2 ! a Cech 2 cocycle z mN S such that [z] = x H ((R S) , mN ) . Now by refining S we2 may assume that it is in the2 standard! form S = (R [T ] / (p (T )))g (t) where p (T ) is a monic polynomial of degree D. So we approximate S by R [t] := R [T ] / (p (T )) = D R. 1 Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 9 / 13 Cohomological Interpretation Now our cocycle z mN (S R S R S) is constant on each connected 3 2 component of S but may vary from one component to another. So we must use an index set that accounts for this. We let J = connected components of S3 D Of course : = ∏ R [t]a = a J 1 R is not usually connected F a J 2 2 3 but for each connected component a of S there is an R [t]a which admits multiplication by the value za of z on that connected component and, as an R module, is free of rank M = D (# (J)) . Then we define an R module isomorphismF · ` = z : R R R S R S z a F S S ! F MJ th by multiplying the a factor in by za. Note that this amounts to a F th diagonal block matrix where the a block is zaID . Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 10 / 13 Cohomological Interpretation We let c ( J z ) : EndR ( ) R S R S EndR ( ) R S R S be the a F ! F algebra isomorphism given by conjugation by `z. Then we get descent data from the diagram End ( ) R R F S S e2 % L End ( ) R c ( J z ) ! F S# a e1 & End ( ) R R F S S th where ei means insert 1S into the i copy of S. Note that End ( ) is the F algebra of M M matrices with d (eij ) = 0 for all d D. Here 2 c ( J z ) = c ` is the patching data used to define L and it preserves a z the action of D since c ` is given by conjugation by an Nth root of z unity on each block in End ( ) . F Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 11 / 13 Cohomological Interpretation It satisfies the cocycle condition 3 End ( ) R F S e3 c ` z % 3 End ( ) R e1 c ` F S # z e2 c ` z & 3 End ( ) R F S This commutes because 1 (e2 (c ( J z ))) (e1 (c ( J z ))) (e3 (c ( J z ))) is conjugation on a a a End ( ) R R by 1 z since z is a 2 cocycle. But this is F S S R S 1End ( ) S 3 which is the cocycle condition for descent. F Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 12 / 13 Cohomological Interpretation Thus the Cech cocycle provides the needed descent data and we immediately see that 2 ¶ ([c ( J z )]) = [z] Hˇ (X , m ) a 2 n where L = [c ( J z )] is the desired D Azumaya algebra.