Flat morphisms with regular fibers do not preserve $F$-rationality

For each positive prime integer $p$ we construct a standard graded $F$-rational ring $R$, over a field $K$ of characteristic $p$, such that $R\otimes_K\overline{K}$ is not $F$-rational. By localizing we obtain a flat local homomorphism $(R, \mathfrak{m}) \to (S, \mathfrak{n})$ such that $R$ is $F$-rational, $S/\mathfrak{m} S$ is regular (in fact, a field), but $S$ is not $F$-rational. In the process we also obtain standard graded $F$-rational rings $R$ for which $R\otimes_K R$ is not $F$-rational.


INTRODUCTION
Let P denote a local property of noetherian rings.The following types of ascent have been studied extensively; recall that for K a field, a noetherian K-algebra A is geometrically regular over K if A ⊗ K L is regular for each finite extension field L of K. (ASC I ) For a flat local homomorphism (R, m) −→ (S, n) of excellent local rings, if R is P and the closed fiber S/mS is regular, then S is P. (ASC II ) For a flat local homomorphism (R, m) −→ (S, n) of excellent local rings, if R is P and the closed fiber S/mS is geometrically regular over R/m, then S is P. Our main interest here is when P is F-rationality, a property rooted in Hochster and Huneke's theory of tight closure [HH1]: a local ring (R, m) of positive prime characteristic is F-rational if R is Cohen-Macaulay and each ideal generated by a system of parameters for R is tightly closed.Smith [Sm2] proved that F-rational rings have rational singularities, while Hara [Har] and Mehta-Srinivas [MS] independently proved that rings with rational singularities have F-rational type.Rational singularities of characteristic zero satisfy (ASC I ), as proven by Elkik [El,Théorème 5].
In the situation of (ASC II ), geometric regularity of the closed fiber R/m −→ S/mS implies that of each fiber k(p) −→ S ⊗ R k(p) for p ∈ Spec R, see [An,Page 297].The ascent (ASC II ) holds for F-rationality; this, and its variations, are due to Vélez [Vé,Theorem 3.1], Enescu [En1,Theorem 2.27], Hashimoto [Has,Theorem 6.4], and Aberbach-Enescu [AE,Theorem 4.3].A common thread amongst these is that each affirmative answer requires assumptions along the lines that the fibers are geometrically regular.The situation is similar for F-injectivity in this regard; a local ring (R, m) of positive prime characteristic is F-injective if the Frobenius action on local cohomology modules and [Has,Corollary 5.7].We present examples demonstrating that the geometric assumptions are indeed required, i.e., that F-rationality and F-injectivity do not satisfy (ASC I ): Theorem 1.1.For each prime integer p > 0, there exists a flat local ring homomorphism (R, m) −→ (S, n) of excellent local rings of characteristic p, such that the ring R is F-rational, S/mS is regular, but S is not F-rational or even F-injective.
Enescu had earlier demonstrated that F-injectivity does not satisfy (ASC I ), though the examples [En2,Page 3075] are not normal; the question of whether normal F-injective rings satisfy (ASC I ) has been raised earlier, e.g., [SZ,Question 8.1], and is settled in the negative by Theorem 1.1.There is a more recent notion, F-anti-nilpotence, developed in the papers [EH, Ma, MQ]; in view of the implications Theorem 1.1 also shows that F-anti-nilpotence does not satisfy (ASC I ).
It is worth mentioning that the rings R in Theorem 1.1 are necessarily not Gorenstein, since F-rational Gorenstein rings are F-regular by [HH2,Theorem 4.2], and F-regularity satisfies (ASC I ) by [Ab,Theorem 3.6].Another subtlety is that such examples can only exist over imperfect fields, since (ASC I ) and (ASC II ) coincide when R/m is a perfect field, and F-rationality satisfies (ASC II ).
Preliminary results are recorded in §2, including an extension of a criterion for Frationality due to Fedder and Watanabe [FW].In §3 we construct two families of examples that each imply Theorem 1.1: the first has the advantage that the proofs are more transparent, though the transcendence degree of the imperfect field over F p increases with the characteristic p; the second family accomplishes the desired with transcendence degree one, independent of the characteristic p > 0, though the calculations are more involved.The examples in §3 are constructed as standard graded rings, with the relevant properties preserved under passing to localizations.In the process, we also obtain standard graded Frational rings R, with the degree zero component being a field K of positive characteristic, such that the enveloping algebra R ⊗ K R is not F-rational.

PRELIMINARIES
Following [Ho,page 125], a local ring of positive prime characteristic is F-rational if it is a homomorphic image of a Cohen-Macaulay ring, and each ideal generated by a system of parameters is tightly closed.It follows from this definition that an F-rational local ring is Cohen-Macaulay, [HH2,Theorem 4.2], so the notion coincides with that in §1.Moreover, an F-rational local ring is a normal domain.A localization of an F-rational local ring at a prime ideal is again F-rational; with this in mind, a noetherian ring of positive prime characteristic-which is not necessarily local-is F-rational if its localization at each maximal ideal (equivalently, at each prime ideal) is F-rational.
For the case of interest in this paper, let R be an N-graded Cohen-Macaulay normal domain, such that the degree zero component is a field K of characteristic p > 0, and R is a finitely generated K-algebra.Then R is F-rational if and only if the ideal generated by some (equivalently, any) homogeneous system of parameters for R is tightly closed; see [HH3,Theorem 4.7] and the remark preceding it.An equivalent formulation in terms of local cohomology, following [Sm1, Proposition 3.3], is described next: Fix a homogeneous system of parameters x 1 , . . ., x d for R, i.e., a sequence of d := dim R homogeneous elements that generate an ideal with radical the homogeneous maximal ideal m of R. The local cohomology module H d m (R) may then be computed using a Čech complex on x 1 , . . ., x d as This module admits a natural Z-grading where the cohomology class (2.0.1) The Frobenius endomorphism F : R −→ R induces a map . This translates as cr p e ∈ (x kp e 1 , . . ., x kp e d )R for all e ∈ N. In particular, R is F-rational precisely when It follows that an F-rational ring must be F-injective.
We next review Veronese subrings.Let S be an N-graded ring for which the degree zero component is a field K, and S is a finitely generated K-algebra.Fix a positive integer n.Then the n-th Veronese subring of S is the ring Set R := S (n) .The extension R ⊆ S is split, so if S is normal ring then so is R. Let m denote the homogeneous maximal ideal of R, and note that mS is primary to the homogeneous maximal ideal n of S. For all i d , and hence that the ring R is Cohen-Macaulay whenever S is.Moreover, by [GW,Theorem 3.1.1]one has Suppose S := K[x 0 , . . ., x d ]/( f ), where f is a homogeneous polynomial that is monic of degree m with respect to the indeterminate x 0 .Then S is free over the polynomial subring K[x 1 , . . ., x d ], with basis {1, x 0 , . . ., x m−1 0 }.The local cohomology module H d n (S), as computed using a Čech complex on x 1 , . . ., x d , thus has a K-basis consisting of elements (2.0.3) where each α i is a nonnegative integer, and α 0 m − 1.When S is graded, by restricting to elements of appropriate degree, one obtains a basis for a graded component of H d n (S), or for the local cohomology H d m (R) of the Veronese subring R. Similarly, for the enveloping algebra S ⊗ K S, one has a K-basis as follows: use y 0 , . . ., y d for the second copy of S, and consider the maximal ideal N := (x 0 , . . ., x d , y 0 , . . ., y d ) of S ⊗ K S. Then the local cohomology module where each α i , β j is a nonnegative integer, α 0 m − 1, and β 0 m − 1.
The following is a variation of [FW,Theorem 2.8] and [HH3, Theorem 7.12], and is used in the proof of Theorem 3.2.
Theorem 2.1.Let S be an N-graded Cohen-Macaulay normal domain, such that the degree zero component is a field K of positive characteristic, and S is a finitely generated Kalgebra.Let n denote the homogeneous maximal ideal of S, and set d := dim S.
Suppose each nonzero element of n has a power that is a test element, and that there exists an integer n > 0 such that the Frobenius action on The hypotheses ensure that S has a homogeneous system of parameters x 1 , . . ., x d where each x i is a test element; we compute local cohomology using a Čech complex on such a homogeneous system of parameters.Suppose the assertion of the theorem is false; then there exists a nonzero homogeneous element η in 0 * H d n (S) with deg η −n.After possibly replacing the x i by powers, we may assume that for s a homogeneous element of S. Since each x i is a test element, one has x i s q ∈ (x q 1 , . . ., x q d ) for each q = p e , and hence , where the equality is because x 1 , . . ., x d is a regular sequence.Since F e (η) is nonzero in view of the injectivity of the Frobenius action on [H d n (S)] −n , one has s q / ∈ (x q 1 , . . ., x q d ).This implies that degs q deg(x 1 • • • x d ) q−1 for each q = p e , which translates as Taking the limit e −→ ∞ gives A ring S is standard graded if it is N-graded, with the degree zero component being a field K, such that S is generated as a K-algebra by finitely many elements of S 1 .
While Theorem 2.1 requires the injectivity of the Frobenius action on [H d n (S)] −n , additional hypotheses enable one to verify the injectivity of Frobenius on one graded component; the following corollary will be used in the proof of Theorem 3.2.Following [GW], the a-invariant of a Cohen-Macaulay graded ring S, as in Theorem 2.1, is Corollary 2.2.Let S be a standard graded Gorenstein normal domain, of characteristic p > 0, such that the homogeneous maximal ideal n is an isolated singular point.Set d := dim S. Suppose a(S) < 0, and that there exists an integer n with −n a(S) such that the Frobenius action Proof.Because n is an isolated singular point, each nonzero element of n has a power that is a test element, and Theorem 2.
) with the standard N-grading, and its p-th Veronese subring R := S (p) .Then: (1) The ring R is F-rational.
(2) The rings R ⊗ K K 1/p and R ⊗ K K are not F-injective, hence not F-rational.
(3) The enveloping algebra R ⊗ K R is not F-injective, hence not F-rational.
Proof.First consider the hypersurface ).The Jacobian criterion shows A x i is regular for each i, so A is normal by Serre's criterion.By inverting an appropriate multiplicative set in A, one obtains the ring S, which therefore is also normal.Since R is a pure subring of the finite extension ring S, it follows that R is normal and Cohen-Macaulay.
Note that S is not F-injective: set n to be the homogeneous maximal ideal of S; computing local cohomology H p n (S) using a Čech complex on the system of parameters x 1 , . . ., x p for S, the cohomology class where each α i is a nonnegative integer, ∑ α i p − 1, and α α α := (α 1 , . . . ,α p ).Since where the latter equality uses the pigeonhole principle.The elements t α 1 1 • • • t α p p of K, as α α α varies subject to the conditions above, are linearly independent over the subfield K p .It follows that for any nonzero K-linear combination η of the elements η α α α , one has F(η) = 0.This proves that the ring R is F-injective.
One may now use Corollary 2.2 to conclude that R is F-rational; alternatively, one can also argue as follows: Equation (3.1.1) shows that the image of [H p m (R)] −p under F lies in the K-span of the cohomology class , so it suffices to verify that µ does not belong to the tight closure of zero in H p m (R).This holds since no nonzero homogeneous form in R annihilates ), since it is a nontrivial linear combination of basis elements as in (2.0.3).However its image under the Frobenius action is which, of course, is zero.Lastly, for (3), write the enveloping algebra S ⊗ K S of S as with the N 2 -grading under which deg x i = (1, 0) and deg y i = (0, 1) for each i.Then Note that R ⊗ K R admits a standard grading; let M denote its homogeneous maximal ideal.
Then M(S ⊗ K S) is primary to the homogeneous maximal ideal N := (x 0 , . . ., x p , y 0 , . . ., y p ) of S ⊗ K S, and The cohomology class is nonzero since it is a nontrivial linear combination of basis elements as in (2.0.4); however, it is readily seen to be in the kernel of the Frobenius action.
Note that R ⊗ K K 1/p and R ⊗ K K in the previous theorem are not reduced: for example, is a nonzero nilpotent element.This gives an alternative proof of (2), since F-injective rings are reduced by [SZ,Remark 2.6].
In the examples provided by Theorem 3.1, the transcendence degree of K over F p increases with p; for the interested reader, the following theorem gets around this, though the proof is perhaps more technical: Theorem 3.2.Fix a prime integer p > 0. Let t be an indeterminate over the field F p and set K := F p (t).Consider the hypersurface with the standard N-grading, and set R := S (p) .Then: (1) The ring R is F-rational.
(2) The rings R ⊗ K K 1/p and R ⊗ K K are not F-injective, hence not F-rational.
(3) The enveloping algebra R ⊗ K R is not F-injective, hence not F-rational.
Proof.We begin with the hypersurface The Jacobian criterion shows that, up to radical, the defining ideal of the singular locus of A contains (w, x, y, z 1 , . . ., z p−1 ).The ring S is obtained from A by inverting an appropriate multiplicative set; it follows that S has an isolated singular point at its homogeneous maximal ideal n.In particular, S is normal by Serre's criterion.
Set m := k + ∑γi, and suppose that the above element belongs to the kernel of the Frobenius action.Then which, therefore, also belongs to the monomial ideal (3.2.4).But then ∑ α+β =k c p α x β p y αp (tx p+1 + xy p ) k ∈ x (k+1)p , y (k+1)p in the polynomial ring K[x, y].This implies that the coefficient of x kp+k y kp in the polynomial above must be zero, i.e., that Since c p α ∈ K p for each α, and k < [K p (t) : K p ] = p, this forces each c α to be zero.But then the element (3.2.2) is zero, so the map (3.2.1) is indeed injective as claimed.This completes the proof of (1).
For (2), let m denote the homogeneous maximal ideal of R, and let R denote either of R ⊗ K K 1/p or R ⊗ K K. Then   w 2 x 2 y ∏ For (3) use w ′ , x ′ , y ′ , z ′ i for the second copy of S, and proceed as in the proof of Theorem 3.1.Using M for the homogeneous maximal ideal of R ⊗ K R, the cohomology class is a nontrivial linear combination of basis elements as in (2.0.4), and is in the kernel of the Frobenius action on H 2p+2 M (R ⊗ K R).It follows that the ring R ⊗ K R is not F-injective.Theorem 1.1 follows readily from the results of this section: Proof of Theorem 1.1.Let K and R be as in Theorem 3.1 or 3.2, and let S := R ⊗ K K 1/p or R ⊗ K K.An example is then obtained after localizing at the homogeneous maximal ideals; note that the closed fiber is the field K 1/p or K in the respective cases.
and (R, m) −→ (S, n) is a flat local map such that S/mS is Cohen-Macaulay and geometrically F-injective over R/m, then S is F-injective; see also[En2, Theorem 4.3] Quinlan-Gallego was supported by the NSF RTG grant DMS #1840190 and NSF postdoctoral fellowship DMS #2203065, Simpson by NSF postdoctoral fellowship DMS #2202890, and Singh by NSF grants DMS #1801285, DMS #2101671, and DMS #2349623.
p y αp w (m+1)p belongs to the idealx (k+1)p , y (k+1)p , z (γ 1 +1)p 1 , . . ., z (γ p−1 +1)p p−1 S.Since w (m+1)p = w p−m w (p+1)m and 1 p − m p, x β p y αp tx p+1 + xy p + ring K[x, y, z 1 , . . ., z p−1 ].Bearing in mind that m = k + ∑ γ i , the terms in the multinomial expansion of (3.2.3) that include the monomial 1 , . . . ,γ p−1∑ α+β =k c p α x β p y αp (tx p+1 + xy p ) k ∏ i z (p+1)γ i i linear combination of basis elements as in (2.0.3).The ring R is not Finjective since under the Frobenius action on H p+1 m (R), this element maps to  1 is applicable.Since S is Gorenstein, each nonzero homogeneous element η of [H d can be chosen to be a product of elements of degree one, therefore η has a nonzero multiple s ′ η in[H d n (S)] −n has a nonzero multiple sη in the socle of H d n (S), which is the graded component [H d n (S)] a(S) .As S is standard graded, such a multiplier s ∈ S n (S)] −n .Since F(s ′ η) is nonzero, so is F(η).It follows that the Frobenius action on [H d n (S)] −n is injective, so Theorem 2.1 implies that the tight closure of zero in H d n (S) is contained in [H d n (S)] >−n .Set R := S (n) .The hypotheses −n a(S) < 0 give EXAMPLES Theorem 3.1.Fix a prime integer p > 0. Let t 1 , . . .,t p be indeterminates over the field F p and set K := F p (t 1 , . . .,t p ).Consider the hypersurface S maps to zero under the Frobenius action on H p n (S).We shall see that the Frobenius action on H p m (R), with m the homogeneous maximal ideal of R, is however injective.First note that [H ′ homogeneous in S, in which case degree considerations imply that s ′ ∈ R. To verify that the Frobenius action F on H p m (R) is injective, it suffices to prove the injectivity of F on the socle [H p m (R)] −p which, following (2.0.3), is the K-vector space spanned by the cohomology classes