Diagonal F-splitting and Symbolic Powers of Ideals

Let $J$ be any ideal in a strongly $F$-regular, diagonally $F$-split ring $R$ essentially of finite type over an $F$-finite field. We show that $J^{s+t} \subseteq \tau(J^{s - \epsilon}) \tau(J^{t-\epsilon})$ for all $s, t, \epsilon>0$ for which the formula makes sense. We use this to show a number of novel containments between symbolic and ordinary powers of prime ideals in this setting, which includes all determinantal rings and a large class of toric rings in positive characteristic. In particular, we show that $P^{(2hn)} \subseteq P^n$ for all prime ideals $P$ of height $h$ in such rings.


Introduction
Given an ideal I in a ring R, we define the n th symbolic power of I to be If I is radical, then one can think of I pnq as roughly being the ideal of elements in R vanishing to "order at least n" along the variety V pIq.For instance, if I is a divisorial ideal I " RpDq, then I pnq " RpnDq; see also [DDS `18, Proposition 2.14].One quickly sees from the definitions that I n is always contained in I pnq , but finding a more precise relationship between ordinary and symbolic powers of ideals is a subtle matter; see [DDS `18, GS21] for excellent overviews of this problem and its history.
Here we address the following question posed in [HKV09].
Question 1.1.Let pR, mq be complete local domain.Does there exist an integer C ě 1 such that p pCnq Ď p n for all p P Spec R and all integers n ě 1?
So far we know this is true, under mild assumptions, for isolated singularities (see [HKV09]), Segre products of polynomial rings (see [CRS20]), and certain Hibi rings (see [PST18]).We also know that this property descends under finite extensions of rings (see [HKV15]).See also Walker's work on monomial ideals and height 1 primes in toric rings [Wal18].
Rings with an affirmative answer to this question are said to have the uniform symbolic topology property, or USTP for short.This is because one can use the symbolic powers of an ideal p to define a topology on R, which is generated by cosets x `ppnq for x P R and n ě 1. Compare this to the p-adic topology, which is generated by the cosets x `pn .Schenzel showed that these two topologies are equivalent, meaning @a P N : Db P N : p pbq Ď p a whenever R is a Noetherian domain; see [Sch85].Swanson later showed that, in this case, one can find a number Cppq, depending on p, where p pCppqnq Ď p n for all n; see [Swa00].In this sense, the two topologies are said to be linearly equivalent.Ein, Lazarsfeld, and Smith kicked off the study of USTP when they showed that one can find a uniform C such that p pCnq Ď p n for all p P Spec R and all n ě 1 whenever R is a regular finitely generated C-algebra; see [ELS01].Analogous results were later shown in positive characteristic by Hochster-Huneke [HH02] and in mixed characteristic by Ma-Schwede [MS18].
Test ideals of pairs, and their characteristic zero/mixed characteristic analogs, have been an important tool in the study of USTP; see [ELS01,HY03,Tak06,CRS20,MS18,Mur21].Each of those proofs relies on some version of the "subadditivity formula," τpa s b t q Ď τpa s qτpb t q.
In regular rings, this formula holds as written, but in other situations one must account for its failure.In [Tak06], Takagi multiplies by the Jacobian ideal.In [CRS20], we use so-called "diagonal Cartier algebras" and show that a ring has USTP if it is diagonally F-regular.Diagonal F-regularity is a strengthening of diagonal F-splitting (and of strong F-regularity), analogously to the way strong F-regularity is a strengthening of F-splitting.So far, however, interesting examples of diagonally F-regular rings have been quite difficult to find.
In this work, we prove that if a ring is only strongly F-regular and diagonally F-split, then it satisfies a weaker form of subadditivity which is enough for USTP.Namely, we show the following.
Theorem A. Let k be an F-finite field of positive characteristic, and let R be a strongly F-regular, diagonally F-split k-algebra essentially of finite type.Let a Ď R be an ideal, and let s and t be positive real numbers with s `t P Z. Then we have a s`t Ď τpa s´ε qτpa t´ε q for all ε P p0, mints, tus.
From this, we follow a line of argument similar to that used in [CRS20] to get various bounds on symbolic powers; see Theorem 4.4 for the strongest statement.In particular, we have the following.

Theorem B.
In the same setting as Theorem A, let p Ď R be a prime ideal and h " ht p. Then p p2hnq Ď p n for all n ą 0.
Observe that Theorem B implies p p2dnq Ď p n for all p and all n, where d is the dimension of R. By work of Ramanathan [Ram87] and Lauritzen-Raben-Pedersen-Thomsen [LRPT06], the hypotheses of these theorems are satisfied by any affine subset of a Schubert subvariety of G{Q, where G is a connected and simply connected linear algebraic group over an algebraically closed field of positive characteristic, and Q is a parabolic subgroup of G containing a Borel subgroup B. In particular, these theorems hold in any determinantal ring krX mˆn s{I t , where X mˆn is an m ˆn matrix of indeterminates, with m ď n, and I t is the ideal of size t minors of X, where t ď m. (We show that k need not be algebraically closed here in Corollary 5.4.)These theorems also hold in diagonally F-split toric rings, which were characterized in [CHP `18].This includes, in particular, all Hibi rings (cf.[PST18, Theorem 3.7]).
By standard "reduction mod p" techniques, the symbolic powers containment of Theorem B holds in the corresponding determinantal and toric rings over fields of characteristic zero; see [HH99, Section 2] (see also [CRS20,Section 6]).

Background on test ideals
We begin with some background on positive-characteristic commutative algebra.For a more in-depth treatment on this material, see the surveys [SZ15] and [ST12], or the recent lecture notes [Hoc22].
Let R be a ring of characteristic p, where p is prime.Then the Frobenius map, is a ring homomorphism.So are its iterates F e pxq x p e for all e P Z ě0 .We write F e ˚to denote restriction of scalars along F e .More precisely, F e ˚is a functor from R-modules to R-modules.If R happens to be a domain, then we can fix an algebraic closure K of fracpRq, and each element of R will have a (unique!) pp e q th root in K.In this case, F e ˚R is nothing but R 1{p e , the set of these pp e q th roots with the usual R-module structure.We say that R is F-finite if F e ˚R is finitely generated over R for some (equivalently, all) e ą 0.
An F-splitting of R is a map of R-modules F e ˚R Ñ R sending F e ˚1 to 1.Such a map splits the natural R-module map R Ñ F e ˚R given by r Þ Ñ rF e ˚1 " F e ˚rp e -in this sense, it splits the Frobenius map.We say R is F-split if it admits an F-splitting F e ˚R Ñ R for some (equivalently, all) e ą 0. Note that this is equivalent to the existence of an R-linear surjection F e ˚R ↠ R. F-split rings are automatically reduced.

Global assumptions
For the rest of this section, R denotes an F-finite, reduced, and Noetherian ring of characteristic p ą 0.

Test ideals and strong F -regularity
Let R ˝be the set of elements not in any minimal prime of R. The reader should be warned that the big test ideal is sometimes denoted by τpRq in the literature, such as in [HT04], whereas the notation τpRq is reserved for the "ordinary" test ideal.These two ideals are conjectured to be the same, however.In this paper we will only work with big test ideals and big test elements.Thus, we will follow earlier authors in omitting the qualifier "big" throughout.

Test ideals of pairs
Test ideals are, in a precise sense, the positive-characteristic analog of multiplier ideals from birational geometry.However, birational geometers have long worked with multiplier ideals not just of schemes X, but of pairs pX, Zq, where Z is a formal Q-linear combination of subschemes of X.This inspired Hara and Yoshida to define the test ideals of pairs: if a is an ideal of R not contained in any minimal prime, and t ě 0 is a real number, we say that c P R ˝is a test element of the pair pR, a t q if, for all d P R ˝, there exist some e ą 0 and some ϕ P Hom R pF e ˚R, Rq with ¯.
Then we can define the test ideal of the pair pR, a t q to be τpR, a t q " where c is any test element of the pair pR, a t q.We will denote the test ideals by τpa t q when the ambient ring R is clear from context.These definitions can certainly feel quite daunting and unmotivated to the uninitiated.Fortunately, all of the facts we will need about test ideals can be summarized as follows.
Proposition 2.1 (Properties of test ideals).Let a be an ideal not contained in any minimal prime of R, and let t ě 0. Then there exists a test element of the pair pR, a t q; see [HT04, Lemma 2.1], cf.[Sch11, Proposition 3.21].Further, we have the following: For all n P Z ě0 , τppa n q t q " τpa nt q; see [HY03, Remark 6.2].(c) If a has a reduction generated by r elements, then τpa t`n q " a n τpa t q for all n P Z ě0 and all t ě r ´1; Remark.Recall that an ideal J Ď I is called a reduction of I if JI n " I n`1 for all n sufficiently large.In Noetherian rings, this is equivalent to saying J " I, where I denotes the integral closure of I. See [SH06, Chapter 8] Proof.We prove part (d), for the reader's convenience.Let d be a test element of the pair pR, a t q.Then, since c is a big test element of R, there exists a ψ : F Note that rtp e sp e 0 is an integer and that rtp e sp e 0 ě tp e p e 0 ą t `pe`e 0 ´1˘.
It follows that rtp e sp e 0 ě rtpp e`e 0 ´1qs.Finally, since the map Using Proposition 2.1, we can prove the following containment, which is key in establishing the connection between test ideals and symbolic powers.
Lemma 2.2.Let p P Spec R be an ideal of height h with infinite residue field R p {pR p ( for instance, p can be any nonmaximal prime ideal of R).Then τppp pN q q t q Ď p ptN tu´h`1q for all integers N ą 0 and all real numbers t ą 0 such that N t ě h ´1.
Proof.We can check this containment locally at p. Localizing, we get τpR, pp pN q q t qR p " τpR p , pp pN q R p q t q " τpR p , pp N R p q t q " τpR p , ppR p q N t q.
As the residue field at p is infinite, we know that pR p has a reduction generated by h elements (see [SH06, Theorem 8.3.7,Corollary 8.3.9]).Since test ideals localize, it follows from Proposition 2.1, part (c), that τpR p , ppR p q N t q Ď τpR p , ppR p q tN tu q " p tN tu´h`1 τpR p , ppR p q h´1 q Ď p tN tu´h`1 R p , as desired.□ D. Smolkin 6 D. Smolkin

Tensor products
Let A be a ring of characteristic p and R an A-algebra.For all R-modules M and N , we have a natural By [Smo20, Lemma 3.9], we have a natural isomorphism

Cartier algebras, compatible splittings, and diagonal F -splitting
In this section we define diagonal F-splitting and the closely related diagonal Cartier algebra.Using the language of diagonal Cartier algebras allows us to state a base-change lemma for diagonal F-splitting (see Corollary 5.4).Without this base-change lemma, we would need to assume that k is algebraically closed in Theorem 5.1.

Cartier algebras
Cartier algebras give us a unified framework for discussing test ideals in a much more general setting than just pairs pR, a t q; see [Sch11].We review some of the basics of this theory.
Let R be a ring of characteristic p.For each e, define C e pRq " Hom R pF e ˚R, Rq.Then C pRq " We define a Cartier algebra on R to be a graded subring D Ď C pRq with degree zero piece D 0 " Hom R pR, Rq.
We say that a Cartier algebra D is F-split if D e contains an F-splitting for some e ą 0. Note that this is equivalent to the existence of a surjective map ϕ P D e for some e: if ϕ is surjective, then ϕpF e ˚xq " 1 for some x P R. Letting m x P Hom R pR, Rq be "multiplication by x," we see that ϕ ¨mx P D e and pϕ ¨mx qpF e ˚1q " 1.Further, if ϕ P D e is an F-splitting, then ϕ n P D ne is an F-splitting for all n ą 0.

Splittings compatible with an ideal
One sort of Cartier algebra, which is particularly relevant to us, is the Cartier algebra of maps compatible with a given ideal.Given a map ϕ : F e ˚R Ñ R and an ideal I Ď R, we say ϕ is compatible with I if ϕpF e ˚I q Ď I.

If we set
C e pR, I ö q " tϕ P C e pRq | ϕpF e ˚I q Ď Iu , then one quickly checks that C pR, I ö q À e C e pR, I ö q is a Cartier algebra on R; see [Smo20, Proposition 3.4].We call this the Cartier algebra on R of maps compatible with I.The map ι e d is a lift of the natural map ι e d : F d ˚pR{I q Ñ F e ˚pR{I q, so ψ ˝ιe d is an element of C d pR, I ö q {I whenever ψ is an element of C e pR, I ö q {I .By the same argument as before, it follows that C e pR, I ö q {I contains an F-splitting for each e ą 0 whenever C pR, I ö q {I is F-split.We further have the following.
Lemma 3.1.Let R be a ring of characteristic p and I an ideal of R. Consider the following two statements: Proof.If ϕ P C e pR, I ö q is an F-splitting, then ϕ is an F-splitting in C e pR, I ö q {I .For the other direction, suppose that ψ P C e pR, I ö q {I is an F-splitting of R{I and suppose that α is an F-splitting of R. Then ψ has some lift p ψ P C e pR, I ö q with p ψpF e ˚1q P 1 `I.Let i " p ψpF e ˚1q ´1.Then p ψ ´iα is an F-splitting and p ψ ´iα P C e pR, I ö q. □

Diagonal F -splitting
Let A be an F-finite ring, and let R be an A-algebra essentially of finite type.Let µ A be the multiplication map, Then R is defined to be diagonally F-split over A if R b A R is F-split compatibly with ker µ A .We define the second diagonal Cartier algebra on R over A to be the Cartier algebra (1) This terminology comes from the geometric picture, where ϕ can be seen as the restriction of ϕ to the subscheme V pIq.The upshot is that if R is diagonally F-split over A, then the Cartier algebra D p2q pR{Aq is F-split, which implies that for all e ą 0, we can find F-splittings ϕ and ϕ making the following diagram commute: Then, by Lemma 3.1, R is diagonally F-split over A if and only if D p2q pR{Aq is F-split.This amends an error in [CRS20, Remark 3.3], where we overlooked the necessity that Frobenius be surjective on the base ring.
Remark 3.3.Diagonal F-splitting was first studied by Ramanathan in [Ram87] (cf.[BK05, Section 1.5]), where he defined the notion for projective varieties over an algebraically closed field.Ramanathan was interested in diagonally F-split varieties because of the cohomological vanishing results they enjoy.For instance, given any ample line bundles L and M on a diagonally F-split projective variety X, the map is surjective.In particular, the section ring À ně0 H 0 pX, L n q is generated in degree 1.We will come back to the global definition of diagonal F-splitting in Section 5.1.

The main theorems
Lemma 4.1.Let R be a Noetherian, F-finite, and strongly F-regular domain.Then for all ψ : F e ˚R Ñ R, all ideals J Ď R, and all s ą 0, we have ψpF e ˚J tsp e u q Ď τpJ s´1 p e q.
Proof.We have an inequality tsp e u ě rsp e s ´1 " rsp e ´1s "

Rˆs
´1 p e ˙pe V , and so ψpF e ˚J tsp e u q Ď ψpF e ˚J rps´1 p e qp e s q.Since R is strongly F-regular, we know that 1 is a test element of R. Then the conclusion follows from Proposition 2.1, part (d).□ Theorem 4.2.Let k be an F-finite field of positive characteristic, and let R be a strongly F-regular k-algebra essentially of finite type with D p2q pR{kq F-split.Let a Ď R be an ideal, and let s, t ą 0 with s `t P Z. Then we have a s`t Ď τpa s´ε qτpa t´ε q for all ε P p0, mints, tus.
Proof.Pick e such that p e ą 1{ε.Since D p2q pR{kq is F-split, there is an F-splitting ϕ : and so we have a s`t Ď τpa s´1 p e q ¨τpa t´1 p e q Ď τpa s´ε q ¨τpa t´ε q. □ The following bit of arithmetic comes up in the proof of the next theorem.We state it on its own for the sake of clarity.Lemma 4.3.For all x P R, there exists some ε ą 0 such that x ´ε ą rxs ´1.
Proof.If x is an integer, then rxs " x and the statement is obvious.Otherwise, rxs ´1 " txu ă x and the statement follows.□ Theorem 4.4.Let k be an F-finite field of positive characteristic, and let R be a strongly F-regular k-algebra essentially of finite type, with D p2q pR{kq F-split.Let p Ď R be a prime ideal, and let h be the height of p. Then we have (4.1)p pN q Ď p prN ss´hq p prN p1´sqs´hq for all N ą 2h and all s P ´h N , 1 ´h N ¯.Further, we have for all n ě 1.
Proof.Note that we must have N ą 2h for the interval ´h N , 1 ´h N ¯to be nonempty, and that the numbers N s ´h and N p1 ´sq ´h are positive.Symbolic powers and ordinary powers of maximal ideals are the same, so we may assume that p is not maximal.Given any such N and s, we have p pN q Ď τ ˆ´p pN q ¯s´ε ˙τ ˆ´p pN q ¯1´s´ε ˙, by Theorem 4.2.Note that N ps ´εq ´h `1 ą rN ss ´h for ε small enough, by Lemma 4.3.As rN ss ´h is an integer, this means that tN ps ´εqu ´h `1 " tN ps ´εq ´h `1u ě rN ss ´h.
It follows from Lemma 2.2 that τ ˆ´p pN q ¯s´ε ˙Ď p ptN ps´εqu´h`1q Ď p prN ss´hq .

Applications to determinantal rings
Our goal in this section is to explain how Theorems 4.2 and 4.4 apply to determinantal rings.In particular, we show the following.
Theorem 5.1.Let k be an F-finite field and X mˆn an m ˆn matrix of indeterminates over k.Choose integers p ď m and q ď n, as well as integers Let I be the ideal generated by the size r i minors of the first u i rows of X as well as the size s j minors of the first v i columns of X, for all i " 1, . . ., p and j " 1, . . ., q (with the convention that any size t minor of a size u ˆv matrix is 0 if t ą mintu, vu).Then R " krX mˆn s{I is strongly F-regular and D p2q pR{kq is F-split.In particular, we have p p2hnq Ď p n for all p P Spec R of height h and for all n.
Proving this theorem is mostly a matter of collecting results found in the literature.The argument goes as follows: Generalized Schubert varieties over algebraically closed fields are known to be globally F-regular and diagonally F-split, and any open affine subscheme of a globally F-regular (respectively, diagonally F-split) scheme is strongly F-regular (respectively, diagonally F-split).Each ring described in Theorem 5.1 is an open affine subscheme of some Schubert subvariety of a Grassmannian, and thus these rings are strongly F-regular and their diagonal Cartier algebras D p2q pR{kq are F-split, at least if the base field k is algebraically closed.It is well known that strong F-regularity can be checked after changing the base field; we show in Corollary 5.4 that the F-splitting of D p2q pR{kq can be checked after base changing from k to any perfect field.Let us begin by reviewing some of this language.

Global definitions
Global F-regularity is, as the name suggests, a global notion of strong F-regularity: a projective variety X over an F-finite field is said to be globally F-regular if it admits an ample line bundle L such that the section ring À ně0 H 0 pX, L n q is strongly F-regular as a ring.This definition is due to Smith in [Smi00].There, she shows that the section ring of any ample line bundle on a globally F-regular variety is strongly F-regular.Further, if X is a globally F-regular variety, then any affine subscheme of X is strongly F-regular (as a ring), though the converse can fail.
Next, we define what it means for a projective variety to be diagonally F-split.Let Y be a scheme of characteristic p.We define the absolute Frobenius morphism on Y to be the map F : Y Ñ Y which is the identity on topological spaces and for which the map on structure sheaves is given by for all open U Ď Y .We say that Y is F-split if there is a map ϕ : F ˚OY Ñ O Y which splits F # .Given any closed subscheme Z Ď Y with ideal sheaf I Z , we say Y is F-split compatibly with Z if there exists such a splitting which further satisfies ϕpF ˚IZ q Ď I Z .
If X is a projective variety over a field k of characteristic p, then the diagonal subscheme ∆ X{k Ď X ˆk X is closed.In the literature, such a variety X is defined to be diagonally If we take G to be GL n pkq, B to be the subgroup of upper-triangular matrices, and P to be the subgroup of G given by P " ␣ pg ij q P GL n pkq | g ij " 0 whenever i ą d and j ď d ( for some fixed d ă n, then G{P is Gr d pk n q, the Grassmannian of d-dimensional subspaces of k n .The Schubert subvarieties of Grassmannians are the so-called "classical" Schubert varieties.ρpMq " pδ 1,2,...,m pMq : ¨¨¨: δ n`1,n`2,...,n`m pMqq, in some fixed order.Note that one of these minors must be nonzero, as M has maximal rank.Further, one checks that two matrices M, N P M have the same row-span precisely when ρpMq " ρpN q.Thus we can identify the Grassmannian Gr m pk m`n q with the image of ρ.This is known as the Plücker embedding. We get a partial ordering ĺ on the set of minors δ b 1 ,.. In this way we also have a partial ordering on the coordinates of P N ´1.Then the Schubert subvarieties of Gr m pk m`n q are the subvarieties cut out by certain subsets of these coordinates of P N ´1, namely the subsets which form a "poset ideal cogenerated by a single element" under this partial ordering; see [BV88,Theorem 5.4].It turns out that Gr m pk m`n q intersected with any standard open affine Dpx i q Ď P N ´1 k is a copy of mn-dimensional affine space.For instance, the inverse image of Dpx N q X Gr m pk m`n q under ρ is the set of matrices in M where the determinant of the m rightmost columns is nonzero.Up to row operations, any such matrix can be uniquely written in the form (5.1) , and conversely any choice of α ij gives an element of ρ ´1 pDpx N qq.Now, the key observation is that every minor of the m ˆn matrix `αij ˘equals a maximal minor of the full m ˆpn `mq matrix in Equation (5.1).This means that for any collection of minors δ 1 , . . ., δ r P krX mˆn s, one can find a corresponding collection of coordinates x i 1 , . . ., x i r on P N ´1 so that Spec pk rX mˆn s {pδ 1 , . . ., δ r qq " V px i 1 , . . ., x i r q X Gr m pk m`n q X Dpx N q.
All that is left to check is that the coordinates vanishing along the varieties in Theorem 5.1 always form a poset ideal cogenerated by a single element, which is done in [BV88, Theorem 5.3].

A base-change lemma for diagonal F-splitting
The above work proves Theorem 5.1 whenever k is algebraically closed.For the general case, we just need to show that the F-splitting of D p2q pR{kq can be checked after base change to an algebraically closed field.We prove a more general fact, which we hope might be of independent use.Proposition 5.3.Let A be an F-finite field, and let R be an A algebra essentially of finite type.Suppose B is a perfect field extension of A, and let If A is perfect, then the converse holds as well.
Proof.Suppose that C pR B , I B ö q {I B is F-split.We wish to find a surjective map ψ P Hom R pF e ˚pR{I q, R{Iq with a lift to R. Let π denote the map π : R ↠ R{I, so that we have F e ˚π : F e ˚R ↠ F e ˚pR{I q.Let pF e ˚πq ˚be the functor Hom R p´, R{Iq applied to F e ˚π, and let π ˚be the functor Hom R pF e ˚R, ´q applied to π. Observe that a map ψ P Hom R pF e ˚pR{I q, R{Iq has a lift to R if and only if pF e ˚πq ˚pψq " ψ ˝Fe ˚π is in the image of π ˚.Further, pF e ˚πq ˚pψq is surjective precisely when ψ is.Thus we wish to show that ImppF e ˚πq ˚q X Impπ ˚q contains a surjection, or equivalently that the "evaluation at 1" map is surjective.To emphasize, this is the key point: C pR, I ö q {I is F-split precisely when the map ev 1 : ImppF

e 0 ˚R 0 ˚dq ¯Ď ϕ ´Fe ˚ψpF e 0
Ñ R with ψpF e 0 ˚dq " c.It follows that ϕpF e ˚ca rtp e s q " ϕ ´Fe ˚artp e s ψpF e ˚artp e sp e 0 dq ¯.

F e`e 0 ˚R
ÝÑ R, F e`e 0 ˚x Þ ÝÑ ϕ `Fe ˚ψpF e 0 ˚xq ȋs R-linear, we have the desired containment.□
The rings described in Theorem 5.1 are open subschemes of Schubert subvarieties of Gr m pk m`n q; see [BV88, Theorem 5.5].For the reader's convenience, we will sketch a proof of this beautiful fact in the remainder of this subsection.A more detailed treatment is found in [BV88, Chapters 4 and 5].Let M Ď A mpm`nq k be the set of mˆpm`nq matrices with full rank.Then every m-dimensional subspace of k m`n can be expressed as the row-span of some M P M .Note that two different matrices have the same row-span if and only if one can be obtained from the other by a sequence of elementary row operations.Given any list of integers 1 ď b 1 ă ¨¨¨ă b m ď n `m, let δ b 1 ,...,b m : M Ñ k be the function sending a matrix M to the determinant of the submatrix of M obtained by taking columns b 1 , b 2 , . . ., b m .Note that this is a maximal minor of M, and there are `m`n m ˘possible choices of such maximal minors.Set N " `m`n m ȃnd consider the map ρ : M Ñ P N ´1 sending each matrix to a list of its maximal minors, .,b m by declaring δ b 1 ,...,b m ĺ δ c 1 ,...,c m ðñ b i ď c i for i " 1, . . ., m.

Proof.
By definition, the Cartier algebraC pR B b B R B , ker µ B ö q {ker µ B is F-split, where µ B is the multiplication map µ B : R B b B R B Ñ R B .But R B b B R B " R b A R b A B,and we can obtain µ B by applying the functor ´bA B to the multiplication map µ A : R b A R Ñ R. By faithful flatness, we see that ker µ B " pker µ A q b A B. Then the result follows from Proposition 5.3.□ For any R-module M, we have F e ˚M " tF e ˚m | m P Mu, where addition and scalar multiplication are defined by F e ˚pr p e mq for all r P R and m, n P M. For any R-linear map ϕ : M Ñ N , the map F e ˚ϕ : F e ˚M Ñ F e Two important notions from Hochster and Huneke's celebrated theory of tight closure are big test elements and big test ideals.A big test element is defined to be any c P R ˝such that, for all d P R ˝, there exist e ą 0 and ϕ P Hom R pF e ˚R, Rq with ϕpF e ˚dq " c.It is far from obvious, but big test elements exist in any F-finite, reduced, Noetherian ring, and the ideal generated by all big test elements of R is called the big test ideal of R, denoted by τpRq.It is given by given by F e ˚m b A F e ˚n Þ Ñ F e ˚pm b A nq.For instance, if A and R are domains, then F e ˚pR b A Rq " R 1{p e b A 1{p e R 1{p e , and this is a homomorphic image of F e ˚R b A F e ˚R " R 1{p e b A R 1{p e .If A " A p (i.e., the Frobenius map A Ñ A is surjective), then Θ is an isomorphism.If A is an F-finite field and R is finitely generated over A, then maps F e ˚pR b A Rq Ñ R b A R can be expressed in terms of maps F e ˚R Ñ R, as follows.Lemma 2.3.Let k be an F-finite field and R a k-algebra essentially of finite type.Then we have a natural inclusion Hom Rb k R pF e ˚pR b k Rq, R b k Rq Ď Hom R pF e ˚R, Rq b k Hom R pF e ˚R, Rq.Proof.We have a natural surjection F e ˚R b k F e ˚an | a P A, m P M, n P N y Ď F e ˚M b A F e ˚N .˚R ↠ F e ˚pR b k Rq, which induces an inclusion One special property of this Cartier algebra is that maps in C e pR, I ö q restrict to maps in C d pR, I ö q when d ă e, in the following sense: for any d ă e, we have a natural map ι e d : F d ˚R Ñ F e ˚R given by ι e d pF d˚xq " F e ˚xp e´d .Observe that ι e d pF d ˚I q Ď F e ˚I .Thus, for any ψ P C e pR, I ö q, we have ψ ˝ιe d P C d pR, I ö q.Note that if R is a domain, then ψ ˝ιe d is literally the restriction of ψ : R 1{p e Ñ R to R 1{p d .We say that R is F-split compatibly with I if the Cartier algebra C pR, I ö q is F-split.If this is the case, then C d pR, I ö q contains an F-splitting for all d ą 0. Indeed, for any d ą 0, we can find some e ą d and an F-splitting ψ P C e pR, I ö q.Then ψ ˝ιe d is an F-splitting in C d pR, I {I is a Cartier algebra on R{I; see [Smo20, Proposition 3.2].We call this the restriction (1) of C pR, I ö q to R{I.Put another way, given a map ψ : F e ˚pR{I q Ñ R{I, we say that a mapp ψ : F e ˚R Ñ R is a lift of ψ to R if these maps fit into a commutative diagram ψ / / R{I,where the downward arrows are the canonical surjections.Then we get C e pR, I ö q {I " tψ P C e pR{Iq | ψ has a lift to Ru.
pa tsp e u b k a ttp e u qq " µ k p p ϕpF e ˚pa tsp e u b k a ttp e u qqq.
˚µk i ϕ 1,i b ϕ 2,i for some ϕ 1,i , ϕ 2,i P Hom R pF e ˚pa tsp e u b k a ttp e u qq Ď τpa s´1 p e q b k τpa t´1 p e q, split compatibly with ∆ X{k .More generally, if Y is any separated scheme over a base scheme S of characteristic p, we can define Y to be diagonallyF-split over S if Y ˆS Y is F-split compatibly with the diagonal subscheme ∆ Y {S Ď Y ˆS Y .Suppose X is a diagonally F-split S-scheme and U is an open subscheme of X.Then U is diagonally F-split over S. Proof.The diagonal ∆ U {S Ď U ˆS U is the same as ∆ X{S X pU ˆS U q. Since X ˆS X is F-split compatibly with ∆ X{S , it follows that U ˆS U is F-split compatibly with ∆ U {S , by [BK05, Lemma 1.1.7].Let G be a connected and simply connected semisimple linear algebraic group over a field k.Let B be a Borel subgroup of G and P Ě B a parabolic subgroup of G. Then B acts algebraically on G{P on the left with finitely many orbits.The closure of any one of these orbits is called a Schubert subvariety of G{P .If k has positive characteristic, then any Schubert subvariety X Ď G{P is globally F-regular; see [LRPT06, Theorem 2.2].If k is further algebraically closed, then any Schubert subvariety X Ď G{P is diagonally e ˚πq ˚q X Impπ ˚q ÝÑ R{I, ϕ Þ ÝÑ ϕpF e ˚1q is surjective.Because R B is faithfully flat over R, the map ev 1 is surjective precisely when ev 1 b R R B is, which is equivalent to pImppF e ˚πq ˚q X Impπ ˚qq b R R B containing a surjection.This same faithful flatness yieldspImppF e ˚πq ˚q X Impπ ˚qq b R R B " ImppF e ˚πq ˚q b R R B X pIm π ˚q b R R B " ImppF e ˚πq ˚bR R B q X Impπ ˚bR R B q " ImpppF e ˚πq b A Bq ˚q X Impπ b A Bq ˚,where ppF e ˚πq b A Bq ˚is the functor Hom R B p´, R B {I B q applied to pF e ˚πq b A B, and pπ b A Bq ˚is the functor Hom R B pF e ˚R b A B, ´q applied to π b A B. By assumption, there exists a surjection ψ : F e ˚pR B {I B q Ñ R B {I B which lifts to R B : Observe that we have a natural R B -linear map p α : F e ˚R b A B Ñ F e ˚RB given by F e ˚r b b Þ Ñ F e ˚pr b b p e q-this is just the composition of the natural maps F e ˚R b A B Ñ F e ˚R b A F e ˚B and F e ˚R b A F e ˚B Ñ F e ˚pR b A Bq described in Section 2. Similarly, we have a natural map α : F e ˚pR{I q b A B Ñ F e ˚pR B {I B q.Because B is perfect, both α and p α are surjective.These maps fit into a larger diagram R B {I B .Thus ψ ˝α ˝ppF e ˚πq b A Bq is a surjection in ImppF e ˚πq b A Bq ˚X Impπ b A Bq ˚, as desired.For the converse, suppose that A is perfect and C pR, I ö q {I is F-split.By the above argument, we have a surjection in ψ P ImppF e ˚πq b A Bq ˚X Impπ b A Bq ˚.Since B is perfect, the Frobenius map B Ñ F e ˚B is an isomorphism.Combining this with the assumption that A is perfect, we see that the maps α and p α are isomorphisms.Setting ψ " χ ˝pF e ˚π b A Bq, we see that χ ˝α´1 is an F-splitting in C pR B , I B ö q {I B .□ Corollary 5.4.Let A be an F-finite field, and let R be an A algebra essentially of finite type.Suppose B is a perfect field extension of A. If D p2q pR B {Bq is F-split, then D p2q pR{Aq is F-split.
˚pR B {I B q ψ / / R B {I B .α / / F e ˚pR B {I B q ψ / /