Differential operators on classical invariant rings do not lift modulo p
aa r X i v : . [ m a t h . A C ] J un DIFFERENTIAL OPERATORS ON CLASSICAL INVARIANT RINGS DO NOTLIFT MODULO p JACK JEFFRIES AND ANURAG K. SINGHA
BSTRACT . Levasseur and Stafford described the rings of differential operators on vari-ous classical invariant rings of characteristic zero; in each of the cases they considered, thedifferential operators form a simple ring. Towards an attack on the simplicity of rings ofdifferential operators on invariant rings of linearly reductive groups over the complex num-bers, Smith and Van den Bergh asked if differential operators on the corresponding rings ofpositive prime characteristic lift to characteristic zero differential operators. We prove that,in general, this is not the case for determinantal hypersurfaces, as well as for Pfaffian andsymmetric determinantal hypersurfaces. We also prove that, with few exceptions, thesehypersurfaces do not admit a mod p lift of the Frobenius endomorphism.
1. I
NTRODUCTION
For a polynomial ring R over a field K of characteristic zero, the ring of K -linear dif-ferential operators on R is the K -algebra generated by R and the K -linear derivations on R ,i.e., the ring D R | K = R h ∂∂ x , . . . , ∂∂ x d i . This noncommutative ring is the well-known Weylalgebra, which enjoys many good ring-theoretic properties: it is left and right Noetherian,and is a simple ring.For commutative rings R and A , where R is an A -algebra, there is a notion, due toGrothendieck [Gr, § A -linear differential operators on R , denoted D R | A ;see §
2. However, in contrast with the case of a polynomial ring, if one takes A to be afield K of characteristic zero, and R to be K [ x , y , z ] / ( x + y + z ) , then D R | K is not left orright Noetherian, nor a finitely generated K -algebra, nor a simple ring; see [BGG].On the other hand, when R is the ring of invariants for a linear action of a reductivegroup on a polynomial ring over a field K of characteristic zero, it is known in many casesthat D R | K is Noetherian, finitely generated, and a simple ring, just as in the polynomialcase [Ka1, LS, MV, Sc2]. Indeed, it is conjectured that for such invariant rings R , the ringof differential operators D R | K is a simple ring [Sc1]. An analogous statement in positivecharacteristic was proved by Smith and Van den Bergh [SV, Theorem 1.3].Quite generally, for A -algebras R and B , there is an injective ring homomorphism D R | A ⊗ A B −→ D R ⊗ A B | B , that is an isomorphism when B is flat over A . In particular, one has an isomorphism D R | Z ⊗ Z Q ∼ = D R ⊗ Z Q | Q , and, for p a prime integer, an injective homomorphism(1.0.1) D R | Z ⊗ Z ( Z / p Z ) ֒ −→ D ( R / pR ) | ( Z / p Z ) . In order to relate rings of differential operators in characteristic zero to their counter-parts in positive characteristic p , one needs to determine whether the map (1.0.1) is an J.J. was supported by NSF grant DMS 1606353 and A.K.S. by NSF grant DMS 1801285. isomorphism, i.e., whether each differential operator on R / pR lifts to a differential op-erator on R . To study the problem of the simplicity of rings of differential operators oncharacteristic zero invariant rings via reduction to positive characteristic, Smith and Vanden Bergh pose the following question, formulated here in equivalent terms: Question 1.1 ([SV, Question 5.1.2]) . Let A be a domain that is finitely generated as analgebra over Z . Suppose R is a finitely generated A -algebra such that R ⊗ A frac ( A ) isthe ring of invariants for a linear action of a reductive group on a polynomial ring ofcharacteristic zero. Does there exist a nonempty open subset of U of Spec A , such that foreach maximal ideal µ ∈ U , each differential operator on D ( R / µ R ) | ( A / µ A ) lifts to D R | A ?We prove that the answer to Question 1.1 is negative: for several classical invariantrings that are hypersurfaces, we construct explicit differential operators, modulo primeintegers p , that do not lift to characteristic zero differential operators. Our main theorem isbelow; we refer the reader to § Theorem 1.2.
Consider the following classical determinantal rings: (a)
Let X be an n × n matrix of indeterminates over Z , with n > . Set R : = Z [ X ] / ( det X ) .Then, for each prime integer p > , the Frobenius trace on R / pR does not lift to adifferential operator on R / p R, nor, a fortiori, to a differential operator on R. (b)
Let X be an n × n alternating matrix of indeterminates over Z , for n > an even inte-ger. Set R : = Z [ X ] / ( pf X ) , where pf X denotes the Pfaffian of X. Then, for each primeinteger p > , the Frobenius trace on R / pR does not lift to a differential operatoron R / p R, nor, a fortiori, to a differential operator on R. (c)
Let X be a × symmetric matrix of indeterminates over Z . Set R : = Z [ X ] / ( det X ) .Then, for prime integers p > , each differential operator on R / pR lifts to a differentialoperator on R. In the case of characteristic , the Frobenius trace on R / R does notlift to a differential operator on R / R, nor, a fortiori, to a differential operator on R.
For R as in (a), (b), or (c) above, the ring R ⊗ Z Q is the invariant ring for an actionof the linearly reductive group GL n − ( Q ) , Sp n − ( Q ) , or O ( Q ) , respectively; see, forexample, § § §
8, for details. In contrast with the cases discussed above, we provethat if R is a normal toric hypersurface over Z , then, for each prime integer p >
0, everydifferential operator on R / pR lifts to a differential operator on R ; see Theorem 5.1. Inparticular, if R is the hypersurface over Z defined by the determinant of a 2 × × R / pR lifts to a differential operator on R ; this addresses the case n = p with local cohomology elementsthat do not lift modulo p , and consequently with nonzero elements in a different local co-homology module that are annihilated by the prime integer p . These ideas were used bythe first author to give positive answers to Question 1.1 in special cases [Je, Theorem 6.3].However, we have attempted to keep the present paper largely self-contained, and focusedon hypersurfaces, where the isomorphisms can be made entirely explicit; it is striking thatthe isomorphism is one of D -modules: IFFERENTIAL OPERATORS ON INVARIANT RINGS 3
Theorem 1.3.
Let A be a commutative ring. Set R : = A [ x , . . . , x n ] / ( f ) , where f is a non-zerodivisor in the polynomial ring A [ x , . . . , x n ] . Set ∆ to be the kernel of the multiplicationmap R ⊗ A R −→ R. Then the local cohomology module H n ∆ ( R ⊗ A R ) , with the natural D R | A -module structure as in § R | A -module of rank one. The techniques and calculations used in our proof of Theorem 1.2 have implications tothe existence of liftings of the Frobenius morphism that have attracted a lot of attention;for example, [Bh] provides a connection between liftability of the Frobenius and infinitelygenerated crystalline cohomology, while [BTLM] proves Bott vanishing for varieties thatadmit a Frobenius lift modulo p . The liftability of the Frobenius is also studied in con-siderable detail in [Zd]; we use results from that paper to determine whether the Frobeniusendomorphism on a determinantal hypersurface of positive prime characteristic lifts to aring endomorphism in characteristic zero: Theorem 1.4.
Consider the following classical determinantal rings: (a) R : = Z [ X ] / ( det X ) , where X is an n × n matrix of indeterminates over Z , with n > ; (b) R : = Z [ X ] / ( pf X ) , where X is an n × n alternating matrix of indeterminates over Z ,and n > is an even integer; (c) R : = Z [ X ] / ( det X ) , where X is an n × n symmetric matrix of indeterminates over Z ,with n > .Let p be a positive prime integer. In cases (a) and (b) , the Frobenius endomorphismon R / pR does not lift to a ring endomorphism on R / p R, nor, a fortiori, to a ring en-domorphism on R. In case (c) , the same conclusion holds if p = , or if p > and n > . If R is a normal toric hypersurface over Z , then, for each prime integer p >
0, the Frobe-nius endomorphism of R / pR lifts to an endomorphism of R , and hence to an endomorphismof R / p R . Specifically, if R is the hypersurface over Z defined by the determinant of a 2 × × R / pR lifts to an endomorphism of R and of R / p R ; this covers thecase n = G is a linearly reductive group over a field K , with a linear action on apolynomial ring K [ xxx ] , then the invariant ring K [ xxx ] G is a direct summand of K [ xxx ] as a K [ xxx ] G -module; many key properties of classical invariant rings including finite generation andthe Cohen-Macaulay property follow from the existence of such a splitting, see for exam-ple [HE, §
2] and [HR]. Indeed, the general linear group, the symplectic group, and theorthogonal group are linearly reductive over fields of characteristic zero; it follows thatdeterminantal rings, Pfaffian determinantal rings, and symmetric determinantal rings, overfields of characteristic zero, are direct summands of polynomial rings. In contrast, weprove that working over the rings of integers Z , or over the ring of p -adic integers d Z ( p ) , thecorresponding determinantal rings are not direct summands of any polynomial ring: Corollary 1.5.
Let R be a classical determinantal ring as in Theorem 1.4. Then R is not adirect summand, as an R-module, of any polynomial ring over Z .Let V : = d Z ( p ) be the p-adic integers; in case (c) , assume further that either n > , orthat p = . Then the ring R ⊗ Z V is not a direct summand, as an R ⊗ Z V -module, of anypolynomial ring over V .
The corollary is immediate from Theorem 1.4 in light of [Zd, Lemma 4.1], where itis observed that the existence of a Frobenius lift modulo p is inherited by a ring thatis a direct summand. Regarding the exceptional case in the corollary above, it is worth JACK JEFFRIES AND ANURAG K. SINGH mention that when V is a discrete valuation ring of mixed characteristic, such that theresidual characteristic is an odd prime integer, and R V the hypersurface over V defined bythe determinant of a 3 × R V is a direct summandof a polynomial ring V [ xxx ] as an R V -module; see Remark 8.2.In § § p -torsion local cohomology elements to specificdifferential operators; Theorem 1.3 is proved in § § §
7, and §
8; for Theorem 1.2, we prove that therelevant p -torsion local cohomology elements are nonzero. We expect that these calcula-tions are of independent interest, adding to the study of integer torsion in local cohomologymodules pursued in [Si, LSW, BBLSZ].2. P RELIMINARIES
Differential operators.
Differential operators on a commutative ring R are definedinductively as follows: for each r ∈ R , the multiplication by r map e r : R −→ R is a differen-tial operator of order 0; for each positive integer n , the differential operators of order lessthan or equal to n are additive maps δ : R −→ R for which each commutator [ e r , δ ] : = e r ◦ δ − δ ◦ e r is a differential operator of order less than or equal to n −
1. If δ and δ ′ are differentialoperators of order at most m and n respectively, then δ ◦ δ ′ is a differential operator oforder at most m + n . Thus, the differential operators on R form a subring D R of End Z ( R ) .We use D nR to denote the differential operators of order at most n .A differential operator δ ∈ D R is a derivation if δ ( r r ) = r δ ( r ) + r δ ( r ) for r i ∈ R . It is readily seen that each element of D R may be expressed uniquely as the sum of anelement of D R and a derivation.When R is an algebra over a commutative ring A , we set D R | A to be the subring of D R that consists of differential operators that are A -linear; note that D R | Z equals D R . When R is an algebra over a perfect field K of positive prime characteristic, then D R | K equals D R ,see for example [Ly, Example 5.1 (c)].If R = A [ x , . . . , x d ] is a polynomial ring over A , then any element of D nR | A can be ex-pressed as an R -linear combination of the differential operators ∂ a ,..., a d , where a i are non-negative integers with a + · · · + a d n , and ∂ a ,..., a d ( x b · · · x b d d ) = d ∏ i = (cid:18) b i a i (cid:19) x b − a · · · x b d − a d d . Note that if A contains the field of rational numbers, then ∂ a ,..., a d = a ! ∂ a ∂ x a · · · a d ! ∂ a d ∂ x a d d . We record an alternative description of D R | A from [Gr, § R an A -algebra, set P R | A : = R ⊗ A R , IFFERENTIAL OPERATORS ON INVARIANT RINGS 5 and consider the P R | A -module structure on End A ( R ) under which r ⊗ r acts on δ to givethe endomorphism e r ◦ δ ◦ e r , where e r i denotes the map that is multiplication by r i . Set ∆ R | A : = ker ( P R | A µ −−−→ R ) , where µ is the A -algebra homomorphism determined by µ ( r ⊗ r ) = r r . The ideal ∆ R | A is generated by elements of the form r ⊗ − ⊗ r . Since ( r ⊗ − ⊗ r )( δ ) = [ e r , δ ] , it follows that an element δ of End A ( R ) is a differential operator of order at most n preciselyif it is annihilated by ∆ n + R | A . By [Gr, Proposition 16.8.8], the A -linear differential operatorson R of order at most n correspond to(2.0.1) Hom R ( P nR | A , R ) ∼ = ann ( ∆ n + R | A , End A ( R )) , where P nR | A : = P R | A / ∆ n + R | A is viewed as a left R -module via r r ⊗
1. To make the isomorphism (2.0.1) explicit,consider the map ρ : R −→ P nR | A with r ⊗ r , in which case, the map ρ ∗ : Hom R ( P nR | A , R ) −→ D nR | A with δ δ ◦ ρ is an isomorphism of P R | A -modules.In addition to the filtration by order, we will have use for another filtration on D R | A .Supposing that ∆ R | A is a finitely generated ideal of P R | A , fix a set of generators z , . . . , z t .Then the sequence of ideals defined by ∆ [ n ] R | A : = ( z n , . . . , z nt ) , for n ∈ N , is cofinal with the sequence ∆ nR | A for n ∈ N . As there is little risk of confusion, we reusethe notation ρ : R −→ P [ n ] R | A for the map with r ⊗ r , and set D [ n ] R | A : = { δ ◦ ρ | δ ∈ Hom R ( P [ n ] R | A , R ) } , where P [ n ] R | A : = P R | A / ∆ [ n + ] R | A . Similarly, we use ρ ∗ for the P R | A -module isomorphism ρ ∗ : Hom R ( P [ n ] R | A , R ) −→ D [ n ] R | A with δ δ ◦ ρ . Note that D [ n ] R | A gives a filtration of D R | A that is cofinal with the filtration by order; thefiltration D [ n ] R | A depends on the choice of ideal generators for ∆ R | A .Let M be an R -module. Then Hom A ( R , M ) has a P R | A -module structure given by ( r ⊗ r ) · δ : = e r ◦ δ ◦ e r . The differential operators from R to M , of order at most n , are D nR | A ( M ) : = ann (cid:0) ∆ n + R | A , Hom A ( R , M ) (cid:1) . Note that one has an isomorphism of P R | A -modules, D nR | A ( M ) ∼ = Hom R ( P nR | A , M ) . JACK JEFFRIES AND ANURAG K. SINGH
Likewise, given generators z , . . . , z t for ∆ R | A , we set D [ n ] R | A ( M ) : = ann (cid:0) ∆ [ n + ] R | A , Hom A ( R , M ) (cid:1) ∼ = Hom R ( P [ n ] R | A , M ) and D R | A ( M ) : = [ n > D nR | A ( M ) . In analogy with the equality D R = D R | Z , we set P R : = P R | Z and ∆ R : = ∆ R | Z , along withthe corresponding notation P nR : = P R / ∆ n + R and P [ n ] R : = P R / ∆ [ n + ] R . Observe that if R hascharacteristic p , we have P R = P R | ( Z / p Z ) , and likewise for the other notions discussed above.2.2. Koszul and local cohomology.
Let z be an element of a ring R . One has maps be-tween the Koszul complexes K • ( z n ; R ) , for n ∈ N , and the ˇCech complex C • ( z ; R ) as below:0 −−−→ R z n − −−−→ R −−−→ y y z −−−→ R z n −−−→ R −−−→ y y / z n −−−→ R −−−→ R z −−−→ . Taking the direct limit of K • ( z n ; R ) yields an isomorphism lim −→ n K • ( z n ; R ) ∼ = C • ( z ; R ) .For zzz : = z , . . . , z t , one similarly obtains a map of complexes K • ( zzz ; R ) : = N i K • ( z i ; R ) −−−→ N i C • ( z i ; R ) = : C • ( zzz ; R ) , and, for each k >
0, an induced map from Koszul cohomology to local cohomology(2.0.2) H k ( zzz ; R ) −−−→ H k ( zzz ) ( R ) . Setting zzz n : = z n , . . . , z nt , one likewise has lim −→ n K • ( zzz n ; R ) ∼ = C • ( zzz ; R ) , andlim −→ n H k ( zzz n ; R ) ∼ = H k ( zzz ) ( R ) for each k > . When k equals t +
1, the map (2.0.2) takes the form H t + ( zzz ; R ) = R ( zzz ) R −→ R z ··· z t ∑ i R z ··· ˆ z i ··· z t = H t + ( zzz ) ( R ) , with 1 (cid:20) z · · · z t (cid:21) , while, if k equals t , the Koszul cohomology element in H t ( zzz ; R ) corresponding to an equa-tion ∑ i z i g i = R maps to " · · · , ( − ) i g i z · · · ˆ z i · · · z t , · · · ∈ H t ( zzz ) ( R ) . We mention that a local cohomology element (cid:20) rz n · · · z nt (cid:21) ∈ H t + ( zzz ) ( R ) is zero if and only if there exists an integer k > r ( z · · · z t ) k ∈ (cid:0) z n + k , . . . , z n + kt (cid:1) R . IFFERENTIAL OPERATORS ON INVARIANT RINGS 7
Suppose the ring R takes the form S / f S , for S a commutative ring, and f ∈ S a regularelement. Let zzz : = z , . . . , z t , as before. The exact sequence0 −−−→ S f −−−→ S −−−→ R −−−→ −−−→ H t ( zzz ) ( R ) δ f −−−→ H t + ( zzz ) ( S ) f −−−→ H t + ( zzz ) ( S ) −−−→ , with δ f denoting the connecting homomorphism. To make the map explicit, suppose g i areelements of S such that ∑ i z ni g i = s f for some n > s ∈ S ; then δ f : " · · · , ( − ) i g i ( z · · · ˆ z i · · · z t ) n , · · · (cid:20) s ( z · · · z t ) n (cid:21) . Bockstein homomorphisms.
We briefly review Bockstein homomorphisms on localcohomology [SW]. Let p be a prime integer that is a regular element on a ring R . Fix anideal a of R . Applying the local cohomology functor H • a ( − ) to0 −−−→ R / pR p −−−→ R / p R −−−→ R / pR −−−→ , one obtains a cohomology exact sequence; the Bockstein homomorphism β p : H k a ( R / pR ) −→ H k + a ( R / pR ) is the connecting homomorphism in the cohomology exact sequence. For an alternativepoint of view, one may take the cohomology exact sequence induced by0 −−−→ R p −−−→ R −−−→ R / pR −−−→ , i.e., the sequence −−−→ H k a ( R / pR ) δ p −−−→ H k + a ( R ) p −−−→ H k + a ( R ) π p −−−→ H k + a ( R / pR ) −−−→ . The Bockstein homomorphism β p above then coincides with the composition H k a ( R / pR ) π p ◦ δ p −−−−−→ H k + a ( R / pR ) . The D -module structure on local cohomology. Let R be an A -algebra, and z anelement of R . Then the D R | A -module structure on R extends uniquely to the localization R z as follows: one defines the action by induction on the order of δ ∈ D R | A and on the powerof the denominator by the rule δ ( r / z n ) : = δ ( r / z n − ) − [ δ , z ]( r / z n ) z . This may be written in closed form: set δ ( ) : = δ and δ ( i + ) : = [ δ ( i ) , e z n ] inductively; then(2.0.3) δ ( r / z n ) = ord ( δ ) ∑ k = ( − ) k δ ( k ) ( r ) z n ( k + ) . For elements zzz of R , the ˇCech complex C • ( zzz ; R ) is a complex of D R | A -modules, hence itscohomology modules H k ( zzz ) ( R ) have a natural D R | A -module structure. This D R | A -modulestructure on local cohomology is compatible with base change in the following sense:If S −→ R is a homomorphism of A -algebras, we observe that D S | A ( R ) has the structure JACK JEFFRIES AND ANURAG K. SINGH of a ( D R | A , D S | A ) -bimodule, where D R | A acts by postcomposition, and D S | A acts by pre-composition; one verifies using the inductive definition of differential operators that theprescribed compositions are indeed elements of D S | A ( R ) . This bimodule structure yields abase change functor from D S | A -modules to D R | A -modules, M D S | A ( R ) ⊗ D S | A M . Lemma 2.1.
Let S = A [ xxx ] be a polynomial ring over A, and R a homomorphic image of S. (a) For each z ∈ S, one has an isomorphism of D R | A -modules D S | A ( R ) ⊗ D S | A S z ∼ = R z . (b) Given elements zzz : = z , . . . , z t of S, one has an isomorphism of D R | A -modulesD S | A ( R ) ⊗ D S | A H t + ( zzz ) ( S ) ∼ = H t + ( zzz ) ( R ) . Proof.
Set I : = ker ( S −→ R ) . Since S is a polynomial ring over A , the functor D S | A ( − ) from S -modules to D S | A -modules is exact [SV, § D S | A -modules0 −−−→ D S | A ( I ) −−−→ D S | A −−−→ D S | A ( R ) −−−→ . The inclusion above may be used to identify D S | A ( I ) with ID S | A , so D S | A ( R ) ∼ = D S | A / ID S | A as right D S | A -modules. Thus, one has an isomorphism of R -modules D S | A ( R ) ⊗ D S | A S z ∼ = S z / IS z ∼ = R z , given by δ ⊗ ( s / z n ) δ ( s / z n ) , where δ denotes the image of δ ∈ D S | A modulo ID S | A . To see that this isomorphism is D R | A -linear, note that each element of D S | A ( R ) ⊗ D S | A S z may be written as 1 ⊗ s / z n , and anyelement γ ∈ D R | A can be written as µ + ID S | A for some µ ∈ D S | A such that µ ( I ) ⊆ I . Thecompatibility condition then boils down to checking that γ ( s / z n ) = µ ( s / z n ) , which is clearfrom the formula (2.0.3), proving (a).Next, consider the right-exact sequence of D S | A -modules ∑ i S z ··· ˆ z i ··· z t −−−→ S z ··· z t −−−→ H t + ( zzz ) ( S ) −−−→ , and the right-exact sequence of D R | A -modules ∑ i R z ··· ˆ z i ··· z t −−−→ R z ··· z t −−−→ H t + ( zzz ) ( R ) −−−→ . Applying D S | A ( R ) ⊗ D S | A ( − ) to the first, we obtain a commutative diagram of D R | A -modules D S | A ( R ) ⊗ D S | A ∑ i S z ··· ˆ z i ··· z t −−→ D S | A ( R ) ⊗ D S | A S z ··· z t −−→ D S | A ( R ) ⊗ D S | A H t + ( zzz ) ( S ) −−→ y y ∑ i R z ··· ˆ z i ··· z t −−→ R z ··· z t −−→ H t + ( zzz ) ( R ) −−→ , where the vertical maps are isomorphisms by (a). This induces the D R | A -module isomor-phism as claimed in (b), (cid:3) Frobenius lifting and p -derivations. Let T be a ring, and let p > T . A lift of the Frobenius endomorphism F of T / pT is a ringhomomorphism Λ p : T −→ T such that the following diagram commutes T Λ p −−−→ T y y T / pT F −−−→ T / pT . IFFERENTIAL OPERATORS ON INVARIANT RINGS 9
Definition 2.2 (Buium [Bu], Joyal [Jo]) . Let T be a ring, and p > p-derivation on T is a map ϕ p : T −→ T that satisfies the following for all a , b ∈ T :(i) ϕ p ( ) = ϕ p ( ab ) = a p ϕ p ( b ) + b p ϕ p ( a ) + p ϕ p ( a ) ϕ p ( b ) , and(iii) ϕ p ( a + b ) = ϕ p ( a ) + ϕ p ( b ) + C p ( a , b ) ,where C p ( x , y ) is the polynomial p ( x p + y p − ( x + y ) p ) regarded as an element of Z [ x , y ] .It follows from the above that ϕ p ( a + pb ) ≡ ϕ p ( a ) + b p mod p . If ϕ p is a p -derivation on T , then the map Λ p : T −→ T given by Λ p ( t ) = t p + p ϕ p ( t ) is a lift of the Frobenius; conversely, if p is a nonzerodivisor on T , and Λ p : T −→ T is alift of the Frobenius, then ϕ p : T −→ T with ϕ p ( t ) = p ( Λ p ( t ) − t p ) is a p -derivation on T . Example 2.3.
Fix a prime integer p >
0, and let S be a polynomial ring over Z in theindeterminates xxx : = x , . . . , x d . We refer to the Z -algebra homomorphism Λ p : S −→ S with Λ p ( x i ) = x pi for each i as the standard lift of the Frobenius with respect to xxx , and the corresponding p -derivation ϕ p : S −→ S with ϕ p ( s ) = Λ p ( s ) − s p p as the standard p-derivation with respect to xxx .We record the following compatibility for p -derivations used in the sequel: Let S and S ′ be polynomial rings over Z in the indeterminates xxx and xxx ′ respectively. Let ϒ : S −→ S ′ bea ring homomorphism such that ϒ ( x ) ∈ xxx ′ ∪{ , } for each x ∈ xxx , i.e., the homomorphism ϒ either takes an indeterminate x ∈ xxx to an indeterminate x ′ ∈ xxx ′ , or specializes it to 0 or 1.Let Λ p and ϕ p denote the standard lift of the Frobenius and the corresponding p -derivationon S with respect to xxx , and likewise let Λ ′ p and ϕ ′ p be the corresponding maps for S ′ withrespect to xxx ′ . Then the following diagrams commute: S ϒ −−−→ S ′ S ϒ −−−→ S ′ y Λ p y Λ ′ p y ϕ p y ϕ ′ p S ϒ −−−→ S ′ , S ϒ −−−→ S ′ . We recall the following criterion, due to Zdanowicz, for the existence of a lift of theFrobenius endomorphism of a hypersurface:
Proposition 2.4 ([Zd, Corollary 4.9]) . Let p be a prime integer, S : = Z [ xxx ] a polynomialring, and ϕ p : S −→ S the standard p-derivation on S with respect to xxx : = x , . . . , x d .For f ∈ S, set R : = S / f S. Suppose that R / p R is flat over Z / p Z .Then the Frobenius endomorphism on the hypersurface R / pR lifts to an endomorphismof R / p R if and only if ϕ p ( f ) ∈ (cid:18) p , f , (cid:16) ∂ f ∂ x (cid:17) p , . . . , (cid:16) ∂ f ∂ x d (cid:17) p (cid:19) S .
3. D
IFFERENTIAL OPERATORS VIA LOCAL COHOMOLOGY
Polynomial rings.
Let A be a commutative ring, and S : = A [ xxx ] the polynomial ringover A in the indeterminates xxx : = x , . . . , x d . Then P S | A is a polynomial ring over A in theindeterminates x i ⊗ ⊗ x i , for 0 i d . Since we view P S | A as an S -module via themap s s ⊗
1, we simply write x i for x i ⊗
1. Set y i : = ⊗ x i , so that P S | A = A [ xxx , yyy ] = S [ yyy ] , where yyy : = y , . . . , y d . The elements y − x , y − x , . . . , y d − x d form a generating set for the ideal ∆ S | A of P S | A . Using this generating set, we consider thesequence of ideals ∆ [ n ] S | A , the rings P [ n ] S | A , and the filtration D [ n ] S | A , as defined in § y − x , y − x , . . . , y d − x d are algebraically independent gen-erators for P S | A as an S -algebra. Thus, P [ n ] S | A is a free S -module with basis ( y − x ) a · · · ( y d − x d ) a d where 0 a i n . Likewise, Hom S ( P [ n ] S | A , S ) is the free S -module with the dual basis (cid:0) ( y − x ) a · · · ( y d − x d ) a d (cid:1) ⋆ , where 0 a i n , and ( − ) ⋆ denotes the corresponding element of the dual basis. Moreover, there is a P S | A -module isomorphism defined S -linearly by the rule γ n : P [ n ] S | A −→ Hom S ( P [ n ] S | A , S )( y − x ) a · · · ( y d − x d ) a d (cid:0) ( y − x ) n − a · · · ( y d − x d ) n − a d (cid:1) ⋆ . Proposition 3.1.
For each n > , one has P S | A -module isomorphismsH d + ( ∆ [ n + ] S | A ; P S | A ) γ n −−−→ Hom S ( P [ n ] S | A , S ) ρ ∗ −−−→ D [ n ] S | A . Proof.
We need only observe that the Koszul cohomology module H d + ( ∆ [ n + ] S | A ; P S | A ) coincides with P [ n ] S | A . The rest is immediate from the preceding discussion. (cid:3) To make the isomorphism ρ ∗ completely explicit, note first that ρ ( f ( xxx )) = f ( yyy ) . Now, ρ ∗ (cid:16)(cid:0) ( y − x ) a · · · ( y d − x d ) a d (cid:1) ⋆ (cid:17)(cid:0) f ( xxx ) (cid:1) = (cid:0) ( y − x ) a · · · ( y d − x d ) a d (cid:1) ⋆ (cid:0) f ( yyy ) (cid:1) = (cid:0) ( y − x ) a · · · ( y d − x d ) a d (cid:1) ⋆ (cid:16) f (cid:0) ( y − x ) + x , . . . , ( y d − x d ) + x d (cid:1)(cid:17) = ∂ a ,..., a d (cid:0) f ( xxx ) (cid:1) , where the last equality uses the Taylor expansion of a polynomial. It follows that ρ ∗ (cid:16)(cid:0) ( y − x ) a · · · ( y d − x d ) a d (cid:1) ⋆ (cid:17) = ∂ a ,..., a d . IFFERENTIAL OPERATORS ON INVARIANT RINGS 11
Proposition 3.2.
There is a P S | A -module isomorphismH d + ∆ S | A ( P S | A ) ρ ∗ ◦ γ −−−−→ D S | A , where γ is the isomorphismH d + ∆ S | A ( P S | A ) = lim −→ n H d + ∆ S | A ( P S | A ) −→ lim −→ n Hom S ( P [ n ] S | A , S ) . Proof.
For each n >
0, one has a commutative diagram H d + ( ∆ [ n + ] S | A ; P S | A ) γ n −−−→ Hom S ( P [ n ] S | A , S ) ρ ∗ −−−→ D [ n ] S | A y ∏ i ( y i − x i ) y y H d + ( ∆ [ n + ] S | A ; P S | A ) γ n + −−−→ Hom S ( P [ n + ] S | A , S ) ρ ∗ −−−→ D [ n + ] S | A , where the canonical surjection P [ n + ] S | A − ։ P [ n ] S | A induces the map in the middle column. Theleft column realizes local cohomology as the direct limit of Koszul cohomology; the mapsin the right column are injective, with D S | A as the directed union. (cid:3) Example 3.3.
The isomorphism ρ ∗ ◦ γ maps the local cohomology element η S : = (cid:20) ( y − x ) · · · ( y d − x d ) (cid:21) in H d + ∆ S | A ( P S | A ) to the differential operator in D S | A that is the identity map. More generally,for integers a i >
0, the image of the local cohomology element " ( y − x ) a + · · · ( y d − x d ) a d + ∈ H d + ∆ S | A ( P S | A ) , under ρ ∗ ◦ γ , is the differential operator ∂ a ,..., a d ∈ D S | A .There are analogous isomorphisms for differential operators from S to M , as below: Proposition 3.4.
For M an S-module and n > , there are P S | A -module isomorphismsH d + ( ∆ [ n + ] S | A ; P S | A ⊗ S M ) γ n −−−→ Hom S ( P [ n ] S | A , M ) ρ ∗ −−−→ D [ n ] S | A ( M ) and H d + ∆ S | A ( P S | A ⊗ S M ) ρ ∗ ◦ γ −−−−→ D S | A ( M ) . The maps γ n and γ are obtained from those in Propositions 3.1 and 3.2 by applying − ⊗ S M.Proof.
The right exactness of − ⊗ S M gives H d + ( ∆ [ n + ] S | A ; P S | A ⊗ S M ) ∼ = H d + ( ∆ [ n + ] S | A ; P S | A ) ⊗ S M ∼ = P [ n ] S | A ⊗ S M . Since P [ n ] S | A is a free S -module, one also has the isomorphismHom S ( P [ n ] S | A , M ) ∼ = Hom S ( P [ n ] S | A , S ) ⊗ S M . Thus, γ n is an isomorphism, obtained from the corresponding isomorphism in Proposi-tion 3.1; ρ ∗ is an isomorphism as discussed in § (cid:3) Hypersurfaces.
As in § S : = A [ xxx ] be a polynomial ring over a commutativering A , and identify P S | A = A [ xxx , yyy ] = S [ yyy ] . Let R = S / ( f ( xxx )) , where f ∈ S is a nonzero polynomial, and identify R ⊗ A R with P R | A = A [ xxx , yyy ]( f ( xxx ) , f ( yyy )) = R [ yyy ]( f ( yyy )) . Then y − x , y − x , . . . , y d − x d serve as generators for ∆ S | A , and their images in P R | A are generators for the ideal ∆ R | A . Proposition 3.5.
One has P S | A -module isomorphismsD [ n ] R | A ∼ = ann (cid:0) f ( yyy ) , D [ n ] S | A ( R ) (cid:1) for each n > , and D R | A ∼ = ann (cid:0) f ( yyy ) , D S | A ( R ) (cid:1) . Proof.
Taking the exact sequence P [ n ] S | A ⊗ S R f ( yyy ) −−−→ P [ n ] S | A ⊗ S R −−−→ P [ n ] R | A −−−→ , and applying Hom R ( − , R ) , one obtains0 −−−→ Hom R ( P [ n ] R | A , R ) −−−→ Hom R ( P [ n ] S | A ⊗ S R , R ) f ( yyy ) −−−→ Hom R ( P [ n ] S | A ⊗ S R , R ) . SinceHom R ( P [ n ] R | A , R ) ∼ = D [ n ] R | A and Hom R ( P [ n ] S | A ⊗ S R , R ) ∼ = Hom S ( P [ n ] S | A , R ) ∼ = D [ n ] S | A ( R ) , the first isomorphisms follow. The second follows by taking the union over n . (cid:3) In light of the previous proposition, we identify D [ n ] R | A with ann (cid:0) f ( yyy ) , D [ n ] S | A ( R ) (cid:1) ; thisidentifies an A -linear differential operator S −→ R that is annihilated by f ( yyy ) with theinduced factorization R −→ R . Next, consider the sequence0 −−−→ P S | A ⊗ S R f ( yyy ) −−−→ P S | A ⊗ S R −−−→ P R | A −−−→ , and the induced Koszul cohomology exact sequence0 −−→ H d ( ∆ [ n + ] R | A ; P R | A ) δ f ( yyy ) −−→ H d + ( ∆ [ n + ] S | A ; P S | A ⊗ S R ) f ( yyy ) −−→ H d + ( ∆ [ n + ] S | A ; P S | A ⊗ S R ) , where the injectivity on the left holds since the generators of the ideal ∆ [ n + ] S | A form a regularsequence on P S | A ⊗ S R . We also use δ f ( yyy ) for the connecting map in the corresponding localcohomology exact sequence, i.e.,0 −−−→ H d ∆ R | A ( P R | A ) δ f ( yyy ) −−−→ H d + ∆ S | A ( P S | A ⊗ S R ) f ( yyy ) −−−→ H d + ∆ S | A ( P S | A ⊗ S R ) . Proposition 3.6.
The mapsH d ( ∆ [ n + ] R | A ; P R | A ) δ f ( yyy ) −−→ H d + ( ∆ [ n + ] S | A ; P S | A ⊗ S R ) γ n −−→ Hom S ( P [ n ] S | A , R ) ρ ∗ −−→ D [ n ] S | A ( R ) induce an isomorphism of P R | A -modulesH d ( ∆ [ n + ] R | A ; P R | A ) ρ ∗ ◦ γ n ◦ δ f ( yyy ) −−−−−−−−−→ D [ n ] R | A , IFFERENTIAL OPERATORS ON INVARIANT RINGS 13 with D [ n ] R | A regarded as a submodule of D [ n ] S | A ( R ) . Passing to the direct limit, the mapH d ∆ R | A ( P R | A ) ρ ∗ ◦ γ ◦ δ f ( yyy ) −−−−−−−−→ D R | A , is an isomorphism, with D R | A regarded as a submodule of D S | A ( R ) .Proof. By Proposition 3.4, the maps γ n and ρ ∗ are P S | A -module isomorphisms. Also, δ f ( yyy ) : H d ( ∆ [ n + ] R | A ; P R | A ) −→ ann (cid:0) f ( yyy ) , H d + ( ∆ [ n + ] S | A ; P S | A ⊗ S R ) (cid:1) is an isomorphism, and D [ n ] R | A = ann (cid:0) f ( yyy ) , D [ n ] S | A ( R ) (cid:1) . This justifies the first isomorphism.The second is the familiar transition from Koszul cohomology to local cohomology. (cid:3) For S : = A [ xxx ] a polynomial ring, Example 3.3 identifies the element of H d + ∆ S | A ( P S | A ) thatcorresponds to the identity map in D S | A ; for a hypersurface R : = A [ xxx ] / ( f ( xxx )) , we nextidentify the element of H d ∆ R | A ( P R | A ) that corresponds to the identity map in D R | A : Example 3.7.
Since µ ( f ( yyy )) = µ ( f ( xxx )) , where µ : P S | A −→ S is the A -algebra homomor-phism determined by x i x i , and y i x i , one has f ( yyy ) − f ( xxx ) = d ∑ i = ( y i − x i ) g i , where g i ∈ P S | A . It follows that ∑ di = ( y i − x i ) g i = P R | A . Thus, we obtain an element η R : = " · · · , ( − ) i g i ∏ j = i ( y j − x j ) , · · · ∈ H d ∆ R | A ( P R | A ) , where the signs adhere to the convention in § η R maps to the identityin D R | A under the isomorphisms in Proposition 3.6. First note that δ f ( yyy ) ( η R ) = (cid:20) ( y − x ) · · · ( y d − x d ) (cid:21) . Using Example 3.3, ρ ∗ ◦ γ maps δ f ( yyy ) ( η R ) to the element of Hom A ( S , R ) corresponding tothe canonical surjection S − ։ R . Thus, ρ ∗ ◦ γ ◦ δ f ( yyy ) ( η R ) is the identity map on R .3.3. Equivalence as D -modules. Let A be a ring. Given A -algebras T and T ′ , one has ahomomorphism D T | A −→ D ( T ⊗ A T ′ ) | A given by D T | A id ⊗ −−−→ D T | A ⊗ A T ′ −−−→ D ( T ⊗ A T ′ ) | T ′ ⊆ D ( T ⊗ A T ′ ) | A . For an ideal I of T ⊗ A T ′ , we regard the local cohomology module H kI ( T ⊗ A T ′ ) as a D T | A -module by restriction of scalars along the map above. Theorem 3.8.
Let A be a commutative ring, and let S : = A [ xxx ] be the polynomial ring over Ain the indeterminates xxx : = x , . . . , x d . Then the mapH d + ∆ S | A ( P S | A ) ρ ∗ ◦ γ −−−−→ D S | A , as in Proposition 3.2, is an isomorphism of D S | A -modules. Proof.
By Proposition 3.2, the displayed isomorphism is P S | A -linear; we need only verifythat it is D S | A -linear. In view of Example 3.3, it suffices to verify that the map " ( y − x ) a + · · · ( y d − x d ) a d + ∂ a ,..., a d is D S | A -linear. In the case S = A [ x ] , i.e., where d =
0, the verification takes the form ∂ b " ( y − x ) a + = " (cid:0) a + bb (cid:1) ( y − x ) a + b + (cid:18) a + bb (cid:19) ∂ a + b = ∂ b ∂ a , with the general case being similar. (cid:3) Theorem 3.9.
Let A be a commutative ring, and let S : = A [ xxx ] be the polynomial ring over Awhere xxx : = x , . . . , x d . Let R = S / f S, for f ∈ S a nonzero polynomial. Then the mapH d ∆ R | A ( P R | A ) ρ ∗ ◦ γ ◦ δ f ( yyy ) −−−−−−−−→ D R | A , as in Proposition 3.6, is an isomorphism of D R | A -modules.Proof. We consider H d + ∆ S | A ( P S | A ⊗ S R ) ∼ = H d + ∆ S | A ( R ⊗ A S ) as a D R | A -module as described at thebeginning of this subsection.Let ρ ∗ ◦ γ : H d + ∆ S | A ( P S | A ) −→ D S | A be the D S | A -module isomorphism from Theorem 3.8.Applying D S | A ( R ) ⊗ D S | A ( − ) to this map, we obtain a D R | A -linear isomorphism D S | A ( R ) ⊗ D S | A H d + ∆ S | A ( P S | A ) −→ D S | A ( R ) ⊗ D S | A D S | A . We claim that this map identifies with the isomorphism H d + ∆ S | A ( P S | A ⊗ S R ) −→ D S | A ( R ) obtained by applying − ⊗ S R , as in Proposition 3.4. Indeed, it is easy to see that D S | A ( R ) ⊗ D S | A H d + ∆ S | A ( P S | A ) ∼ = D ( S ⊗ A S ) | ( A ⊗ A S ) ( R ⊗ A S ) ⊗ D ( S ⊗ AS ) | ( A ⊗ AS ) H d + ∆ S | A ( P S | A ) , and the latter identifies with H d + ∆ S | A ( P S | A ⊗ S R ) by Lemma 2.1.We have the commutative diagram0 −−−→ H d ∆ R | A ( P R | A ) δ f ( yyy ) −−−→ H d + ∆ S | A ( P S | A ⊗ S R ) f ( yyy ) −−−→ H d + ∆ S | A ( P S | A ⊗ S R ) y ∼ = y ∼ = y ∼ = −−−→ D R | A −−−→ D S | A ( R ) f ( yyy ) −−−→ D S | A ( R ) , where all of maps in the rightmost square are D R | A -linear; the first vertical isomorphism isthe map from Proposition 3.6. It follows that this map is a D R | A -linear isomorphism. (cid:3) Example 3.10.
Let S : = A [ x , . . . , x d ] be a polynomial ring over a commutative ring A . Fixthe N -grading on S where S = A and deg x i = i . Let R : = S / ( f ( xxx )) , for f ∈ S anonzero homogeneous polynomial. We explicitly describe the element of H d ∆ R | A ( P R | A ) thatcorresponds to the Euler operator E : = d ∑ i = x i ∂∂ x i ∈ D R | A . IFFERENTIAL OPERATORS ON INVARIANT RINGS 15
With the notation as in Example 3.7, let f ( yyy ) − f ( xxx ) = ∑ di = ( y i − x i ) g i with g i ∈ P S | A , inwhich case ∑ di = ( y i − x i ) g i = P R | A . Under the isomorphism of the previous theorem,the cohomology class η R ∈ H d ∆ R | A ( P R | A ) of the ˇCech cocycle g ∏ j = ( y j − x j ) , − g ∏ j = ( y j − x j ) , · · · , ( − ) d g d ∏ j = d ( y j − x j ) ! corresponds to the identity element in D R | A . As the isomorphism is one of D R | A modules,the element of H d ∆ R | A ( P R | A ) that corresponds to the Euler operator E is the cohomologyclass of the element obtained by applying E , considered as an operator on the xxx variables,componentwise to the above cocycle. Example 3.11.
Consider R : = A [ x ] / ( x n ) , for n a positive integer. Then H ∆ R | A ( P R | A ) = P R | A = A [ x , y ] / ( x n , y n ) . In P S | A one has y n − x n = ( y − x ) ∑ n − k = y n − − k x k , so the element of H ∆ R | A ( P R | A ) that corre-sponds to the identity in D R | A is n − ∑ k = y n − − k x k , while the element corresponding to the Euler operator is E (cid:16) n − ∑ k = y n − − k x k (cid:17) = n − ∑ k = ky n − − k x k . Example 3.12.
Set R : = A [ x , x ] / ( x + x ) . Then the identity in D R | A corresponds to (cid:20) y + x y − x , − ( y + x ) y − x (cid:21) ∈ H ∆ R | A ( P R | A ) , and the Euler operator to " y x + y x ( y − x ) , − ( y x + y x )( y − x ) ∈ H ∆ R | A ( P R | A ) . Frobenius trace.
Suppose A is a field of characteristic p >
0. Let S : = A [ x , . . . , x d ] be a polynomial ring, and R : = S / ( f ( xxx )) a graded hypersurface. Let e be a positive inte-ger. The rings R and R p e are Gorenstein, so the graded analogue of local duality impliesthat Hom R pe ( R , R p e ) is a cyclic R -module. To specify a generator, first set Φ eS : = ∂ p e − ,..., p e − . It is a key point that Φ eS : S −→ S p e , and that it is S p e -linear. Next, consider the composition S f pe − −−−→ S Φ eS −−−→ S p e π −−−→ R p e where π is the canonical surjection. Since f S is contained in its kernel, the compositionfactors through a map Φ eR : R −→ R p e , that we define to be the e-th Frobenius trace of R . When e =
1, we refer to Φ R : R −→ R p as the Frobenius trace map. In general, the e -th Frobenius trace Φ eR is an element of D [ p e ] R | A ,and is an R -module generator for Hom R pe ( R , R p e ) .We claim that Φ eR , viewed as a differential operator in D R | A , corresponds to the imageof the element η R from Example 3.7 under the e -th iterate of the Frobenius action F on thelocal cohomology module H d ∆ R | A ( P R | A ) , i.e., ( ρ ∗ ◦ γ ◦ δ f ( yyy ) )( F e ( η R )) = Φ eR . To see this, consider η S : = (cid:20) ( y − x ) · · · ( y d − x d ) (cid:21) ∈ H d + ∆ S | A ( P S | A ) as in Example 3.3, and note that F e ( η S ) = " ( y − x ) p e · · · ( y d − x d ) p e . By Example 3.3, ( ρ ∗ ◦ γ )( F e ( η S )) = ∂ p e − ,..., p e − , which equals Φ eS . Likewise, if η S is the image of η S in H d + ∆ S | A ( P S | A ⊗ S R ) , then ( ρ ∗ ◦ γ )( F e ( η S )) = π ◦ Φ eS . With η R as in Example 3.7, one has F e ( η R ) = " · · · , ( − ) i g p e i ∏ j = i ( y j − x j ) p e , · · · ∈ H d ∆ R | A ( P R | A ) , so δ f ( yyy ) ( F e ( η R )) = " f ( yyy ) p e − ∏ di = ( y i − x i ) p e = f ( yyy ) p e − F e ( η S ) in H d + ∆ S | A ( P S | A ⊗ S R ) . Thus, since ρ ∗ ◦ γ is P S | A -linear, one has ( ρ ∗ ◦ γ ◦ δ f ( yyy ) )( F e ( η R )) = f ( yyy ) p e − π ◦ Φ eS , which coincides with Φ eR in D R | A , regarded as a submodule of D S | A ( R ) .3.5. Lifting differential operators modulo p . Let A , S , and R be as in § A is acommutative ring, S : = A [ x , . . . , x d ] is a polynomial ring, and R : = S / ( f ( xxx )) is a hypersur-face. Suppose p > P R | A . The exact sequence0 −−−→ P R | A p −−−→ P R | A −−−→ P ( R / pR ) | ( A / pA ) −−−→ −−→ H d ∆ R | A ( P R | A ) −−→ H d ∆ R | A ( P ( R / pR ) | ( A / pA ) ) δ p −−→ H d + ∆ R | A ( P R | A ) p −−→ H d + ∆ R | A ( P R | A ) with δ p denoting the connecting homomorphism. In particular, δ p induces an isomorphismcoker (cid:16) H d ∆ R | A ( P R | A ) −−−→ H d ∆ R | A ( P ( R / pR ) | ( A / pA ) ) (cid:17) −−−→ ann (cid:0) p , H d + ∆ R | A ( P R | A ) (cid:1) . Acknowledging the abuse of notation, we denote the inverse of this isomorphism by δ − p ,so as to obtain: IFFERENTIAL OPERATORS ON INVARIANT RINGS 17
Proposition 3.13.
There is an isomorphism of P R | A -modules ann (cid:0) p , H d + ∆ R | A ( P R | A ) (cid:1) −−−→ coker (cid:16) D R | A −−−→ D ( R / pR ) | ( A / pA ) (cid:17) , given by ρ ∗ ◦ γ ◦ δ f ( yyy ) ◦ δ − p . In particular, given an element ν ∈ H d ∆ R | A ( P ( R / pR ) | ( A / pA ) ) , the corresponding differential operator ( ρ ∗ ◦ γ ◦ δ f ( yyy ) )( ν ) ∈ D ( R / pR ) | ( A / pA ) lifts to an ele-ment of D R | A if and only if δ p ( ν ) = .Proof. One has a commutative diagram H d ∆ R | A ( P R | A ) −−−→ H d ∆ R | A ( P ( R / pR ) | ( A / pA ) ) y y D R | A −−−→ D ( R / pR ) | ( A / pA ) where the vertical maps are the isomorphisms ρ ∗ ◦ γ ◦ δ f ( yyy ) of Proposition 3.6. Combiningwith the isomorphismann (cid:0) p , H d + ∆ R | A ( P R | A ) (cid:1) δ − p −−−→ coker (cid:16) H d ∆ R | A ( P R | A ) −−−→ H d ∆ R | A ( P ( R / pR ) | ( A / pA ) ) (cid:17) , the assertion follows. (cid:3) Next, we establish a similar criterion for when a differential operator D ( R / pR ) | ( A / pA ) liftsto D ( R / p R ) | ( A / p A ) . Start with the exact sequence0 −−−→ P ( R / pR ) | ( A / pA ) p −−−→ P ( R / p R ) | ( A / p A ) −−−→ P ( R / pR ) | ( A / pA ) −−−→ −−→ H d ∆ R | A ( P ( R / p R ) | ( A / p A ) ) −−→ H d ∆ R | A ( P ( R / pR ) | ( A / pA ) ) β p −−→ H d + ∆ R | A ( P ( R / pR ) | ( A / pA ) ) p −−→ H d + ∆ R | A ( P ( R / p R ) | ( A / p A ) ) −−→ . The connecting homomorphism β p in the sequence above is the Bockstein homomorphismon local cohomology, see § β p induces an isomorphismcoker (cid:16) H d ∆ R | A ( P ( R / p R ) | ( A / p A ) ) −−−→ H d ∆ R | A ( P ( R / pR ) | ( A / pA ) ) (cid:17) −−−→ image ( β p ) , and we denote the inverse of this isomorphism by β − p . Then, along the same lines asProposition 3.13, we have: Proposition 3.14.
There is an isomorphism of P R | A -modules image ( β p ) −−−→ coker (cid:16) D ( R / p R ) | ( A / p A ) −−−→ D ( R / pR ) | ( A / pA ) (cid:17) induced by ρ ∗ ◦ γ ◦ δ f ( yyy ) ◦ β − p . In particular, for an element ν ∈ H d ∆ R | A ( P ( R / pR ) | ( A / pA ) ) , the corresponding differential operator ( ρ ∗ ◦ γ ◦ δ f ( yyy ) )( ν ) ∈ D ( R / pR ) | ( A / pA ) lifts to an ele-ment of D ( R / p R ) | ( A / p A ) if and only if β p ( ν ) = . Lifting Frobenius trace.
Let S : = Z [ xxx ] be a polynomial ring over Z in the indetermi-nates xxx : = x , . . . , x d . Identifying P S with Z [ xxx , yyy ] as before, note that P S may also be viewedas a polynomial ring over Z in the indeterminates x , . . . , x d , y − x , . . . , y d − x d . Fix a prime integer p >
0, and let Λ p : P S −→ P S denote the standard lift of Frobenius withrespect to the indeterminates above, i.e., Λ p is the endomorphism of P S with Λ p ( x i ) = x pi and Λ p ( y i − x i ) = ( y i − x i ) p for each i . Let ϕ p denote the corresponding p -derivation on P S .Let R = S / ( f ( xxx )) be a graded hypersurface. Assume that p is regular on P R , in whichcase there is an exact sequence(3.14.1) H d ∆ R ( P R ) −−→ H d ∆ R ( P R / pP R ) δ p −−→ H d + ∆ R ( P R ) p −−→ H d + ∆ R ( P R ) π p −−→ H d + ∆ R ( P R / pP R ) . Let η R ∈ H d ∆ R ( P R ) be the element that corresponds to the identity map in D R , as in Exam-ple 3.7, and let η R denote its image in H d ∆ R ( P R / pP R ) . We claim that δ p (cid:0) F ( η R ) (cid:1) = " ϕ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) ∏ di = ( y i − x i ) p as elements of H d + ∆ R ( P R ) . To see this, take g i ∈ P S with f ( yyy ) − f ( xxx ) = ∑ di = ( y i − x i ) g i as inExample 3.7. Then, in P S , one has ϕ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) = p (cid:16) Λ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) − (cid:0) f ( yyy ) − f ( xxx ) (cid:1) p (cid:17) = p (cid:16) ∑ i Λ p ( y i − x i ) Λ p ( g i ) − (cid:0) f ( yyy ) − f ( xxx ) (cid:1) p (cid:17) = p (cid:16) ∑ i ( y i − x i ) p Λ p ( g i ) − (cid:0) f ( yyy ) − f ( xxx ) (cid:1) p (cid:17) = p (cid:16) ∑ i ( y i − x i ) p (cid:0) g pi + p ϕ p ( g i ) (cid:1) − (cid:0) f ( yyy ) − f ( xxx ) (cid:1) p (cid:17) = ∑ i ( y i − x i ) p ϕ p ( g i ) + p (cid:16) ∑ i ( y i − x i ) p g pi − (cid:0) f ( yyy ) − f ( xxx ) (cid:1) p (cid:17) . It follows that the image of ϕ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) in P R , i.e., modulo the ideal ( f ( xxx ) , f ( yyy )) , is ∑ i ( y i − x i ) p ϕ p ( g i ) + p ∑ i ( y i − x i ) p g pi . Hence, in H d + ∆ R ( P R ) , one has " ϕ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) ∏ i ( y i − x i ) p = " ∑ i ( y i − x i ) p ϕ p ( g i ) + p ∑ i ( y i − x i ) p g pi ∏ i ( y i − x i ) p = " p ∑ i ( y i − x i ) p g pi ∏ i ( y i − x i ) p = δ p (cid:0) F ( η R ) (cid:1) , which proves the claim. Similarly, with β p denoting the Bockstein homomorphism H d ∆ R ( P R / pP R ) π p ◦ δ p −−−−−→ H d + ∆ R ( P R / pP R ) IFFERENTIAL OPERATORS ON INVARIANT RINGS 19 with δ p and π p as in (3.14.1), one see that β p (cid:0) F ( η R ) (cid:1) = " ϕ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) ∏ di = ( y i − x i ) p as elements of H d + ∆ R ( P R / pP R ) .Combining the preceding calculations with Propositions 3.13 and 3.14, we obtain: Theorem 3.15.
Let S : = Z [ xxx ] be a polynomial ring in the indeterminates xxx : = x , . . . , x d .Fix a nonzero homogeneous polynomial f ( xxx ) in S, and set R : = S / ( f ( xxx )) . Let p be a primeinteger that is a regular element on P R . Let Λ p be the endomorphism of P S : = Z [ xxx , yyy ] with Λ p ( x i ) = x pi and Λ p ( y i − x i ) = ( y i − x i ) p for each i, and let ϕ p denote the corresponding p-derivation.Then the Frobenius trace map Φ R / pR on R / pR, as in § " ϕ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) ∏ di = ( y i − x i ) p ∈ H d + ∆ R ( P R ) is zero; similarly, Φ R / pR lifts to a differential operator on R / p R if and only if the imageof the displayed element in H d + ∆ R ( P R / pP R ) is zero. Remark 3.16.
For S and R as in the previous theorem set a : = deg f ( xxx ) − ( d + ) . Then,with the grading shifts as below, Theorem 3.9 provides degree-preserving isomorphisms D R ∼ = H d ∆ R ( P R )( a ) and D R / pR ∼ = H d ∆ R ( P R / pP R )( a ) . In particular, the identity element in the ring D R or in the ring D R / pR corresponds to acohomology class in [ H d ∆ R ( P R )] a or [ H d ∆ R ( P R / pP R )] a respectively; confer Example 3.7.It is a straightforward calculation that the Frobenius trace map Φ R / pR on R / pR is adifferential operator of degree ap − a . It follows that Φ R / pR corresponds to a cohomologyclass in [ H d ∆ R ( P R / pP R )] ap as, indeed, is evident by the calculation in § Remark 3.17.
Consider R : = S / f S , where S : = Z [ x , x , x ] and f ( xxx ) : = x + x + x .Let p = §
3] and [Je, Example 6.6], the Frobenius tracemap Φ R / pR on R / pR does not lift to a differential operator on R . Using Theorem 3.15, itfollows that the local cohomology element ζ p : = " ϕ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) ∏ ( y i − x i ) p in H ∆ R ( P R ) is nonzero. Note that ζ p is annihilated by p in view of (3.14.1). Hence, themodule H ∆ R ( P R ) contains a nonzero p -torsion element for each prime integer p = R : = S / f S , where S : = Z [ x , x , x , x ] and f ( xxx ) : = x + x + x + x .Recent work of Mallory [Ma, Theorem 1.2] shows that R has no differential operators ofnegative degree. For each prime integer p =
3, the Frobenius trace on R / pR has negativedegree, namely 1 − p , and hence does not lift to R . It follows that H ∆ R ( P R ) contains anonzero p -torsion element for each prime integer p =
4. I
NTEGER TORSION IN LOCAL COHOMOLOGY
Our approach to constructing mod p differential operators that do not lift, based on theresults of the previous section, is via constructing p -torsion elements in local cohomologymodules of the form H d + ∆ R ( P R ) . Towards this, the following lemma will prove useful; thenotation is as in § Lemma 4.1.
Let S : = Z [ xxx ] be a polynomial ring in the indeterminates xxx : = x , . . . , x d .Consider a polynomial f ( xxx ) ∈ S of the formf ( xxx ) = m ∑ i = x i f i , where m d and each f i ∈ S. Set R : = S / ( f ( xxx )) . Fix a prime integer p, and let ϕ p denotethe p-derivation of P S as in Theorem 3.15; this restricts to the standard p-derivation of Swith respect to xxx.If the local cohomology element " ϕ p (cid:0) f ( xxx ) (cid:1) ( x · · · x m ) p in H m + ( x ,..., x m ) ( R ) is nonzero, then so isthe element " ϕ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) ∏ di = ( y i − x i ) p in H d + ∆ R ( P R ) .Similarly, if the image of " ϕ p (cid:0) f ( xxx ) (cid:1) ( x · · · x m ) p in H m + ( x ,..., x m ) ( R / pR ) is nonzero, then so is theimage of " ϕ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) ∏ i ( y i − x i ) p in H d + ∆ R ( P R / pP R ) .Proof. Recall the notation yyy : = y , . . . , y d . We identify P S with the polynomial ring Z [ xxx , yyy ] and P R with Z [ xxx , yyy ] / ( f ( xxx ) , f ( yyy )) , so that ∆ R is the ideal generated by the elements y − x , y − x , . . . , y d − x d . Suppose the displayed element of H d + ∆ R ( P R ) is zero. Then there exists an integer k > P S one has(4.1.1) Λ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) − (cid:0) f ( yyy ) − f ( xxx ) (cid:1) p p ( y − x ) k · · · ( y d − x d ) k ∈ (cid:0) f ( yyy ) , f ( xxx ) , ( y − x ) p + k , . . . , ( y d − x d ) p + k (cid:1) P S . Let P ′ denote the image of P S under the specialization x i i m . Notethat f ( xxx ) P S Λ p −−−→ P S y y P ′ Λ p −−−→ P ′ , the ideal membership (4.1.1) specializes to give(4.1.2) Λ p (cid:0) f ( yyy ) (cid:1) − (cid:0) f ( yyy ) (cid:1) p p y k · · · y km ( y m + − x m + ) k · · · ( y d − x d ) k ∈ (cid:0) f ( yyy ) , y p + k , . . . , y p + km , ( y m + − x m + ) p + k , . . . , ( y d − x d ) p + k (cid:1) P ′ . IFFERENTIAL OPERATORS ON INVARIANT RINGS 21
The elements y m + − x m + , . . . , y d − x d are algebraically independent over Z [ yyy ] / (cid:0) f ( yyy ) , y p + k , . . . , y p + km (cid:1) , so y m + − x m + , . . . , y d − x d , as well as their powers, form a regular sequence in the ring P ′ / (cid:0) f ( yyy ) , y p + k , . . . , y p + km (cid:1) P ′ . Using this, (4.1.2) implies Λ p (cid:0) f ( yyy ) (cid:1) − (cid:0) f ( yyy ) (cid:1) p p y k · · · y km ∈ (cid:0) f ( yyy ) , y p + k , . . . , y p + km , ( y m + − x m + ) p , . . . , ( y d − x d ) p (cid:1) P ′ . Next, specialize x i y i for m + i d , to obtain Λ p (cid:0) f ( yyy ) (cid:1) − (cid:0) f ( yyy ) (cid:1) p p y k · · · y km ∈ (cid:0) f ( yyy ) , y p + k , . . . , y p + km (cid:1) Z [ yyy ] , where Λ p is the standard lift of Frobenius on Z [ yyy ] with respect to yyy . Renaming y i x i for each i , the above reads ϕ p (cid:0) f ( xxx ) (cid:1) x k · · · x km ∈ (cid:0) f ( xxx ) , x p + k , . . . , x p + km (cid:1) Z [ xxx ] , implying that " ϕ p (cid:0) f ( xxx ) (cid:1) ( x · · · x m ) p ∈ H m + ( x ,..., x m ) ( R ) is zero. The proof of the final assertion is similar. (cid:3) A question of Huneke [Hu, Problem 4] asks whether local cohomology modules of Noe-therian rings have finitely many associated prime ideals. This was answered in the negativeby the second author in [Si, § R over Z for which a local cohomol-ogy module H k a ( R ) has p -torsion elements for each prime integer p . From this, it followsthat H k a ( R ) has infinitely many associated primes; this is extended to several natural fami-lies of hypersurfaces by Theorems 6.1, 6.2, and 7.1. On the other hand, for a polynomialring over Z , or, more generally, a smooth Z -algebra, the answer to Huneke’s question ispositive; this follows from the following result, which we use in the next section: Theorem 4.2. [BBLSZ, Theorem 3.1]
Let S be a smooth Z -algebra, and a an ideal of Sgenerated by elements fff = f , . . . , f t . Let k be a nonnegative integer.If a prime integer is a nonzerodivisor on the Koszul cohomology module H k ( fff ; S ) , thenit is also a nonzerodivisor on the local cohomology module H k a ( S ) .
5. T
ORIC HYPERSURFACES
The semigroups Γ that we consider here will be finitely generated, commutative, andcontain an identity element. For A a commutative ring, the semigroup ring A [ Γ ] is thefree A -module with basis Γ , with multiplication of elements of Γ determined by the semi-group operation. By an affine semigroup Γ , we mean a subsemigroup of N d for some d ; inthis case, fixing indeterminates x , . . . , x d over A , one may identify A [ Γ ] with the subring A (cid:2) x h · · · x h d d | ( h , . . . , h d ) ∈ Γ (cid:3) of the polynomial ring A [ x , . . . , x d ] . An affine semigroup Γ is normal if for γ , γ ∈ Γ ,if k ( γ − γ ) ∈ Γ for a positive integer k , then γ − γ ∈ Γ . We informally refer to the semigroup ring A [ Γ ] , where Γ is a normal affine semigroup, as a toric A -algebra. In viewof [Ho, Proposition 1], each normal affine semigroup Γ admits an embedding in N m , forsome m , such that the corresponding inclusion of rings(5.0.1) A [ Γ ] ֒ −→ A [ N m ] is split as a map of A [ Γ ] -modules; moreover, if A is a ring with an infinite unit group A × ,the embedding (5.0.1) realizes A [ Γ ] as the invariant ring of an action of a product of copiesof A × on the polynomial ring A [ N m ] . Theorem 5.1.
Let R be a hypersurface over Z that is toric. Then, for each prime p, eachdifferential operator on R / pR lifts to a differential operator on R; thus, the natural mapD R ⊗ Z ( Z / p Z ) −→ D R / pR is an isomorphism.Proof. Let d denote the relative dimension of R over Z , i.e., dim R = d +
1. Fix a primeinteger p >
0. In view of Proposition 3.13, it suffices to show that the local cohomologymodule H d + ∆ R ( P R ) has no nonzero p -torsion elements. Fix an embedding R ֒ −→ T : = Z [ xxx ] as in (5.0.1) that is R -split. Then the inclusion P R ֒ −→ P T = Z [ xxx , yyy ] is P R -split, and it followsthat the induced map H d + ∆ R ( P R ) −→ H d + ∆ R ( P T ) is injective. Thus, it suffices to show that the module H d + ∆ R ( P T ) has no nonzero p -torsionelements. Fix a set of monomials µ ( xxx ) , . . . , µ d ( xxx ) that generate R as a Z -algebra. Then µ ( yyy ) − µ ( xxx ) , . . . , µ d ( yyy ) − µ d ( xxx ) serve as generators for the ideal ∆ R . By Theorem 4.2, applied to the polynomial ring P T , itsuffices to verify that the Koszul cohomology module H d + ( ∆ R ; P T ) ∼ = P T ∆ R P T has no nonzero p -torsion elements. The ideal ∆ R P T is generated by differences of monomi-als, hence the ring P T / ∆ R P T is isomorphic to a (possibly nonaffine) semigroup ring over Z by [Gi, Theorem 7.11], and thus a free Z -module. (cid:3) For toric hypersurfaces R , while the map D R −→ D R / pR is indeed surjective, a differen-tial operator on R / pR need not lift to a differential operator on R of the same order: Example 5.2.
Let R : = Z [ x , x x , x ] , i.e., R is the second Veronese subring of the poly-nomial ring T : = Z [ x , x ] . Fix the standard N -grading on R , i.e., with deg x i x j = K be a field of characteristic other than 2. Then the group {± } acts on thepolynomial ring T ⊗ Z K with invariant ring R ⊗ Z K , and the ring of differential opera-tors D R ⊗ Z K | K is generated by elements of D T ⊗ Z K | K that have even degree. Note that thederivations x i ∂∂ x j have degree 0; there are no derivations of negative degree, confer [Ka2].Since D R ⊆ D R ⊗ Z Q | Q , it follows that R has no derivations of negative degree. In contrast,the ring R / R has a derivation of degree −
1, namely x − ∂∂ x = x − ∂∂ x , IFFERENTIAL OPERATORS ON INVARIANT RINGS 23 which is the endomorphism of R / R with x i y j x i + y j + x i y j for i , j ∈ N . This derivation lifts to the differential operator ∂∂ x ∂∂ x in D R .6. Q UADRATIC FORMS AND P FAFFIANS
The main focus of this section is Pfaffian hypersurfaces, but we begin with the analogueof Theorem 1.2 for quadratic hypersurfaces, where the arguments are most transparent:
Theorem 6.1.
Let x , . . . , x m be indeterminates over Z , where m > , and setR : = Z [ x , . . . , x m ] / ( m ∑ i = x i x m + i ) . Then, for each prime integer p > , the Frobenius trace on R / pR does not lift to a differ-ential operator on R / p R, nor, a fortiori, to a differential operator on R.Proof.
Set S : = Z [ x , . . . , x m ] . Fix p , and let ϕ p be the standard p -derivation on S withrespect to xxx . In view of Theorem 3.15 and Lemma 4.1, it suffices to prove that the element (cid:20) ϕ p ( ∑ mi = x i x m + i )( x · · · x m ) p (cid:21) ∈ H m ( x ,..., x m ) ( R / pR ) is nonzero. Indeed, if it were zero, then there exists an integer k > ϕ p ( m ∑ i = x i x m + i )( x · · · x m ) k ∈ (cid:0) x p + k , . . . , x p + km (cid:1) R / pR . The image of ϕ p ( ∑ x i x m + i ) in R equals p ∑ x pi x pm + i , regarded as an element of R , giving(6.1.1) (cid:16) p m ∑ i = x pi x pm + i (cid:17) ( x · · · x m ) k ∈ (cid:0) x p + k , . . . , x p + km (cid:1) R / pR . Let e i ∈ Z m denote the i -th standard basis vector. Consider the Z m -grading of R / pR wheredeg x i = e i , deg x m + i = − e i , for 1 i m . The element on the left hand side in (6.1.1) then has degree ( k , . . . , k ) . Hence,in a homogeneous equation for the ideal membership (6.1.1), the coefficient of x p + ki on theright has degree ( k , . . . , k , − p , k , . . . , k ) , and therefore the coefficient must be a multiple of x k · · · x ki − x ki + · · · x km x pm + i , i.e., (cid:16) p m ∑ i = x pi x pm + i (cid:17) ( x · · · x m ) k ∈ (cid:0) x k · · · x km x pm + x p + k , . . . , x k · · · x km − x p m x p + km (cid:1) R / pR . Canceling the term ( x · · · x m ) k , the above display implies that(6.1.2) 1 p m ∑ i = x pi x pm + i ∈ (cid:0) ( x x m + ) p , . . . , ( x m x m ) p (cid:1) in the ring ( R / pR ) ( ,..., ) = Z / p Z [ x x m + , . . . , x m x m ] / ( m ∑ i = x i x m + i ) . But ( R / pR ) ( ,..., ) may be identified with the polynomial ring Z / p Z [ z , . . . , z m − ] , where z i : = x i x m + i for 1 i m − , in which case, (6.1.2) reads z p + · · · + z pm − + ( − ) p ( z + · · · + z m − ) p p ∈ ( z p , . . . , z pm − ) . This is readily seen to be false, for example by examining the coefficient of z p − z . (cid:3) We now turn to Pfaffian hypersurfaces. Let n be an even integer with n >
4, and let Z be an ( n − ) × n matrix of indeterminates over an infinite field K . Set T : = K [ Z ] . Thesymplectic group Sp n − ( K ) acts K -linearly on T by the rule M : Z MZ for M ∈ Sp n − ( K ) . By [DCP, § K -algebra generated by the entries ofthe product matrix Z tr Ω Z , where Ω is the size n − (cid:18) I − I (cid:19) , with I the identity. This invariant ring is isomorphic to K [ X ] / ( pf X ) for X an n × n alternat-ing matrix of indeterminates, and pf X its Pfaffian. When K has characteristic zero, the ringof differential operators on the invariant ring is described explicitly in [LS, IV 1.9, Case C].For M an alternating matrix, the cofactor expansion for Pfaffians takes the formpf M = ∑ j > ( − ) j m j pf M j , where M j is the submatrix obtained by deleting the first and j -th rows and columns. For t an even integer, we use Pf t ( M ) to denote the ideal generated by the Pfaffians of the t × t diagonal submatrices of M . It follows from the cofactor expansion that Pf t ( M ) ⊆ Pf t − ( M ) .The Pfaffian of a 4 × X is the quadratic form x x − x x + x x , which, aside from the change of notation, coincides with the case m = Theorem 6.2.
Let X be an n × n alternating matrix of indeterminates over Z , where n isan even integer with n > . Set S : = Z [ X ] and R : = S / ( pf X ) . Fix a prime integer p > ,and let ϕ p be the standard p-derivation on S with respect to the indeterminates X. Then (cid:20) ϕ p ( pf X )( x · · · x n ) p (cid:21) ∈ H n − ( x ,..., x n ) ( R ) is a nonzero p-torsion element; moreover, its image in H n − ( x ,..., x n ) ( R / pR ) is nonzero.In particular, the local cohomology module H n − ( x ,..., x n ) ( R ) contains a nonzero p-torsionelement for each prime integer p > . IFFERENTIAL OPERATORS ON INVARIANT RINGS 25
Proof.
As noted above, the case n = n , with n = H n − ( x ,..., x n ) ( R / pR ) if and only if thereexists an integer k >
0, such that in the polynomial ring S one has(6.2.1) ϕ p ( pf X )( x · · · x n ) k ∈ (cid:0) p , pf X , x p + k , . . . , x p + k n (cid:1) S . We know this cannot happen for n =
4. Suppose (6.2.1) holds for an even integer n > x n − , n
1, and x i j i = , . . . , n − j = n − , n , in which case,the image of X is x x x . . . x , n − x , n − x , n − x n − x x x . . . x , n − x , n − − x − x x . . . x , n − x , n − − x − x x . . . x , n − x , n − − x , n − − x , n − − x , n − − x , n − . . . x n − , n − − x , n − − x , n − − x , n − − x , n − . . . − x n − , n − − x , n − . . . − x n . . . − . The upper left ( n − ) × ( n − ) submatrix is unchanged; denote this by X ′ . Elementaryrow and column operations transform the matrix displayed above to (cid:18) X ′ Ω (cid:19) , where Ω : = (cid:18) − (cid:19) , which shows that the Pfaffian of the image of X after specialization equals pf X ′ . Hencethe ideal membership (6.2.1) specializes to ϕ p ( pf X ′ )( x · · · x n ) k ∈ (cid:0) p , pf X ′ , x p + k , . . . , x p + k n (cid:1) S ′ , with S ′ denoting the image of S under the specialization. The indeterminates x , n − and x n do not occur in the polynomial pf X ′ , and hence form a regular sequence on S ′ / (cid:0) p , pf X ′ , x p + k , . . . , x p + k , n − (cid:1) S ′ . Using this, we obtain ϕ p ( pf X ′ )( x · · · x , n − ) k ∈ (cid:0) p , pf X ′ , x p + k , . . . , x p + k , n − , x p , n − , x p n (cid:1) S ′ . Next specialize x , n − x n
0, so as to obtain ϕ p ( pf X ′ )( x · · · x , n − ) k ∈ (cid:0) p , pf X ′ , x p + k , . . . , x p + k , n − (cid:1) S ′′ , where S ′′ is the image of S ′ under the specialization, equivalently, the polynomial ringover Z in the indeterminates occurring in X ′ . But this violates the inductive hypothesis. (cid:3) We next show that Pfaffian hypersurfaces do not admit a lift of Frobenius modulo p : Theorem 6.3.
Let X be an n × n alternating matrix of indeterminates over Z , where n isan even integer with n > . Set S : = Z [ X ] and R : = S / ( pf X ) . Fix a prime integer p > .Then the Frobenius endomorphism on R / pR does not lift to an endomorphism of R / p R. Proof.
We proceed by induction on even integers n >
4, along the same lines as in theproof of Theorem 6.2. The case n = R defined by x x − x x + x x . While the case of diagonal quadratic forms of rank at least 5, in odd characteristic, iscovered by [Zd, Theorem 4.15], we give a different argument to avoid characteristic re-strictions. Suppose that the Frobenius endomorphism on R / pR lifts to R / p R . Then, byProposition 2.4, one has(6.3.1) 1 p (cid:0) x p x p − x p x p + x p x p (cid:1) ∈ (cid:0) x p , x p , x p , x p , x p , x p (cid:1) R / pR . Using the grading deg x = e , deg x = − e , deg x = e , deg x = − e , deg x = e , deg x = − e , as in the proof of Theorem 6.1, we may work in the subring ( R / pR ) ( , , ) = Z / p Z [ x x , x x , x x ] / ( x x − x x + x x ) , which, in turn, may be identified with the polynomial ring Z / p Z [ z , z ] , where z : = x x and z : = x x . But then (6.3.1) reads 1 p (cid:0) z p − z p + ( z − z ) p (cid:1) ∈ (cid:0) z p , z p (cid:1) R / pR , which is seen to be false by examining the coefficient of z p − z .For the inductive step, let R : = S / ( pf X ) be the hypersurface defined by the Pfaffianof an n × n alternating matrix X , for an even integer n >
6. Suppose that the Frobeniusendomorphism on R / pR lifts to R / p R . Then, by Proposition 2.4, ϕ p ( pf X ) ∈ (cid:18)(cid:16) ∂ pf X ∂ x i j (cid:17) p : 1 i < j n (cid:19) S / ( p , pf X ) S . Each partial derivative ∂ pf X / ∂ x i j is, up to sign, the Pfaffian of the ( n − ) × ( n − ) sub-matrix of X obtained by deleting rows i , j and columns i , j . Using the notation a [ p ] : = ( a p | a ∈ a ) for ideals a in a ring of prime characteristic p >
0, the preceding ideal membership mayhence be written as ϕ p ( pf X ) ∈ Pf n − ( X ) [ p ] S / ( p , pf X ) S . Applying the specialization in the proof of Theorem 6.2, X may be replaced by X ′′ : = (cid:18) X ′ Ω (cid:19) , where Ω : = (cid:18) − (cid:19) , implying that ϕ p ( pf X ′ ) ∈ Pf n − ( X ′′ ) [ p ] S ′ / ( p , pf X ′ ) S ′ . But Pf n − ( X ′′ ) = Pf n − ( X ′ ) = (cid:18)(cid:16) ∂ pf X ′ ∂ x i j (cid:17) : 1 i < j n − (cid:19) , contradicting the inductive hypothesis. (cid:3) IFFERENTIAL OPERATORS ON INVARIANT RINGS 27
7. D
ETERMINANTAL HYPERSURFACES
Let K be an infinite field. Let Y and Z be n × ( n − ) and ( n − ) × n matrices ofindeterminates respectively, and set T : = K [ Y , Z ] . The general linear group GL n − ( K ) acts K -linearly on T where, for M ∈ GL n − ( K ) , one has M : ( Y Y M − Z MZ . By [DCP, § K by the entries of theproduct matrix Y Z . This invariant ring is isomorphic to K [ X ] / ( det X ) , where X is an n × n matrix of indeterminates. When K has characteristic zero, the ring of differential operatorson the invariant ring is described explicitly in [LS, IV 1.9, Case A].We complete the proof of Theorem 1.2 (a). In view of Theorem 3.15 and Lemma 4.1, itsuffices to prove: Theorem 7.1.
Let X be an n × n matrix of indeterminates over Z , where n > . Fix aprime integer p > , and let ϕ p be the standard p-derivation on S : = Z [ X ] with respectto X. Take a to be the ideal ( x , . . . , x n , x , . . . , x n ) R, where R : = S / ( det X ) . Then (cid:20) ϕ p ( det X )( x · · · x n x · · · x n ) p (cid:21) ∈ H n − a ( R ) is a nonzero p-torsion element; moreover, its image in H n − a ( R / pR ) is nonzero.In particular, the local cohomology module H n − a ( R ) contains a nonzero p-torsionelement for each prime integer p > .Proof. We first show that the proof of the theorem reduces to the case n =
3. Fix p .Suppose that the image of the displayed element in H n − a ( R / pR ) is zero. Then thereexists an integer k > S one has(7.1.1) ϕ p ( det X )( x · · · x n x · · · x n ) k ∈ (cid:0) p , det X , x p + k , . . . , x p + k n , x p + k , . . . , x p + k n ) S . For i = , . . . , n , specialize x ii x i j j = i . The image of X after special-ization has the form (cid:18) X ′ ∗ I (cid:19) , where X ′ denotes the upper left 3 × X , and I is the size n − X specializes to det X ′ , and ϕ p ( det X ) to ϕ p ( det X ′ ) . Hence the idealmembership (7.1.1) specializes to ϕ p ( det X ′ )( x · · · x n x · · · x n ) k ∈ (cid:0) p , det X ′ , x p + k , . . . , x p + k n , x p + k , . . . , x p + k n (cid:1) S ′ , with S ′ the image of S . Since the indeterminates x , . . . , x n , x , . . . , x n do not occur inthe polynomials det X ′ and ϕ p ( det X ′ ) , the ideal membership above implies ϕ p ( det X ′ )( x x x x ) k ∈ (cid:0) p , det X ′ , x p + k , x p + k , x p + k , x p + k (cid:1) S ′ . Specializing x i j i > j >
4, we reduce to the case n =
3; specifically,it suffices to consider X : = x x x x x x x x x and S = Z [ X ] , in which case (7.1.1) reads(7.1.2) ϕ p ( det X )( x x x x ) k ∈ (cid:0) p , det X , x p + k , x p + k , x p + k , x p + k (cid:1) S . Specialize x
0. Using X ′′ for the image of X , one hasdet X ′′ = − ( x x ) x + ( x x ) x + ( x x ) x − ( x x ) x . In the hypersurface R ′ : = Z [ x , x , x , x , x , x , x , x ] / ( det X ′′ ) , set λ : = p (cid:0) − x p x p x p + x p x p x p + x p x p x p − x p x p x p (cid:1) . Then (7.1.2) implies that λ ( x x x x ) k ∈ (cid:0) x p + k , x p + k , x p + k , x p + k (cid:1) R ′ / pR ′ . Take the Z -grading on R ′ / pR ′ defined bydeg x = (0, 0, 0, 1, 0)deg x = (1, 0, −
1, 0, − x = (1, −
1, 0, 0, − x = (0, 0, 0, 0, 1)deg x = (1, 0, − −
1, 0)deg x = (1, −
1, 0, −
1, 0)deg x = (0, 1, 0, 0, 0)deg x = (0, 0, 1, 0, 0)and note that λ is homogeneous of degree ( p , , , , ) . Hencedeg λ ( x x x x ) k = ( p + k , − k , − k , − k , − k ) . Fix a homogeneous equation λ ( x x x x ) k = α x p + k + β x p + k + γ x p + k + δ x p + k with α , β , γ , δ in R ′ / pR ′ . The element α has degree ( k , − k , p − k , − k , p − k ) . Let µ be a monomial of the above degree. Examining the second and fourth componentsof the degree, the exponents on x and x in µ add up to at least 2 k , as do the exponentson x and x . Bear in mind the first component, µ must be a multiple of x k . A similaranalysis for β , γ , and δ shows that λ ( x x x x ) k ∈ (cid:0) x k x p + k , x k x p + k , x k x p + k , x k x p + k (cid:1) R ′ / pR ′ . Next, specialize x and x to 1 and x and x to −
1, in which case one has(7.1.3) λ ′ ( x x x x ) k ∈ (cid:0) x k x p + k , x k x p + k , x k x p + k , x k x p + k (cid:1) B , where λ ′ denote the image of λ under the specialization, and B is the image of R ′ / pR ′ , i.e., B : = Z / p Z [ x , x , x , x ] / ( x + x + x + x ) , IFFERENTIAL OPERATORS ON INVARIANT RINGS 29 which we identify with the polynomial ring Z / p Z [ x , x , x ] . With this identification, λ ′ = p (cid:0) x p + x p + x p + x p (cid:1) = p (cid:0) ( − x − x − x ) p + x p + x p + x p (cid:1) = ± x p − x + · · · is a polynomial in Z / p Z [ x , x , x ] in which each indeterminate occurs with exponentsstrictly less than p . The ring B is a free module over its subring B p = Z / p Z [ x p , x p , x p ] , with a basis given by monomials in x , x , x , with each exponent less than p . Let π : B −→ B p be the B p -linear map that sends x p − x to 1 and other basis elements to 0. Specifically, π ( λ ′ ) = ± . Consider (7.1.3) where, without loss of generality, the exponent k is taken to be a powerof p , and apply π . Then, in the ring B p , one has(7.1.4) ( x x x x ) k ∈ (cid:0) x k x p + k , x k x p + k , x k x p + k , x k x p + k (cid:1) , where we retain the notation x p = − ( x p + x p + x p ) for the sake of symmetry; note that B p may be regarded as a polynomial ring in any three of the elements x p , x p , x p , x p . The ideal membership (7.1.4) implies the existence of elements a , b , c , d in B p with ( x x x x ) k = ax k x p + k + bx k x p + k + cx k x p + k + dx k x p + k . Rearranging terms, one has x k (( x x x ) k − ax k x p − dx p + k ) ∈ ( x k x p + k , x k x p + k ) . But x k is a nonzerodivisor in the ring B p / ( x k x p + k , x k x p + k ) , so the above implies that ( x x x ) k ∈ ( x k x p , x k x p + k , x k x p + k , x p + k ) . Similarly, using that x k is a nonzerodivisor modulo in B p / ( x k x p , x p + k ) , one obtains ( x x ) k ∈ ( x k x p , x k x p , x p + k , x p + k ) . Continuing, x k is a nonzerodivisor in B p / ( x k x p , x p + k ) , yielding x k ∈ ( x k x p , x p , x p , x p + k ) , and finally, with x k being a nonzerodivisor in B p / ( x p , x p ) , one obtains the contradiction1 ∈ ( x p , x p , x p , x p ) . (cid:3) Determinantal hypersurfaces, in general, do not admit a lift of Frobenius modulo p : Theorem 7.2.
Let X be an n × n matrix of indeterminates over Z , where n > . Fix aprime integer p > . Set S : = Z [ X ] and R : = S / ( det X ) . Then the Frobenius endomorphismon R / pR does not lift to an endomorphism of R / p R. Proof.
As in the proof of Theorem 7.1, one first reduces to the case n = R / pR lifts to R / p R . Then, by Proposition 2.4, ϕ p ( det X ) ∈ (cid:18)(cid:16) ∂ det X ∂ x i j (cid:17) p : 1 i , j n (cid:19) S / ( p , det X ) S . Each partial derivative above is, up to sign, the determinant of an ( n − ) × ( n − ) subma-trix of X , so the above may be restated as(7.2.1) ϕ p ( det X ) ∈ I n − ( X ) [ p ] S / ( p , det X ) S , where I n − ( X ) denotes the ideal generated by the size n − X . Applying thespecialization S −→ S ′ in the proof of Theorem 7.1, one has X (cid:18) X ′ ∗ I (cid:19) , with I denoting the identity matrix of size n −
3. Hence I n − ( X ) S ′ = I ( X ′ ) , and the idealmembership (7.2.1) specializes to ϕ p ( det X ′ ) ∈ I ( X ′ ) [ p ] S ′ / ( p , det X ′ ) S ′ , which is essentially the n = n =
3. Specializing x and x to 0, the resulting matrix X ′′ : = x x x x x x x has determinant x x x − x x x − x x x , and the ideal membership implies(7.2.2) ϕ p ( det X ′′ ) ∈ I ( X ′′ ) [ p ] in the ring Z / p Z [ x , x , x , x , x , x , x ] / ( det X ′′ ) . Using the Z -grading in the proof of Theorem 7.1, det X ′′ has degree ( p , , , , ) , so weobtain an ideal membership in the subring generated by elements of degree ( ∗ , , , , ) ,namely the ring Z / p Z [ x x x , x x x , x x x ] / ( x x x − x x x − x x x ) ∼ = Z / p Z [ x x x , x x x ] . Working with the degree of each generator of I ( X ′′ ) , the statement (7.2.2) gives1 p (cid:0) ( x x x + x x x ) p − ( x x x ) p − ( x x x ) p (cid:1) ∈ (cid:0) x x x , x x x (cid:1) [ p ] in the ring above, which is readily seen to be false. (cid:3)
8. S
YMMETRIC DETERMINANTAL HYPERSURFACES
Let Z be an ( n − ) × n matrix of indeterminates over an infinite field K . Set T : = K [ Z ] .The orthogonal group O n − ( K ) acts K -linearly on T by the rule M : Z MZ for M ∈ O n − ( K ) . By [DCP, § K -algebra generated by the entriesof the product matrix Z tr Z , and is isomorphic to K [ X ] / ( det X ) for X an n × n symmetric IFFERENTIAL OPERATORS ON INVARIANT RINGS 31 matrix of indeterminates. When K has characteristic zero, the ring of differential operatorson this invariant ring is described explicitly in [LS, IV 1.9 Case B]. Theorem 8.1.
Let X be a symmetric × matrix of indeterminates over Z , and set R tobe the hypersurface Z [ X ] / ( det X ) . Then, for each odd prime integer p, the mapD R −→ D R / pR is surjective. The map displayed above is not surjective when p =
2, see Theorem 8.4.
Proof.
In view of Proposition 3.13, it suffices to show that each odd prime p acts injectivelyon the local cohomology module H ∆ R ( P R ) . This is unaffected by inverting the integer 2, sowe work instead with the rings Z = Z [ / ] and R = Z [ X ] / ( det X ) . Let T : = Z [ u , v , w , x , y , z ] be a polynomial ring, and T = T [ / ] . It is readily checked thatthe symmetric matrix ux uy + vx uz + wxuy + vx vy vz + wyuz + wx vz + wy wz has determinant 0. By a dimension argument, it follows that R is isomorphic to the subring Z [ ux , vy , wz , uy + vx , uz + wx , vz + wy ] . of T and, indeed, we identify R with this subring.We claim that R is a direct summand of T as an R -module. To see this, first con-sider the Z -grading on T where the indeterminates u , v , w have degree 1, and x , y , z havedegree −
1. It follows that the degree 0 component of T , i.e., the ring B : = Z [ ux , uy , uz , vx , vy , vz , wx , wy , wz ] is a direct summand of T as a B -module. Next, let G : = h σ i be a group of order 2 actingon T , where σ is the involution with u x , v y , w z . The action of G on T restricts to an action on the subring B , and R ⊆ B G . We claim that equality holds in the above display. The equation ( uy ) − ( uy )( uy + vx ) + ( ux )( vy ) = uy is integral over R ; similarly one sees that B is integral over R . Moreover, itis readily checked that at the level of fraction fields one hasfrac ( R )( uy ) = frac ( B ) , so that [ frac ( B ) : frac ( R )]
2. Hencefrac ( R ) = frac (cid:0) B G (cid:1) . Each element of B G is integral over R , and belongs to the fraction field of R ; but R isnormal, so we conclude that R = B G .Since 2 is a unit in R , the Reynolds operator B −→ R shows that R is a direct sum-mand of B as an R -module. It follows that R is a direct summand of T as an R -module.At this stage, one may invoke [Je, Theorem 6.3] to conclude that the map D R −→ D R / pR is surjective for almost all integer primes p ; we shall, however, go further and prove thatthe map is surjective for each odd prime integer p . Fix such a prime p ; it suffices to provethat p acts injectively on the local cohomology module H ∆ ( R ⊗ Z R ) , where ∆ is the diagonal ideal in R ⊗ Z R . Since R is a direct summand of T as an R -module, it follows that R ⊗ Z R is a direct summand of T ⊗ Z T as an R ⊗ Z R -module.It suffices to show that p acts injectively on H ∆ ( C ) , where C : = T ⊗ Z T is identified with Z [ u , v , w , x , y , z , u ′ , v ′ , w ′ , x ′ , y ′ , z ′ ] and ∆ is the ideal of C generated by the elements ux − u ′ x ′ , vy − v ′ y ′ , wz − w ′ z ′ , uy + vx − u ′ y ′ − v ′ x ′ , uz + wx − u ′ z ′ − w ′ x ′ , vz + wy − v ′ z ′ − w ′ y ′ . Consider the ideals of C as below: p : = I (cid:18) u v w x ′ y ′ z ′ u ′ v ′ w ′ x y z (cid:19) and q : = I (cid:18) u v w u ′ v ′ w ′ x ′ y ′ z ′ x y z (cid:19) . Since p and q are generated by minors of matrices of indeterminates, each is prime; more-over, p and q each contain ∆ . It is a straightforward—albeit slightly tedious—verificationthat p q ⊆ ∆ . It then follows that p ∩ q = rad ∆ . Since H k p ( C ) = = H k q ( C ) for integers k other than 5 and 9, the Mayer-Vietoris sequence −−→ H p ( C ) ⊕ H q ( C ) −−→ H ∆ ( C ) −−→ H p + q ( C ) −−→ H p ( C ) ⊕ H q ( C ) −−→ gives an isomorphism H ∆ ( C ) ∼ = H p + q ( C ) . It now suffices to check that p acts injectivelyon H p + q ( C ) . We claim that the ideal ( p + q ) C / pC has height 7. To see this, note that thereis an isomorphism C / ( p + q + pC ) −→ Z / p Z [ x , x ] Z / p Z [ y , y ] Z / p Z [ z , z , z ] , with u x y z , v x y z , w x y z , u ′ x y z , v ′ x y z , w ′ x y z , x x y z , y x y z , z x y z , x ′ x y z , y ′ x y z , z ′ x y z . It follows that H p + q ( C / pC ) =
0, which gives the desired injectivity using the exactness of −−−→ H p + q ( C / pC ) −−−→ H p + q ( C ) p −−−→ H p + q ( C ) −−−→ H p + q ( C / pC ) . (cid:3) Remark 8.2.
Let R be the hypersurface Z [ X ] / ( det X ) , where X is a symmetric 3 × R may be identified with Z [ Z tr Z ] , for Z a 2 × T : = Z [ Z ] . Using arguments from the previous proof, we show that R is adirect summand of T as an R -module. IFFERENTIAL OPERATORS ON INVARIANT RINGS 33
Take i = √− C . It suffices to prove that R [ i ] is a direct summand of T [ i ] asan R [ i ] -module: indeed, if ρ : T [ i ] −→ R [ i ] is a splitting of R [ i ] ֒ −→ T [ i ] , then12 ( ρ + ρ ) : T −→ R , with ρ denoting the complex conjugate, is a splitting of R ֒ −→ T . In the ring T [ i ] , set u = z + iz , x = z − iz , v = z + iz , y = z − iz , w = z + iz , z = z − iz , so that T [ i ] = Z [ i ][ u , v , w , x , y , z ] . But R [ i ] is the Z [ i ] -algebra generated by the entries of Z tr Z = z + z z z + z z z z + z z z z + z z z + z z z + z z z z + z z z z + z z z + z = ux uy + vx uz + wxuy + vx vy vz + wyuz + wx vz + wy wz . The proof of Theorem 8.1 shows that R [ i ] is a direct summand of T [ i ] as an R [ i ] -module.To study the case of symmetric matrices in characteristic 2, we record the followingvariant of Lemma 4.1: Lemma 8.3.
Let S : = Z [ xxx ] be a polynomial ring in the indeterminates xxx : = x , . . . , x d . Fixa prime integer p > , and let f ( xxx ) ∈ S be a polynomial of the formf ( xxx ) = g ( xxx ) + ph ( xxx ) , where g ( xxx ) ∈ ( x , . . . , x m ) S for some fixed integer m < d, and h ( xxx ) ∈ S is a polynomialin the indeterminates x m + , . . . , x d . Set R : = S / ( f ( xxx )) , and let ϕ p denote the p-derivationof P S as in Theorem 3.15. Then, if the local cohomology element " ϕ p (cid:0) g ( xxx ) (cid:1) ( x · · · x m ) p ∈ H m + ( x ,..., x m ) ( R / pR ) is nonzero, so is the element " ϕ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) ∏ di = ( y i − x i ) p ∈ H d + ∆ R ( P R / pP R ) . Proof.
Suppose the displayed element of H d + ∆ R ( P R / pP R ) is zero. Then there exists aninteger k > Λ p (cid:0) f ( yyy ) − f ( xxx ) (cid:1) − (cid:0) f ( yyy ) − f ( xxx ) (cid:1) p p ( y − x ) k · · · ( y d − x d ) k ∈ (cid:0) p , f ( yyy ) , f ( xxx ) , ( y − x ) p + k , . . . , ( y d − x d ) p + k (cid:1) P S . Specialize x i i m , in which case f ( xxx ) specializes to ph ( xxx ) , so Λ p (cid:0) f ( yyy ) − ph ( xxx ) (cid:1) − (cid:0) f ( yyy ) − ph ( xxx ) (cid:1) p p y k · · · y km ( y m + − x m + ) k · · · ( y d − x d ) k ∈ (cid:0) p , f ( yyy ) , y p + k , . . . , y p + km , ( y m + − x m + ) p + k , . . . , ( y d − x d ) p + k (cid:1) . Since y m + − x m + , . . . , y d − x d are algebraically independent over Z [ yyy ] / ( p , f ( yyy ) , y p + k , . . . , y p + km ) , it follows that Λ p (cid:0) f ( yyy ) − ph ( xxx ) (cid:1) − (cid:0) f ( yyy ) − ph ( xxx ) (cid:1) p p y k · · · y km ∈ (cid:0) p , f ( yyy ) , y p + k , . . . , y p + km , ( y m + − x m + ) p , . . . , ( y d − x d ) p (cid:1) . Next, specialize x i y i for m + i d . Then ph ( xxx ) specializes to ph ( yyy ) , giving Λ p (cid:0) g ( yyy ) (cid:1) − (cid:0) g ( yyy ) (cid:1) p p y k · · · y km ∈ (cid:0) p , f ( yyy ) , y p + k , . . . , y p + km (cid:1) Z [ yyy ] , where Λ p is the standard lift of Frobenius on Z [ yyy ] with respect to yyy . Renaming y i x i for each i , it follows that " ϕ p (cid:0) g ( xxx ) (cid:1) ( x · · · x m ) p ∈ H m + ( x ,..., x m ) ( R / pR ) is zero. (cid:3) Theorem 8.4.
Let X be a symmetric × matrix of indeterminates over Z , and set R tobe the hypersurface Z [ X ] / ( det X ) . Then the Frobenius trace on R / R does not lift to adifferential operator on R / R.Proof.
Set S : = Z [ X ] . Modulo the ideal ( , x , x , x ) , the image of X is a 3 × X ∈ ( , x , x , x ) S . Indeed, det X = g ( xxx ) + h ( xxx ) , where g ( xxx ) : = x x x − x x − x x − x x and h ( xxx ) : = x x x . In light of Lemma 8.3, it suffices to check that " ϕ (cid:0) g ( xxx ) (cid:1) ( x x x ) ∈ H ( x , x , x ) ( R / R ) is nonzero. We shall go a step further and prove that while det X ≡ g ( xxx ) mod 2, " ϕ (cid:0) det X (cid:1) ( x x x ) = = " ϕ (cid:0) g ( xxx ) (cid:1) ( x x x ) . It is a straightforward calculation that modulo the ideal ( x , x , x ) R / R , one has ϕ (cid:0) det X (cid:1) ≡ x x x x + x x x and ϕ (cid:0) g ( xxx ) (cid:1) ≡ x x x x . In the ring R / R , we set a : = x x , b : = x x , c : = x x , d : = x x x , e : = x x x . Note that a + b + c = d and abc = de in R / R . Working modulo ( x , x , x ) R / R , ϕ (cid:0) det X (cid:1) x x x ≡ ( ab + e ) d ≡ ab ( d + c ) ≡ ab ( a + b ) ≡ a + b + ( a + b ) . Since a ∈ ( x ) R / R , and b ∈ ( x ) R / R , and a + b = c + d ∈ ( x ) R / R , it followsthat ϕ (cid:0) det X (cid:1) x x x ∈ ( x , x , x ) R / R . This proves the first assertion. IFFERENTIAL OPERATORS ON INVARIANT RINGS 35
Next, suppose (cid:20) ϕ (cid:0) g ( xxx ) (cid:1) ( x x x ) (cid:21) =
0. Then there exists an integer k > x x x x ( x x x ) k ∈ (cid:0) x + k , x + k , x + k (cid:1) R / R . Consider the Z -grading on R / R defined bydeg x = ( , ) , deg x = ( − , ) , deg x = ( , ) , deg x = ( , − ) , deg x = ( − , − ) , deg x = ( , ) . The element on the left in (8.4.1) has degree ( , ) , so we work in the subring [ R / R ] ( , ) ,which is the Z / Z -algebra generated by x x , x x , x x , x x x , x x x . Using c = d − a − b , this subring may be identified with B : = Z / Z [ a , b , d , e ] / (cid:0) ab ( d − a − b ) − de (cid:1) . In the hypersurface B , (8.4.1) implies that abd k ∈ ( a , d ) + k + ( b , d ) + k + ( c , d ) + k ⊆ ( a , b , d k + ) . But the image of abd k is nonzero in B / ( a , b , d k + , ab − e ) = Z / Z [ a , b , d , e ] / ( a , b , d k + , ab − e ) , which is a contradiction. (cid:3) Lastly, we examine the existence of Frobenius lifts for hypersurfaces defined by deter-minants of symmetric matrices of indeterminates:
Theorem 8.5.
Let X be an n × n symmetric matrix of indeterminates over Z . Set S : = Z [ X ] and R : = S / ( det X ) . (a) If n = and p is an odd prime integer, then the Frobenius endomorphism on R / pRlifts to an endomorphism of R / p R. (b) If n > , then the Frobenius endomorphism on R / R does not lift to an endomorphismof R / R. (c) For n > , and p an odd prime integer, the Frobenius endomorphism on R / pR doesnot lift to an endomorphism of R / p R.Proof. (a) In the case n =
3, the ring R [ / ] is a direct summand, as an R [ / ] -module, ofa polynomial ring over Z [ / ] , see Remark 8.2. But then, for p odd, the ring R / p R is adirect summand, as an R / p R -module, of a polynomial ring over Z / p Z . The existence ofa Frobenius lift now follows using [Zd, Lemma 4.1].In the remaining cases, in view of Proposition 2.4, we need to verify that(8.5.1) ϕ p ( det X ) / ∈ (cid:18) p , det X , (cid:16) ∂ det X ∂ x i j (cid:17) p : 1 i j n (cid:19) S . A partial derivative of the form ∂ det X / ∂ x ii is the determinant of a size n − X , whereas, for i < j , the partial derivative ∂ det X / ∂ x i j is, aside from a sign,twice the determinant of a size n − p equals 2, and where p is an odd prime. For (b), suppose (8.5.1) fails for some n >
3. In view of the above paragraph, one has(8.5.2) ϕ ( det X ) ∈ (cid:18) , det X , (cid:16) ∂ det X ∂ x ii (cid:17) p : 1 i n (cid:19) S . Specialize the symmetric matrix X as X (cid:18) X ′ I (cid:19) , where X ′ : = x x x x x x , and I is the size n − X specializes to det X ′ = x x x ,so ϕ ( det X ) specializes to ϕ ( det X ′ ) = (cid:16) x x x − ( x x x ) (cid:17) = − x x x . With S ′ denoting the image of S , the ideal membership (8.5.2) implies that x x x ∈ (cid:16) x , x , x (cid:17) [ ] S ′ / S ′ , which is a contradiction.For (c), suppose (8.5.1) fails for some n >
4, and p odd. Then(8.5.3) ϕ p ( det X ) ∈ I n − ( X ) [ p ] S / ( p , det X ) S , where I n − ( X ) denotes the ideal generated by the size n − X . Specialize X (cid:18) X ′ I (cid:19) , where X ′ : = x x x x x x x x x x x x , and I is the size n − I n − ( X ) S ′ = I ( X ′ ) , with S ′ denoting the imageof S , and (8.5.3) specializes to(8.5.4) ϕ p ( det X ′ ) ∈ I ( X ′ ) [ p ] S ′ / ( p , det X ′ ) S ′ . Consider the Z -grading on S ′ / ( p , det X ′ ) S ′ defined bydeg x = e + e , deg x = e − e , deg x = e + e , deg x = e − e , deg x = e + e , deg x = e − e , under which det X ′ has degree ( , , , ) , and ϕ p ( det X ′ ) has degree ( , , , p ) . Let ∆ i j denote the determinant of the submatrix of X ′ obtained by deleting the i -th row and j -thcolumn. Then deg ∆ i j = e − deg x i j for i < j , whereas deg ∆ = ( − , − , , ) , deg ∆ = ( , − , − , ) deg ∆ = ( − , , − , ) , deg ∆ = ( , , , ) . Hence a homogeneous equation for (8.5.4) forces ϕ p ( det X ′ ) ∈ (cid:0) x pi j ∆ pi j : 1 i < j (cid:1) S ′ / ( p , det X ′ ) S ′ . It is readily checked that x ∆ = x ∆ , x ∆ = x ∆ , x ∆ = x ∆ , IFFERENTIAL OPERATORS ON INVARIANT RINGS 37 so(8.5.5) ϕ p ( det X ′ ) ∈ (cid:0) x p ∆ p , x p ∆ p ) S ′ / ( p , det X ′ ) S ′ . Since the elements in question have degree of the form ( , , , ∗ ) , this ideal membershipholds in the subring [ S ′ / ( p , det X ′ ) S ′ ] ( , , , ∗ ) , which may be identified with B : = Z / p Z [ a , b , c ] / ( a + b + c − ab − ac − bc ) , where a : = x x , b : = x x , and c : = x x . In this ring, (8.5.5) implies that1 p (cid:0) a p + b p + c p − a p b p − a p c p − b p c p (cid:1) ∈ (cid:0) a − ab − ac , b − ab − bc (cid:1) [ p ] B . Enlarge the ring B by adjoining u : = √ a and v : = √ b , in which case the defining equationof B factors as (cid:0) c − ( u + v ) (cid:1)(cid:0) c − ( u − v ) (cid:1) We work modulo the first factor c − ( u + v ) , i.e., in the polynomial ring Z / p Z [ u , v ] , where c is identified with ( u + v ) . The ideal membership then implies that1 p (cid:16) u p + v p + ( u + v ) p − u p v p − u p ( u + v ) p − v p ( u + v ) p (cid:17) is a linear combination of ( u v + u v ) p and ( u v + uv ) p with coefficients from Z / p Z .This is not possible, for example by examining the coefficient of u p − v p + ; the interestedreader—if one remains—may verify that this coefficient is 8. (cid:3) A CKNOWLEDGMENTS
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