For a polynomial ring R over a field K of characteristic zero, the ring of K-linear differential 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⟨ ∂ ∂x 0 , . . . , ∂ ∂x d ⟩. This noncommutative ring is the wellknown Weyl algebra, 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 to Grothendieck [9, §16.8], of the ring of 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 a field K of characteristic zero, and R to be K[x, y, z]/(x 3 + y 3 + z 3 ), then D R|K is not left or right Noetherian, nor a finitely generated K-algebra, nor a simple ring; see [2].
On the other hand, when R is the ring of invariants for a linear action of a reductive group on a polynomial ring over a field K of characteristic zero, it is known in many cases that D R|K is Noetherian, finitely generated, and a simple ring, just as in the polynomial case [16,18,22,26]. Indeed, it is conjectured that for such invariant rings R, the ring of differential operators D R|K is a simple ring [25]. An analogous statement in positive characteristic was proved by Smith and Van den Bergh [30,Theorem 1.3].
Quite generally, for A-algebras R and B, there is a ring homomorphism
Suppose A is a Noetherian ring, R is a finitely generated A-algebra, and B is flat over A; then the map above is an isomorphism. In particular, when R is a finitely generated Z-algebra, one has an isomorphism
and, for p a prime integer that is a nonzerodivisor on R, an injective homomorphism
(1.0.1)
In order to relate rings of differential operators in characteristic zero to their counterparts in positive characteristic p, one needs to determine whether the map (1.0.1) is an isomorphism, i.e., whether each differential operator on R/pR lifts to a differential operator on R. To study the problem of the simplicity of rings of differential operators on characteristic zero invariant rings via reduction to positive characteristic, Smith and Van den Bergh pose the following question, formulated here in equivalent terms: Question 1.1 ( [30,Question 5.1.2]). Let A be a domain that is finitely generated as an algebra over Z. Suppose R is a finitely generated A-algebra such that R ⊗ A frac(A) is the ring of invariants for a linear action of a reductive group on a polynomial ring of characteristic zero. Does there exist a nonempty open subset of U of Spec A, such that for each maximal ideal µ ∈ U , each differential operator on D (R/µR) | (A/µA) lifts to D R|A ?
We prove that the answer to Question 1.1 is negative: for several classical invariant rings that are hypersurfaces, we construct explicit differential operators, modulo prime integers p, that do not lift to characteristic zero differential operators. Our main theorem is below; we refer the reader to §3.4 for the definition of Frobenius trace.
(a) Let X be an n × n matrix of indeterminates over Z, with n 3. Set R := Z[X]/(det X). Then, for each prime integer p > 0, the Frobenius trace on R/pR does not lift to a differential operator on R/p 2 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 4 an even integer. Set R := Z[X]/(pf X), where pf X denotes the Pfaffian of X. Then, for each prime integer p > 0, the Frobenius trace on R/pR does not lift to a differential operator on R/p 2 R, nor, a fortiori, to a differential operator on R. (c) Let X be a 3 ×3 symmetric matrix of indeterminates over Z. Set R := Z[X]/(det X).
Then, for prime integers p > 2, each differential operator on R/pR lifts to a differential operator on R. In the case of characteristic 2, the Frobenius trace on R/2R does not lift to a differential operator on R/4R, 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 action of the linearly reductive group GL n-1 (Q), Sp n-2 (Q), or O 2 (Q), respectively; see, for example, §7, §6, §8, for details. In contrast with the cases discussed above, we prove that if R is a toric Z-algebra, then, for each prime integer p > 0, every differential operator on R/pR lifts to a differential operator on R; see Theorem 5.2. In particular, if R is the hypersurface over Z defined by the determinant of a 2 × 2 matrix of indeterminates or a symmetric 2 × 2 matrix of indeterminates, then every differential operator on R/pR lifts to a differential operator on R; this addresses the case n = 2 in the context of Theorem 1.2 (a) and (c) above.
Our approach is based on the paper [14] by the first author, where it was established that there is an isomorphism between rings of differential operators (considered as modules over the enveloping algebra) and certain local cohomology modules; see also [1] where related isomorphisms are established under different hypotheses. The isomorphism between rings of differential operators and local cohomology modules identifies differential operators that do not lift modulo a prime integer p with local cohomology elements that do not lift modulo p, and consequently with nonzero elements in a different local cohomology module that are annihilated by the prime integer p. These ideas were used by the first author to give positive answers to Question 1.1 in special cases [14,. However, we have attempted to keep the present paper largely self-contained, and focused on hypersurfaces, where the isomorphisms can be made entirely explicit; it is striking that the isomorphism is one of D-modules: Theorem 1.3. Let A be a commutative ring. Set R := A[x 0 , . . . , x n ]/(f ), where f is a nonzerodivisor in the polynomial ring A[x 0 , . . . , x n ]. Set ∆ to be the kernel of the multiplication map R ⊗ A R -→ R. Then the local cohomology module H n ∆ (R ⊗ A R), with the natural D R|A -module structure as in §3. 3, is a free D R|A -module of rank one.
The techniques and calculations used in our proof of Theorem 1.2 have implications to the existence of liftings of the Frobenius morphism that have attracted a lot of attention; for example, [3] provides a connection between liftability of the Frobenius and infinitely generated crystalline cohomology, while [6] proves Bott vanishing for varieties that admit a Frobenius lift modulo p 2 . The liftability of the Frobenius is also studied in considerable detail in [31]; we use results from that paper to determine whether the Frobenius endomorphism on a classical invariant ring of positive prime characteristic lifts to a ring endomorphism in characteristic zero: Theorem 1.4. Consider the following classical invariant rings, modeled over Z:
(a) R := Z[X]/I t (X), where X is an m × n matrix of indeterminates, I t (X) the ideal generated by the size t minors of X, and min{m, n} t 3. (b) R := Z[X]/ Pf t (X), where X is an n × n alternating matrix of indeterminates, Pf t (X) the ideal generated by the Pfaffians of the size t principal submatrices of X, for t even, and n t 4. (c) R := Z[X]/I t (X), where X is a symmetric n × n matrix of indeterminates, I t (X) the ideal generated by the size t minors of X, and n t 3.
Let p be a positive prime integer. In cases (a) and (b), the Frobenius endomorphism on R/pR does not lift to a ring endomorphism of R/p 2 R, nor, a fortiori, to a ring endomorphism of R. In case (c), the same conclusion holds if t 4 or if p = 2.
If R is a normal affine semigroup ring over Z, then, for each prime integer p > 0, the Frobenius endomorphism on R/pR lifts to an endomorphism of R, and hence to an endomorphism of R/p 2 R. Specifically, if R is defined by the size 2 minors of a matrix of indeterminates, or of a symmetric matrix of indeterminates, then the Frobenius endomorphism on R/pR lifts to an endomorphism of R and of R/p 2 R; this explains the case t = 2 in the context of Theorem 1.4 (a) and (c).
Recall that if G is a linearly reductive group over a field K, with a linear action on a polynomial ring K[x], then the invariant ring K[x] G is a direct summand of K[x] as a K[x] G -module; many key properties of classical invariant rings including finite generation and the Cohen-Macaulay property follow from the existence of such a splitting, see for example [11, §2] and [12]. Indeed, the general linear group, the symplectic group, and the orthogonal group are linearly reductive over fields of characteristic zero; it follows that determinantal rings, Pfaffian determinantal rings, and symmetric determinantal rings, over fields of characteristic zero, are direct summands of polynomial rings. In contrast, we prove that working over the ring of integers Z, or over the ring of p-adic integers Z (p) , the corresponding rings are typically not direct summands of any polynomial ring: Corollary 1.5. Let R be as in Theorem 1.4 (a), (b), or (c). Then R is not a direct summand, as an R-module, of any polynomial ring over Z.
Let V := Z (p) be the p-adic integers; in case (c), assume further that either t 4, or that p = 2. Then the ring R ⊗ Z V is not a direct summand, as an R ⊗ Z V -module, of any polynomial ring over V .
The corollary is immediate from Theorem 1.4 since the existence of a Frobenius lift modulo p 2 is inherited by a ring that is a direct summand, see Proposition 2.2. Regarding the case t = 3 in Theorem 1.4 (c) and the corollary, if V is a discrete valuation ring of mixed characteristic, such that the residual characteristic is an odd prime integer, then a symmetric determinantal ring of the form V [X]/I 3 (X) is a direct summand of a polynomial ring over V , see Remark 8.3 and §9.
In §2 we record basic facts about differential operators and Koszul and local cohomology modules from perspectives needed later in the paper; §3 includes the aforementioned explicit isomorphisms, relating specific p-torsion local cohomology elements to specific differential operators; Theorem 1.3 is proved in §3.3. The proofs of the three cases of Theorems 1.2 and 1.4 are completed in §6, §7, and §8, while Theorem 1.4 is proved in §9. For Theorem 1.2, we prove that the relevant p-torsion local cohomology elements are nonzero. We expect that these calculations are of independent interest, adding to the study of integer torsion in local cohomology modules pursued in [27,20,4].
Differential operators on a commutative ring R are defined inductively as follows: for each r ∈ R, the multiplication by r map r : R -→ R is a differential operator of order 0; for each positive integer n, the differential operators of order less than or equal to n are additive maps δ : R -→ R for which each commutator
is a differential operator of order less than or equal to n -1. If δ and δ ′ are differential operators of order at most m and n respectively, then δ • δ ′ is a differential operator of order at most m + n. Thus, the differential operators on R form a subring D R of End Z (R). We use D n R to denote the differential operators of order at most n.
It is readily seen that each element of D 1 R may be expressed uniquely as the sum of an element of D 0 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
Note that if A contains the field of rational numbers, then
We record an alternative description of D R|A from [9, §16.8]. For R an A-algebra, set
and consider the P R|A -module structure on End A (R) under which r 1 ⊗ r 2 acts on δ to give the endomorphism r 1 • δ • r 2 , where r i denotes the map that is multiplication by r i . Set
where µ is the A-algebra homomorphism determined by µ(r
it follows that an element δ of End A (R) is a differential operator of order at most n precisely if it is annihilated by ∆ n+1 R|A . By [9,Proposition 16.8.8], the A-linear differential operators on R of order at most n correspond to
where
is viewed as a left R-module via r -→ r ⊗1. To make the isomorphism (2.0.1) explicit, consider the map
in which case, the map
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 0 , . . . , z t . Then the sequence of ideals defined by
is cofinal with the sequence ∆ n R|A for n ∈ N. As there is little risk of confusion, we reuse the notation ϱ : R -→ P
R|A for the map with r -→ 1 ⊗ r, and set
where
Similarly, we use ϱ * for the P R|A -module isomorphism
R|A gives a filtration of D R|A that is cofinal with the filtration by order; the filtration D
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
The differential operators from R to M , of order at most n, are
Note that one has an isomorphism of P R|A -modules,
Likewise, given generators z 0 , . . . , z t for ∆ R|A , we set
and
In analogy with the equality D R = D R|Z , we set P R := P R|Z and ∆ R := ∆ R|Z , along with the corresponding notation P n R := P R /∆ n+1 R and P
[n]
. Observe that if R has characteristic p, we have P R = P R|(Z/pZ) , and likewise for the other notions discussed above.
Let z be an element of a ring R. One has maps between the Koszul complexes K • (z n ; R), for n ∈ N, and the Čech complex C • (z; R) as below:
Taking the direct limit of K • (z n ; R) yields an isomorphism lim --→n K • (z n ; R) ∼ = C • (z; R). For z := z 0 , . . . , z t , one similarly obtains a map of complexes
and, for each k 0, an induced map from Koszul cohomology to local cohomology
Setting z n := z n 0 , . . . , z n t , one likewise has lim
for each k 0.
When k equals t + 1, the map (2.0.2) takes the form
while, if k equals t, the Koszul cohomology element in H t (z; R) corresponding to an equation i z i g i = 0 in R maps to
We mention that a local cohomology element
is zero if and only if there exists an integer k 0 such that
Suppose the ring R takes the form S/f S, for S a commutative ring, and f ∈ S a regular element. Let z := z 0 , . . . , z t , as before. The exact sequence
with δ f denoting the connecting homomorphism. To make the map explicit, suppose g i are elements of S such that
for some n 1 and an element s ∈ S; then
We briefly review Bockstein homomorphisms on local cohomology [28]. Let p be a prime integer that is a regular element on a ring R. Fix an ideal a of R. Applying the local cohomology functor
one obtains a cohomology exact sequence; the Bockstein homomorphism
is the connecting homomorphism in the cohomology exact sequence. For an alternative point of view, one may take the cohomology exact sequence induced by
i.e., the sequence
The Bockstein homomorphism β p above then coincides with the composition
Let R be an A-algebra, and z an element 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 power of the denominator by the rule
This may be written in closed form: set δ (0) := δ and δ (i+1) := [δ (i) , z n ] inductively; then
For elements z of R, the Čech complex C • (z; R) is a complex of D R|A -modules, hence its cohomology modules H k (z) (R) have a natural D R|A -module structure. This D R|Amodule structure 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 of a (D R|A , D S|A )-bimodule, where D R|A acts by postcomposition, and D S|A acts by precomposition; one verifies using the inductive definition of differential operators that the prescribed compositions are indeed elements of D S|A (R). This bimodule structure yields a base change functor from D S|A -modules to D R|A -modules, (a) For each z ∈ S, one has an isomorphism of
(b) Given elements z := z 0 , . . . , z t of S, one has an isomorphism of D R|A -modules
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 [30, §2.2], so there is an exact sequence of right D S|A -modules
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
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 any element γ ∈ D R|A can be written as µ + ID S|A for some µ ∈ D S|A such that µ(I) ⊆ I. The compatibility condition then boils down to checking that γ(s/z n ) = µ(s/z n ), which is clear from the formula (2.0.3), proving (a).
Next, consider the right-exact sequence of D S|A -modules
and the right-exact sequence of
Applying D S|A (R) ⊗ D S|A (-) to the first, we obtain a commutative diagram of D R|Amodules
where the vertical maps are isomorphisms by (a). This induces the D R|A -module isomorphism as claimed in (b),
Let T be a ring, and let p > 0 be a prime integer that is not a unit in T . A lift of the Frobenius endomorphism F on T /pT is a ring homomorphism Λ p : T -→ T such that the following diagram commutes
As we show next, the existence of a Frobenius lift modulo p 2 is inherited by rings that are direct summands; see also [31,Lemma 4.1].
Proposition 2.2. Let T be a ring, and let p > 0 be a prime integer that is not a unit in T . Let R be a subring of T that is a direct summand of T as an R-module.
If the Frobenius endomorphism on T /pT lifts to an endomorphism of T /p 2 T , then the Frobenius endomorphism on R/pR lifts to an endomorphism of R/p 2 R.
Proof. Note that R/p 2 R is a direct summand of T /p 2 T ; after a change of notation, we may assume that p 2 = 0 in R and T . We use ι : R -→ T for the inclusion, and ϱ : T -→ R to denote an R-linear splitting.
Let Λ p : T -→ T denote a Frobenius lift; we claim that
is an endomorphism of R. Since each map is additive, so is the composition. Given elements r i in R, there exist elements
It is readily verified that ϱ • Λ p • ι induces the Frobenius endomorphism on R/pR. Definition 2.3 (Buium [7], Joyal [15]). Let T be a ring, and p > 0 be a prime integer. A p-derivation on T is a map ϕ p : T -→ T that satisfies the following for all a, b ∈ T :
It follows from the above that
If ϕ p is a p-derivation on T , then the map Λ p : T -→ T given by Λ p (t) = t p + pϕ p (t) is a lift of the Frobenius; conversely, if p is a nonzerodivisor on T , and
Example 2.4. Fix a prime integer p > 0, and let S be a polynomial ring over Z in the indeterminates x := x 0 , . . . , x d . We refer to the Z-algebra homomorphism
as the standard lift of the Frobenius with respect to x, and the corresponding p-derivation
as the standard p-derivation with respect to x.
We record the following compatibility for p-derivations used in the sequel: Let S and S ′ be polynomial rings over Z in the indeterminates x and x ′ respectively. Let Υ : S -→ S ′ be a ring homomorphism such that Υ(x) ∈ x ′ ∪ {0, 1} for each x ∈ x, i.e., the homomorphism Υ either takes an indeterminate x ∈ x to an indeterminate x ′ ∈ x ′ , or specializes it to 0 or 1. Let Λ p and ϕ p denote the standard lift of the Frobenius and the corresponding p-derivation on S with respect to x, and likewise let Λ ′ p and ϕ ′ p be the corresponding maps for S ′ with respect to x ′ . Then the following diagrams commute:
We recall the following criterion, due to Zdanowicz, for the existence of a lift of the Frobenius endomorphism on a hypersurface: Proposition 2.5 ([31,Corollary 4.9]). Let p be a prime integer, S := Z[x] a polynomial ring, and ϕ p : S -→ S the standard p-derivation on S with respect to
Then the Frobenius endomorphism on the hypersurface R/pR lifts to an endomorphism of R/p 2 R if and only if
Let A be a commutative ring, and S := A[x] the polynomial ring over A in the indeterminates x := x 0 , . . . , x d . Then P S|A is a polynomial ring over A in the indeterminates x i ⊗1 and 1 ⊗x i , for 0 i d. Since we view P S|A as an S-module via the map s -→ s ⊗1, we simply write x i for x i ⊗1. Set y i := 1 ⊗ x i , so that
where y := y 0 , . . . , y d . The elements
form a generating set for the ideal ∆ S|A of P S|A . Using this generating set, we consider the sequence of ideals ∆ S|A , as defined in §2.1. Note that the elements y 0x 0 , y 1x 1 , . . . , y dx d are algebraically independent generators for P S|A as an S-algebra. Thus, P
S|A is a free S-module with basis
where 0 a i n.
Likewise, Hom S (P
S|A , S) is the free S-module with the dual basis
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
Proposition 3.1. For each n 0, one has P S|A -module isomorphisms
Proof. We need only observe that the Koszul cohomology module
S|A . The rest is immediate from the preceding discussion.
To make the isomorphism ϱ * completely explicit, note first that ϱ(f (x)) = f (y). Now,
where the last equality uses the Taylor expansion of a polynomial. It follows that
There is a P S|A -module isomorphism
where γ is the isomorphism
Proof. For each n 0, one has a commutative diagram
where the canonical surjection
S|A induces the map in the middle column. The left column realizes local cohomology as the direct limit of Koszul cohomology; the maps in the right column are injective, with D S|A as the directed union.
Example 3.3. The isomorphism ϱ * • γ maps the local cohomology element
∆ 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
There are analogous isomorphisms for differential operators from S to M , as below:
Proposition 3.4. For M an S-module and n 0, there are P S|A -module isomorphisms
and
The maps γ n and γ are obtained from those in Propositions 3.1 and 3.2 by applying
Proof. The right exactness of -⊗ S M gives
S|A is a free S-module, one also has the isomorphism Hom S (P
Thus, γ n is an isomorphism, obtained from the corresponding isomorphism in Proposition 3.1; ϱ * is an isomorphism as discussed in §2.1. The transition from Koszul cohomology to local cohomology follows as in the proof of Proposition 3.2.
As in §3.1, let S := A[x] be a polynomial ring over a commutative ring A, and identify
, where f ∈ S is a nonzero polynomial, and identify R ⊗ A R with
Then y 0x 0 , y 1x 1 , . . . , y dx 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 isomorphisms
for each n 0, and
Proof. Taking the exact sequence
and applying Hom R (-, R), one obtains
Hom R (P
and Hom R (P
the first isomorphisms follow. The second follows by taking the union over n.
In light of the previous proposition, we identify D
[n]
R|A with ann f (y), D
[n]
and the induced Koszul cohomology exact sequence
where the injectivity on the left holds since the generators of the ideal ∆
S|A form a regular sequence on P S|A ⊗ S R. We also use δ f (y) for the connecting map in the corresponding local cohomology exact sequence, i.e.,
Proposition 3.6. The maps
induce an isomorphism of P R|A -modules
R|A regarded as a submodule of D
[n] S|A (R). Passing to the direct limit, the map
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,
is an isomorphism, and D
[n]
S|A (R) . This justifies the first isomorphism. The second is the familiar transition from Koszul cohomology to local cohomology.
For S := A[x] a polynomial ring, Example 3.3 identifies the element of H d+1 ∆ S|A (P S|A ) that corresponds to the identity map in D S|A ; for a hypersurface R := A[x]/(f (x)), we next identify the element of H d ∆ R|A (P R|A ) that corresponds to the identity map in D R|A :
, where µ : P S|A -→ S is the A-algebra homomorphism determined by x i -→ x i , and y i -→ x i , one has
where g i ∈ P S|A . It follows that d i=0 (y i -x i )g i = 0 in P R|A . Thus, we obtain an element
where the signs adhere to the convention in §2.2. We claim that η R maps to the identity in D R|A under the isomorphisms in Proposition 3.6. First note that
.
) is the identity map on R.
Let A be a commutative ring. Given A-algebras T and T ′ , one has a homomorphism
For an ideal I of T ⊗ A T ′ , we regard the local cohomology module H k I (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[x] be the polynomial ring over A in the indeterminates x := x 0 , . . . , x d . Then the map
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 verify that it is D S|A -linear. In view of Example 3.3, it suffices to verify that the map
, where d = 0, the verification takes the form
with the general case being similar.
Theorem 3.9. Let A be a commutative ring, and let S := A[x] be the polynomial ring over A where x := x 0 , . . . , x d . Let R = S/f S, for f ∈ S a nonzero polynomial. Then the map
as in Proposition 3.6, is an isomorphism of D R|A -modules.
Proof. We consider
as a D R|A -module as described at the beginning of this subsection.
Let ϱ * • γ : H d+1 ∆ 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
We claim that this map identifies with the isomorphism
obtained by applying -⊗ S R, as in Proposition 3.4. Indeed, it is easy to see that
and the latter identifies with H d+1 ∆ S|A (P S|A ⊗ S R) by Lemma 2.1. We have the commutative diagram
where all of maps in the rightmost square are D R|A -linear; the first vertical isomorphism is the map from Proposition 3.6. It follows that this map is a D R|A -linear isomorphism.
Example 3.10. Let S := A[x 0 , . . . , x d ] be a polynomial ring over a commutative ring A.
Fix the N-grading on S where S 0 = A and deg
for f ∈ S a nonzero homogeneous polynomial. We explicitly describe the element of H d ∆ R|A (P R|A ) that corresponds to the Euler operator
With the notation as in Example 3.7, let f (y
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 cohomology class of the element obtained by applying E, considered as an operator on the x variables, componentwise to the above cocycle.
Example 3.11. Consider R := A[x]/(x n ), for n a positive integer. Then
In P S|A one has y n -
while the element corresponding to the Euler operator is
). Then the identity in D R|A corresponds to
and the Euler operator to
Suppose A is a field of characteristic p > 0. Let S := A[x 0 , . . . , x d ] be a polynomial ring, and R := S/(f (x)) a graded hypersurface. Let e be a positive integer. The rings R and R p e are Gorenstein, so duality implies that Hom R p e (R, R p e ) is a cyclic R-module. To specify a generator, first set
It is a key point that Φ e S : S -→ S p e , and that it is S p e -linear. Next, consider the composition
where π is the canonical surjection. Since fS is contained in its kernel, the composition factors through a map
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 e R is an element of
R|A , and is an R-module generator for Hom R p e (R, R p e ). We claim that Φ e R , viewed as a differential operator in D R|A , corresponds to the image of the element η R from Example 3.7 under the e-th iterate of the Frobenius action F on the local cohomology module
To see this, consider
as in Example 3.3, and note that
With η R as in Example 3.7, one has
which coincides with Φ e R in D R|A , regarded as a submodule of D S|A (R).
Let A, S, and R be as in §3.2, i.e., A is a commutative ring, S := A[x 0 , . . . , x d ] is a polynomial ring, and R := S/(f (x)) is a hypersurface. Suppose p > 0 is a prime integer that is regular on P R|A . The exact sequence
with δ p denoting the connecting homomorphism. In particular, δ p induces an isomorphism
Acknowledging the abuse of notation, we denote the inverse of this isomorphism by δ -1 p , so as to obtain: Proposition 3.13. There is an isomorphism of P R|A -modules
given by
In particular, given an element
Proof. One has a commutative diagram
where the vertical maps are the isomorphisms ϱ * •γ •δ f (y) of Proposition 3.6. Combining with the isomorphism
the assertion follows.
Next, we establish a similar criterion for when a differential operator
and the corresponding cohomology exact sequence
The connecting homomorphism β p in the sequence above is the Bockstein homomorphism on local cohomology, see §2.3. The map β p induces an isomorphism coker
and we denote the inverse of this isomorphism by β -1 p . Then, along the same lines as Proposition 3.13, we have: Proposition 3.14. There is an isomorphism of P R|A -modules
In particular, for an element
if and only if β p (ν) = 0.
Let S := Z[x] be a polynomial ring over Z in the indeterminates x := x 0 , . . . , x d . Identifying P S with Z[x, y] as before, note that P S may also be viewed as a polynomial ring over Z in the indeterminates
Fix a prime integer p > 0, and let Λ p : P S -→ P S denote the standard lift of Frobenius with respect to the indeterminates above, i.e., Λ p is the endomorphism of P S with Λ p (x i ) = x p i and Λ p (y ix i ) = (y ix i ) p for each i. Let ϕ p denote the corresponding p-derivation on P S .
Let R = S/(f (x)) be a graded hypersurface. Assume that p is regular on P R , in which case there is an exact sequence
) be the element that corresponds to the identity map in D R , as in Example 3.7, and let η R denote its image in
as elements of H d+1 ∆ R (P R ). To see this, take g i ∈ P S with f (y) -f (x) = d i=0 (y i -x i )g i as in Example 3.7. Then, in P S , one has
It follows that the image of
Hence, in H d+1 ∆ R (P R ), one has
which proves the claim. Similarly, with β p denoting the Bockstein homomorphism
with δ p and π p as in (3.14.1), one see that
Combining the preceding calculations with Propositions 3.13 and 3.14, we obtain:
Theorem 3.15. Let S := Z[x] be a polynomial ring in the indeterminates x := x 0 , . . . , x d . Fix a nonzero homogeneous polynomial f (x) in S, and set R := S/(f (x)). Let p be a prime integer that is a regular element on P R . Let Λ p be the endomorphism of
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 §3. 4
is zero; similarly, Φ R/pR lifts to a differential operator on R/p 2 R if and only if the image of the displayed element in
Remark 3.16. For S and R as in the previous theorem set a := deg f (x) -(d + 1). Then, with the grading shifts as below, Theorem 3.9 provides degree-preserving isomorphisms
In
Note that ζ p is annihilated by p in view of (3.14.1). Hence, the module H 3 ∆ R (P R ) contains a nonzero p-torsion element for each prime integer p ̸ = 3. Likewise, consider R := S/f S, where S := Z[x 0 , x 1 , x 2 , x 3 ] and f (x) := x 3 0 + x 3 1 + x 3 2 + x 3 3 . Recent work of Mallory [21,Theorem 1.2] shows that R has no differential operators of negative degree. For each prime integer p ̸ = 3, the Frobenius trace on R/pR has negative degree, namely 1p, and hence does not lift to R. It follows that H 4 ∆ R (P R ) contains a nonzero p-torsion element for each prime integer p ̸ = 3.
Our approach to constructing mod p differential operators that do not lift, based on the results of the previous section, is via constructing p-torsion elements in local cohomology modules of the form H d+1 ∆ R (P R ). Towards this, the following lemma will prove useful; the notation is as in §3.6.
where m d and each f i ∈ S. Set R := S/(f (x)). Fix a prime integer p, and let ϕ p denote the p-derivation of P S as in Theorem 3.15; this restricts to the standard pderivation of S with respect to x.
Similarly, if the image of
Proof. Recall the notation y := y 0 , . . . , y d . We identify P S with the polynomial ring Z[x, y] and P R with Z[x, y]/(f (x), f (y)), so that ∆ R is the ideal generated by the elements
Suppose the displayed element of H d+1 ∆ R (P R ) is zero. Then there exists an integer k 0 such that in the ring P S one has
Let P ′ denote the image of P S under the specialization x i -→ 0 for 0 i m. Note that f (x) -→ 0 under this specialization. Since one has a commutative diagram
the ideal membership (4.1.1) specializes to give
Next, specialize x i -→ y i for m + 1 i d, to obtain
where Λ p is the standard lift of Frobenius on Z[y] with respect to y. Renaming y i -→ x i for each i, the above reads
implying that
is zero. The proof of the final assertion is similar.
A question of Huneke [13, Problem 4] asks whether local cohomology modules of Noetherian rings have finitely many associated prime ideals. This was answered in the negative by the second author in [27, §4]: there is hypersurface R over Z for which a local cohomology module H k a (R) has p-torsion elements for each prime integer p. From this, it follows that H k a (R) has infinitely many associated primes; this is extended to several natural families of hypersurfaces by Theorems 6.1, 6.2, and 7.1. On the other hand, for a polynomial ring over Z, or, more generally, a smooth Z-algebra, the answer to Huneke's question is positive; this follows from the following result, which we use in the next section: Theorem 4.2. [4, Theorem 3.1] Let S be a smooth Z-algebra, and a an ideal of S generated by elements f = f 1 , . . . , f t . Let k be a nonnegative integer.
If a prime integer is a nonzerodivisor on the Koszul cohomology module H k (f ; S), then it is also a nonzerodivisor on the local cohomology module H k a (S).
We consider the question whether differential operators lift modulo p for invariant rings of tori. An m × n integer matrix M determines a map
and hence a normal affine semigroup
Let A be a commutative ring. The semigroup ring A[Λ] is the subring of the polynomial ring A[N n ] := A[x 1 , . . . , x n ] that is generated, as an A-algebra, by monomials with exponent vectors in Λ. The ring A[N n ] has an N m -grading where the degree of x i is the
When A has an infinite group of units A × , the ring A[Λ] is the invariant ring for an action of a product of copies of A × on A[N n ], hence the name.
Example 5.1. The integer matrix M := (1 1 -2 ) yields the semigroup Λ generated by 2 0 1 ,
Theorem 5.2. Let R be a toric Z-algebra. Then, for each prime p, each differential operator on R/pR lifts to a differential operator on R; thus, the natural map
for Λ as in the earlier discussion. Recall that R is a direct summand of S := Z[N n ], and let ϱ : S -→ R be a splitting. Then ϱ induces a splitting ϱ of the inclusion R/pR ⊆ S/pS, with the commutative diagram
which is the second Veronese subring of the polynomial ring
Consider R with the standard N-grading, i.e., where the Z-algebra generators deg x i x j have degree 1. The group {±1} acts on the polynomial ring T ⊗ Z C = C[x 1 , x 2 ] with invariant ring R ⊗ Z C, and the ring of differential operators D R⊗ Z C|C is generated by the elements of D T ⊗ Z C|C that have even degree. Note that the derivations x i ∂ ∂x j have degree 0; there are no derivations of negative degree, confer [17]. Since
it follows that R has no derivations of negative degree. In contrast, the ring R/2R has a derivation of degree -1, namely
which is the endomorphism of R/2R with
for i, j ∈ N. This derivation lifts to the differential operator
The main focus of this section is Pfaffian hypersurfaces, but we begin with the analogue of Theorem 1.2 for quadratic hypersurfaces, where the arguments are most transparent: Theorem 6.1. Let x 1 , . . . , x 2m be indeterminates over Z, where m 3, and set
Then, for each prime integer p > 0, the Frobenius trace on R/pR does not lift to a differential operator on R/p 2 R, nor, a fortiori, to a differential operator on R.
Proof. Set S := Z[x 1 , . . . , x 2m ]. Fix p, and let ϕ p be the standard p-derivation on S with respect to x. In view of Theorem 3.15 and Lemma 4.1, it suffices to prove that the element
is nonzero. Indeed, if it were zero, then there exists an integer k 0 such that
Let e i ∈ Z m denote the i-th standard basis vector. Consider the Z m -grading of R/pR where
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
on the right has degree (k, . . . , k, -p, k, . . . , k), and therefore the coefficient must be a multiple of
in the ring (R/pR) (0,...,0) = Z/pZ[x 1 x m+1 , . . . , x m x 2m ]/( m i=1
x i x m+i ).
But (R/pR) (0,...,0) may be identified with the polynomial ring Z/pZ[z 1 , . . . , z m-1 ], where
in which case, (6.1.2) reads
This is readily seen to be false, for example by examining the coefficient of z p-1
We now turn to Pfaffian hypersurfaces. Let n be an even integer with n 4, and let Z be an (n -2) × n matrix of indeterminates over an infinite field K. Set T := K[Z]. The symplectic group Sp n-2 (K) acts K-linearly on T by the rule
By [8, §6] or [10, Theorem 5.1], the invariant ring for this action is the K-algebra generated by the entries of the product matrix Z tr Ω Z, where Ω is the size n -2 standard symplectic block matrix 0 I -I 0 , with I the identity. This invariant ring is isomorphic to K[X]/(pf X) for X an n × n alternating matrix of indeterminates, and pf X its Pfaffian. When K has characteristic zero, the ring of differential operators on the invariant ring is described explicitly in [18, IV 1.9, Case C].
For M an alternating matrix, the cofactor expansion for Pfaffians takes the form
where M 1j 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 principal submatrices of M . It follows from the cofactor expansion that Pf t (M ) ⊆ Pf t-2 (M ). The Pfaffian of a 4 × 4 alternating matrix of indeterminates X is the quadratic form Theorem 6.2. Let X be an n × n alternating matrix of indeterminates over Z, where n is an even integer with n 4. Set S := Z[X] and R := S/(pf X). Fix a prime integer p > 0, and let ϕ p be the standard p-derivation on S with respect to the indeterminates X. Then
is a nonzero p-torsion element; moreover, its image in H n-1 (x 12 ,...,x 1n ) (R/pR) is nonzero. In particular, the local cohomology module H n-1 (x 12 ,...,x 1n ) (R) contains a nonzero ptorsion element for each prime integer p > 0.
Proof. As noted above, the case n = 4 is settled by the proof of Theorem 6.1. We proceed by induction on n, with n = 4 serving as the base case.
The image of the displayed element is zero in H n-1 (x 12 ,...,x 1n ) (R/pR) if and only if there exists an integer k 0, such that in the polynomial ring S one has
We know this cannot happen for n = 4. Suppose (6.2.1) holds for an even integer n 6. Specialize x n-1,n -→ 1, and x ij -→ 0 for i = 2, . . . , n -2 and j = n -1, n, in which case, the image of X is
The upper left (n -2) × (n -2) submatrix is unchanged; denote this by X ′ . Elementary row and column operations transform the matrix displayed above to
where Ω := 0 1 -1 0 , which shows that the Pfaffian of the image of X after specialization equals pf X ′ . Hence the ideal membership (6.2.1) specializes to
with S ′ denoting the image of S under the specialization. The indeterminates x 1,n-1 and x 1n do not occur in the polynomial pf X ′ , and hence form a regular sequence on
Using this, we obtain
Next specialize x 1,n-1 -→ 0 and x 1n -→ 0, so as to obtain
where S ′′ is the image of S ′ under the specialization, equivalently, the polynomial ring over Z in the indeterminates occurring in X ′ . But this violates the inductive hypothesis.
We next show that Pfaffian hypersurfaces do not admit a lift of Frobenius modulo p 2 : Theorem 6.3. Let X be an n × n alternating matrix of indeterminates over Z, where n is an even integer with n 4. Set S := Z[X] and R := S/(pf X). Fix a prime integer p > 0. Then the Frobenius endomorphism on R/pR does not lift to an endomorphism of R/p 2 R.
Proof. We proceed by induction on even integers n 4, along the same lines as in the proof of Theorem 6.2. The case n = 4 is the hypersurface R defined by
x 12 x 34x 13 x 24 + x 14 x 23 .
While the case of diagonal quadratic forms of rank at least as in the proof of Theorem 6.1, we may work in the subring (R/pR) (0,0,0) = Z/pZ[x 12 x 34 , x 13 x 24 , x 14 x 23 ]/(x 12 x 34x 13 x 24 + x 14 x 23 ), which, in turn, may be identified with the polynomial ring Z/pZ[z 1 , z 2 ], where z 1 := x 12 x 34 and z 2 := x 13 x 24 .
But then (6.3.1) reads
which is seen to be false by examining the coefficient of z p-1 1 z 2 . For the inductive step, let R := S/(pf X) be the hypersurface defined by the Pfaffian of an n × n alternating matrix X, for an even integer n 6. Suppose that the Frobenius endomorphism on R/pR lifts to R/p 2 R. Then, by Proposition 2.5,
Each partial derivative ∂ pf X/∂x ij is, up to sign, the Pfaffian of the (n -2) × (n -2) submatrix of X obtained by deleting rows i, j and columns i, j. Using the notation
for ideals a in a ring of prime characteristic p > 0, the preceding ideal membership may hence be written as
Applying the specialization in the proof of Theorem 6.2, X may be replaced by
contradicting the inductive hypothesis.
Let K be an infinite field. Let Y and Z be n × (n -1) and (n -1) × n matrices of indeterminates respectively, and set T := K[Y, Z]. The general linear group GL n-1 (K) acts K-linearly on T where, for M ∈ GL n-1 (K), one has
By [8, §3] or [10, Theorem 4.1], the invariant ring for this action is generated over K by the entries of the product 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 operators on the invariant ring is described explicitly in [18, IV 1.9, Case A].
We complete the proof of Theorem 1.2 (a). In view of Theorem 3.15 and Lemma 4.1, it suffices to prove: Theorem 7.1. Let X be an n × n matrix of indeterminates over Z, where n 3. Fix a prime integer p > 0, and let ϕ p be the standard p-derivation on S := Z[X] with respect to X. Take a to be the ideal (x 12 , . . . , x 1n , x 22 , . . . , x 2n ) R, where R := S/(det X). Then
In particular, the local cohomology module H 2n-2 a (R) contains a nonzero p-torsion element for each prime integer p > 0.
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 2n-2 a (R/pR) is zero. Then there exists an integer k 0 such that in the polynomial ring S one has
12 , . . . , x p+k 1n , x p+k 22 , . . . , x p+k 2n )S. (7.1.1) For i = 4, . . . , n, specialize x ii -→ 1 and x ij -→ 0 for j ̸ = i. The image of X after specialization has the form
where X ′ denotes the upper left 3 × 3 submatrix of X, and I is the size n -3 identity matrix. Note that det X specializes to det X ′ , and ϕ p (det X) to ϕ p (det X ′ ). Hence the ideal membership (7.1.1) specializes to
with S ′ the image of S. Since the indeterminates x 14 , . . . , x 1n , x 24 , . . . , x 2n do not occur in the polynomials det X ′ and ϕ p (det X ′ ), the ideal membership above implies
Specializing x ij -→ 0 for i 4 and also for j 4, we reduce to the case n = 3; specifically, it suffices to consider
Take the Z 5 -grading on R ′ /pR ′ defined by
and note that λ is homogeneous of degree (p, 0, 0, 0, 0). Hence deg λ(x 12 x 13 x 22 x 23 ) k = (p + 4k, -2k, -2k, -2k, -2k).
Fix a homogeneous equation
Let µ be a monomial of the above degree. Examining the second and fourth components of the degree, the exponents on x 13 and x 23 in µ add up to at least 2k, as do the exponents on Proof. As in the proof of Theorem 7.1, one first reduces to the case n = 3 as follows.
Suppose that the Frobenius endomorphism on R/pR lifts to R/p 2 R. Then, by Proposition 2.5,
Each partial derivative above is, up to sign, the determinant of an (n -1) × (n -1) submatrix of X, so the above may be restated as
where I n-1 (X) denotes the ideal generated by the size n -1 minors of X. Applying the specialization S -→ S ′ in the proof of Theorem 7.1, one has
with I denoting the identity matrix of size n -3. Hence I n-1 (X)S ′ = I 2 (X ′ ), and the ideal membership (7.2.1) specializes to
which is essentially the n = 3 case.
Assume n = 3. Specializing x 13 and x 31 to 0, the resulting matrix Using the Z 5 -grading in the proof of Theorem 7.1, det X ′′ has degree (p, 0, 0, 0, 0), so we obtain an ideal membership in the subring generated by elements of degree ( * , 0, 0, 0, 0), namely the ring in the ring above, which is readily seen to be false.
Let Z be an (n -1) ×n matrix of indeterminates over an infinite field K of characteristic other than two. Set T := K[Z]. The orthogonal group O n-1 (K) acts K-linearly on T with
By [8, §5], the invariant ring for this action is the K-algebra generated by the entries of the product matrix Z tr Z, and is isomorphic to K[X]/(det X) for X an n × n symmetric matrix of indeterminates; see [24, §5] for the case where K has characteristic two. When K has characteristic zero, the ring of differential operators on this invariant ring is described explicitly in [18, IV 1.9 Case B].
Theorem 8.1. Let X be a symmetric 3 × 3 matrix of indeterminates over Z, and set R to be the hypersurface Z[X]/(det X). Then, for each odd prime integer p, the map
The map displayed above is not surjective when p = 2, see Theorem 8.5.
Proof. In view of Proposition 3.13, it suffices to show that each odd prime p acts injectively on the local cohomology module H 6 ∆ R (P R ). This is unaffected by inverting the integer 2, so we work instead with the rings Z 2 = Z[1/2] and
Let T := Z[u, v, w, x, y, z] be a polynomial ring, and of T 2 and, indeed, we identify R 2 with this subring. We claim that R 2 is a direct summand of T 2 as an R 2 -module. To see this, first consider the Z-grading on T 2 where the indeterminates u, v, w have degree 1, and x, y, z have degree -1. It follows that the degree 0 component of T 2 , i.e., the ring uy, uz, vx, vy, vz, wx, wy, wz], is a direct summand of T 2 as a B-module. Next, let G := ⟨σ⟩ be a group of order 2 acting on T 2 , where σ is the involution with
The action of G on T 2 restricts to an action on the subring B, and
We claim that equality holds in the above display. The equation (uy) 2 -(uy)(uy + vx) + (ux)(vy) = 0
shows that uy is integral over R 2 ; similarly one sees that B is integral over R 2 . Moreover, it is readily checked that at the level of fraction fields one has frac(R 2 )(uy) = frac(B),
Each element of B G is integral over R 2 , and belongs to the fraction field of R 2 ; but R 2 is normal, so we conclude that R 2 = B G . Since 2 is a unit in R 2 , the Reynolds operator B -→ R 2 shows that R 2 is a direct summand of B as an R 2 -module. It follows that R 2 is a direct summand of T 2 as an R 2 -module. At this stage, one may invoke [14,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 that the map is surjective for each odd prime integer p. Fix such a prime p; it suffices to prove that p acts injectively on the local cohomology module
where
It suffices to show that p acts injectively on H 6 ∆ (C), where
and ∆ is the ideal of C generated by the elements
Consider the ideals of C as below:
Since p and q are generated by minors of matrices of indeterminates, each is prime; moreover, p and q each contain ∆. It is a straightforward-albeit slightly tediousverification that p 2 q ⊆ ∆. It then follows that p ∩ q = rad ∆.
Since H k p (C) = 0 = H k q (C) for integers k other than 5 and 9, the Mayer-Vietoris sequence
gives an isomorphism H 6 ∆ (C) ∼ = H 7 p+q (C). It now suffices to check that p acts injectively on H 7 p+q (C). We claim that the ideal (p + q)C/pC has height 7. To see this, note that there is an isomorphism
with # denoting the Segre product of graded rings, given by
It follows that H 6 p+q (C/pC) = 0, which gives the desired injectivity using the exactness of
Remark 8.2. Over a field K of odd characteristic, the orthogonal group O 2 (K) is linearly reductive: to see this, first enlarge K so that it contains i := √ -1. The special orthogonal group SO 2 (K) is then isomorphic to a torus K × , as seen by conjugating elements of
Since SO 2 (K) is a normal subgroup of O 2 (K) with index 2, the claim follows by [23, §3].
If Z is 2 × n matrix of indeterminates over a field K of odd characteristic, the fact that O 2 (K) is linearly reductive may be used to conclude that K[Z tr Z] is a direct summand of the polynomial ring K[Z] as a K[Z tr Z]-module. However, given the arithmetic nature of the paper, it is more advantageous to construct an explicit splitting over Z 2 [i], as we carry out in the remark below, and more generally in the proof of Theorem 9.1 (c).
with ϱ denoting the complex conjugate, is a splitting of R 2 -→ T 2 . In the ring T 2 [i], set
To study the case of symmetric matrices in characteristic 2, we record the following variant of Lemma 4.1: Lemma 8.4. Let S := Z[x] be a polynomial ring in the indeterminates x := x 0 , . . . , x d . Fix a prime integer p > 0, and let f (x) ∈ S be a polynomial of the form
where g(x) ∈ (x 0 , . . . , x m )S for some fixed integer m < d, and h(x) ∈ S is a polynomial in the indeterminates x m+1 , . . . , x d . Set R := S/(f (x)), and let ϕ p denote the p-derivation of P S as in Theorem 3.15. Then, if the local cohomology element
is nonzero, so is the element
Proof. Suppose the displayed element of H d+1 ∆ R (P R /pP R ) is zero. Then there exists an integer k 0 such that
Specialize x i -→ 0 for 0 i m, in which case f (x) specializes to ph(x), so
Next, specialize x i -→ y i for m + 1 i d. Then ph(x) specializes to ph(y), giving
where Λ p is the standard lift of Frobenius on Z[y] with respect to y. Renaming y i -→ x i for each i, it follows that ϕ p g(x) and h(x) := x 12 x 13 x 23 .
In light of Lemma 8.4, it suffices to check that
is nonzero. We shall go a step further and prove that while det X ≡ g(x) mod 2,
It is a straightforward calculation that modulo the ideal (x The element on the left in (8.5.1) has degree (0, 0), so we work in the subring [R/2R] (0,0) , which is the Z/2Z-algebra generated by In the remaining cases, in view of Proposition 2.5, we need to verify that ϕ p (det X) / ∈ p, det X, ∂ det X ∂x ij p : 1 i j n S. (8.6.1)
A partial derivative of the form ∂ det X/∂x ii is the determinant of a size n -1 principal submatrix of X, whereas, for i < j, the partial derivative ∂ det X/∂x ij is, aside from a sign, twice the determinant of a size n -1 submatrix. This necessitates the distinction between the cases where p equals 2, and where p is an odd prime. For (b), suppose (8.6.1) fails for some n 3. In view of the above paragraph, one has [2] S ′ /2S ′ , which is a contradiction. For (c), suppose (8.6.1) fails for some n 4, and p odd. Then ϕ p (det X) ∈ I n-1 (X) [p] S/(p, det X)S, (8.6.3) where I n-1 (X) denotes the ideal generated by the size n -1 minors of X. Specialize under which det X ′ has degree (0, 0, 0, 4), and ϕ p (det X ′ ) has degree (0, 0, 0, 4p). Let ∆ ij denote the determinant of the submatrix of X ′ obtained by deleting the i-th row and Since the elements in question have degree of the form (0, 0, 0, * ), this ideal membership holds in the subring [S ′ /(p, det X ′ )S ′ ] (0,0,0, * ) , which may be identified with We work modulo the first factor c -(u + v) 2 , i.e., in the polynomial ring
where c is identified with (u + v) 2 . The ideal membership then implies that 1
is a linear combination of (u 2 v 2 + u 3 v) p and (u 2 v 2 + uv 3 ) p with coefficients from Z/pZ. This is not possible, for example by examining the coefficient of u 3p-1 v p+1 ; the interested reader-if one remains-may verify that this coefficient is 8.
We extend some earlier results to rings defined by minors and Pfaffians of arbitrary size; the following theorem subsumes Theorem 1.4 and Corollary 1.5. (a) Let R := V [X]/ Pf t (X), where X is an n × n alternating matrix of indeterminates, and Pf t (X) the ideal generated by the Pfaffians of the size t principal submatrices of X, where t is even, and n t 2. Then R is a direct summand of a polynomial ring over V , as an R-module, if and only t = 2. (b) Let R := V [X]/I t (X), where X is an m × n matrix of indeterminates, and I t (X) the ideal generated by the size t minors of X, where min{m, n} t 2. Then R is a direct summand of a polynomial ring over V , as an R-module, if and only t = 2. (c) Let R := V [X]/I t (X), where X is a symmetric n × n matrix of indeterminates, and I t (X) the ideal generated by the size t minors of X, where n t 2. Then R is a direct summand of a polynomial ring over V , as an R-module, if and only t = 2 or if t = 3 and the prime p is odd in the case V = Z (p) .
In each of the case above where R is not a direct summand of a polynomial ring, the Frobenius endomorphism on R/pR does not lift to an endomorphism of R/p 2 R.
Proof. In each case, when proving that R is not a direct summand of any polynomial ring over V , or that the Frobenius endomorphism on R/pR does not lift to an endomorphism of R/p 2 R, it suffices to consider the case where V = Z (p) . Next, reduce to the case where X is a t × t matrix, since such a matrix may be enlarged by adding additional rows or columns of indeterminates, with a retraction from the larger determinantal ring to the hypersurface obtained by killing the additional indeterminates. Thus, we may assume in each case that R is a hypersurface over V . The three cases then follow from Theorem 6.3, Theorem 7.2, and Theorem 8.6 respectively.
It remains to settle the cases where R is a direct summand of a polynomial ring over V . When t = 2 in (a), one simply has R = V . In case (b), when t = 2, consider the polynomial ring T := V [y 1 , . . . , y m , z 1 , . . . , z n ]. It is readily seen that x ij -→ y i z j induces an inclusion of V -algebras. The ring T admits a Z-grading where V has degree 0, each y i has degree 1, and each z j has degree -1. The projection to the degree 0 component
J.Jeffries, A.K. Singh / Advances in Mathematics 432 (2023) 109276