LECTURES AT UIC WENLIANG ZHANG

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1 LECTUES AT UIC WENLIANG ZHANG Throughout this lecture, p enotes a prime number an enotes a commutative noetherian ring of characteristic p. Definition 0.1. The Frobenius enomorphism of is the map f : r rp. Since has characteristic p, it follows from the binomial formula that (a+b) p = a p +b p an consequently f is a ring homomorphism, an so are its iterates f e for each integer e 1. We will use F e to enote the -moule that is the same as as an abelian group an whose -moule structure is efine via F e :. The Peskine-Szpiro functor, enote by F, is efine via F (M) := F M. The e-th iteration of F is enote by F e, which is clearly given by F e (M) = F e M. Exercise 0.2. Prove that (1) F e (l ) = l for all free moules l ; (2) F e (/I) = /I [pe] for each ieal I of ; (3) F e ( g) = g for each element g ; (4) F commutes with localization, i.e, F (M) p = Fp (M p ). Let a = (g 1,..., g n ) be an ieal of an ϕ : S be a ring homomorphism between commutative noetherian rings of characteristic p. Tensoring ϕ with the Čech complex Č(; g) prouces a map of complexes: Č(; g) Č(S; ϕ(g)). Therefore, there is a homomorphism of -moules H i a() H i ϕ(a)s (S). Specializing in the case when S = an ϕ = f, we have the Frobenius action on H i a(): H i a() H i f(a) () = Hi a(), where the equality follows from f(a) = (g p 1,..., gp n) which has the same raical as a. We will enote this Frobenius action by F. Example 0.3. Let = k[x 1,..., x ] (or = k[[x 1,..., x ]]) an a = (x 1,..., x ) where k is a fiel of characteristic p. The Frobenius action on H a () is given by where c k. [ c 1 ] [ xe x e 1 1 x pe 1 c p 1 x pe ],

2 Example 0.4. Let (, m) be a noetherian local ring an x 1,..., x be a system of parameters, where = im(). Then each element of Hm() r can be written as [ x e 1 1 x e ] for some r an positive integers e i. The Frobenius action on H m() is given by [ r 1 ] [ xe x e 1 x pe 1 r p 1 x pe ]. One may notice that, in the case when = k[x 1,..., x ], one has is a free moule over f() = k p [x p 1,..., xp ]; hence f : is flat. More generally, we have the following theorem ue to Kunz [Kun69]. Theorem 0.5 (Kunz). The following are equivalent: (1) is regular; (2) f e is a flat ring homomorphism for each e 1; (3) f e is a flat ring homomorphism for some e 1. Theorem 0.6. Assume that is regular an a is an ieal of. Then F e (H i a()) = H i a(). Proof. Since H i a() = lim n Ext i (/ a n, ), we have F e H i a() = F e lim n Ext i (/ a n, ) = lim n F e Ext i (/ a n, ) = lim Ext i (F e n / a n, F e ) since F e is flat = lim n Ext i (/(a [pe] ) n, ) by Exercise 0.2 = H i a [pe ]() = H i a() Alternatively, one may also prove it via the Čech complex characterization of Hi a(), using the facts that F e is flat an that F e ( g) = g. Exercise 0.7. Prove that, if is regular, then F (E) = E for each injective -moule E. 1. Vanishing of local cohomology moules Exercise 1.1. Let (, m) be a regular local ring of characteristic p. Ann (Ha()) i 0. Prove Ha() i = 0. Assume that Exercise 1.2. Let p be a prime ieal of a regular ring of characteristic p. Prove that p [pe] is p-primary. Theorem 1.3 (Peskine-Szpiro [PS73]). Let (, m) be a regular local ring of characteristic p of imension n an let I be an ieal of. Assume that epth(/i) c. Then H n i I () = 0 for i c 1. 2

3 Proof. Since epth(/i) c, we have Hm(/I) i = 0 for i c 1. By local uality, Ext n i (/I, ) = 0. By Theorem 0.5, we have Extn i (/I[pe], ) = F e Ext n i (/I, ) = 0. Hence H n i I () = lim e Ext n i (/I[pe], ) = 0. Example 1.4. Let = k[[x ij ]] with i = 1, 2 an j = 1, 2, 3 where k is a fiel. Let I be generate by the 2 2 minors of (x ij ). Then /I has epth 4 by [HE71]. When k has characteristic p, by Theorem 1.3, we have HI i () 0 if an only if i = 2. In contrast, when k has characteristic 0, one has HI 3 () 0 ([HS77, page 75]). Exercise 1.5. Let (, m) be a complete regular local ring an E be the injective hull of the resiue fiel / m. Denote the Matlis ual functor Hom (, E) by D( ). Prove that there is a functorial -moule isomorphism for all artinian -moules M. τ : D(F (M)) = F (D(M)) Let A be a noetherian commutative ring of characteristic p. Let A{f} be the subring of Hom Z (A, A) generate by multiplications by elements of A an f : A a ap A, or equivalently A{f} = A f (fa a p f). An A{f}-moule M is an A-moule M equippe with a Z-linear map f : M M such that f(am) = a p f(m) for all a A an m M. Let (, m) be a complete regular local ring an A be a homomorphic image of. Let M be an A{f}-moule. One can check that α : F (M) r m rf(m) M is an -moule homomorphism. Now, assume that M is artinian. Taking the Matlis ual of α an applying Exercise 1.5, we have an -moule homomorphism β = τ D(α) : D(M) F (D(M)), an hence we have a irect system of -moules: D(M) β F (D(M)) F (β) F 2 (D(M)) Definition 1.6 (Lyubeznik s H,A functor [Lyu97]). Let (, m) be a complete regular local ring an A be a homomorphic image of. For each artinian A{f}-moule, we efine H,A (M) := lim(d(m) β F (D(M)) F (β) F(D(M)) 2 ) Example 1.7. Let (, m) be a complete regular local ring of imension n an A be a homomorphic image of. Let I be the kernel of A. From our iscussion, we know that Hm(A) i is naturally an A{f}-moule. We woul like to unerstan H,A (Hm(A)). i By local uality, we have D(Hm(A)) i = Ext n i (/I, ) an hence the irect system Ext n i which is the same as (/I, ) β F (Ext n i (/I, )) F (β) F(Ext 2 n i (/I, )) Ext n i (/I, ) Extn i (/I[p], ) Ext n i (/I[p2], ) 3

4 whose irect limit is H n i I (). Therefore, H,A (Hm(A)) i = H n i I (). Theorem 1.8 (Theorem 1.1 in [Lyu06]). Let (, m) be a complete regular local ring of imension n an A be a homomorphic image of. Let I be the kernel of A. Then () = 0 if an only if the Frobenius action on Hm(A) i is nilpotent. H n i I Proof. First note that, since H n i i () = lim e Ext n i (/I[pe], ) an F(Ext e n i (/I, )) = Ext n i (/I[pe], ), we see that Hi n i () = 0 if an only if there is t such that ϕ t : Ext n i (/I, ) Extn i (/I[pt], ) is the 0 map. With notation as in Example 1.7, ϕ t = 0 if an only if F t 1 (β) F (β) β = 0. Since Ext n i (/I, ) is finitely generate (or equivalently Hi m(/i) is artinian), by Matlis uality, F t 1 (β) F (β) β = 0 if an only if the following composition of maps is 0 F(H t m(/i)) i F t 1 (α) F t 1 (Hi m(/i)) F (Hm(/I)) i α Hm(/I). i Writing F t (Hi m(/i)) as F F H i m(/i), one can check that α F t 1 (α)(r t r 1 z) = r t r p t 1 rpt 1 f t (z) for all r i F an z Hm(/I). i It is now straightforwar to check that α F t 1 (α) = 0 if an only if f t = 0, i.e. the Frobenius action f : Hm(/I) i Hm(/I) i is nilpotent. This completes the proof. In light of Theorem 1.8, we introuce the notion of F -epth: Definition 1.9. Let (A, m) be a noetherian local ring of characteristic p. The F -epth of A is the smallest integer j such that the Frobenius on H j m(a) is nilpotent. Exercise Prove that F -epth(a) im(a). As an application of Theorem 1.8, we will state (without proofs) a result on cohomological imension of an open subset of P n k, where k is a fiel of characteristic p. To this en, let s recall the efinition of cohomological imension. Definition Let Y P n k be a close subscheme an U be the complement of Y in P n k. The cohomological imension of U, enote by c(u), is the largest integer i such there is a quasi-coherent sheaf F on U such that H i (U, F) 0. emark Set = k[x 0,..., x n ] an I to be the homogeneous efining ieal of Y. Then it is prove in [Har68, pp ] that c(u) < c if an only if H j I () = 0 for all j > c. Theorem 1.13 (Corollary 5.4 in [Lyu06]). Let Y, U be the same as above. For each c 2, we have c(u) < n c if an only if the following 3 conitions hol: 4

5 (1) Y is geometrically connecte an im(y ) > 0; (2) F -epth(o Y,y ) c for every close point y Y ; (3) the Frobenius action on H i (Y, O Y ) is nilpotent for each 1 j c Associate primes an Bass numbers of local cohomology moules A question of Huneke [Hun92, Problem 4] asks whether local cohomology moules of Noetherian rings have finitely many associate prime ieals. The answer is negative in general; the first counterexample was given by Singh [Sin00, 4]. There are positive answers in some cases; for instance, when the ring is a regular ring of characteristic p. we want to iscuss the approach in [HS93]. ecall that, when A is a commutative noetherian local ring an M is an A-moule, the Bass number µ i (p, M) (with respect to a prime ieal p of A) is efine as µ i (p, M) := im κ(p) (Ext i A p (κ(p), M p )), where κ(p) = A p / p A p. Alternatively, let E i (M) enote the i-th item in the minimal injective resolution of M. One has a ecomposition E i (M) = λ Λ E(A/ p λ ). Then for each prime p, one has µ i (p, M) is the cainality of the set {λ Λ p λ = p}. Exercise 2.1. Prove that p Ass A (M) if an only if µ 0 (p, M) > 0. Exercise 2.2. Let (, m) be a noetherian local ring an let {I j } j Z>0 be a irect system of injective -moules with limit I. Assume that there is an integer µ such that each I j has at most µ copies of E(/ m). Prove I also has at most µ copies of E(/ m). Theorem 2.3 (Huneke-Sharp). Let (, m) be a regular local ring of characteristic p an I be an ieal of. Then µ i (p, H j I ()) µi (p, Ext j (/I, )) for each prime ieal p an all integers i, j. In particular, µ i (p, H j I ()) is finite. Proof. After localizing at p, we may an we o assume that p = m. Let 0 Ext j (/I, ) E0 E 1 be a minimal injective resolution of Ext j (/I, ). For each positive integer t, we will enote F t (Ej ) by E j t. By Exercise 0.7, we know that Ej t = E j for each t. Applying the exact functor F t to this injective resolution of Extj (/I, ), we get an exact sequence 0 Ext j (/I[pt], ) E 0 t E 1 t 5

6 which is a minimal injective resolution of Ext j (/I[pt], ). The functoriality of F an stanar results on injective resolutions prouce a commutative iagram 0 Ext j (/I, ) E 0 E j 0 Ext j (/I[p], ) E 0 1 E j Ext j (/I[pt], ) E 0 t E j t. Taking irect limits an keeping in min that a irect of injective moules is still injective, we get an exact sequence 0 H j I () E0 E 1 which is a minimal injective resolution of H j I (). It follows from Exercise 2.2 an E j t = E j that µ i (p, H j I ()) µi (p, Ext j (/I, ))... As an immeiate consequence of Theorem 2.3 an Exercise 2.1, we have: Theorem 2.4 (Huneke-Sharp). Let be a regular ring of characteristic p an I be an ieal of. Then Ass (H j I ()) Ass (Ext j (/I, )). In particular, Ass (H j I ()) is a finite set. There are still many interesting question regaring finiteness properties of local cohomology moules in characteristic p when is not regular. Before we procee to iscussions of non-regular rings, I like to iscuss Lyubeznik s F -moule theory, which is a generalization of what we have seen so far. 3. F -moules Definition 3.1. Let be a regular ring of characteristic p. An F -moule is an -moule M equippe with an -linear isomorphism θ M : M F (M). A homomorphism between F -moules (M, θ M ) an (N, θ N ) is a homomorphism ϕ : M N such that the following is a commutative iagram M N ϕ θ M F (M) F (ϕ) θ N F (N). 6

7 Example 3.2. It follows from Exercise 0.2 an Theorem 0.6 that l, g, an H j a() are F -moules. From what we have seen, the morphism Ext j (/ a, ) F (Ext j (/ a, )) plays a key role in unerstaning Ha(). j The analog of such a morphism for an F -moule is calle a generating morphism. Definition 3.3. Let be a regular ring of characteristic p an M be an F -moule. A generating morphism of M is an an -moule homomorphism M F (M), with some -moule M, such that such that M is the irect of the top row of the following commutative iagram: M β F (M) F (β) F 2 2 (M)F (β) β F (β) F 2 (β) F (M) F (β) F 2 (M) F 2 (β) F 3 (M)F 3 (β) an the structure isomorphism θ : M F (M ) is inuce from the vertical arrows in the iagram. emark 3.4. Since tensor prouct commutes with irect limits, so the irect limit of the bottom row is inee F (M ). It is clear that each F -moule M has a generating morphism, the structure isomorphism. Example 3.5. The morphism Ext j (/ a, ) F (Ext j (/ a, )) is clearly a generating morphism for the F -moule Ha(). j Given f, there is a map of Koszul complexes K (f; ) = 0 0 f p 1 K (f p ; ) = 0 f p 0 where K enotes the Koszul complex. Let f = f 1,..., f n be a sequence of elements of. egaring K (f; ) as the tensor proucts one obtains a map of complexes an inuce maps on cohomology moules K (f 1 ; ) K (f n ; ), K (f; ) K (f p ; ), H j (f; ) β H j (f p ; ), where a is the ieal generate by f. Since H j a() is the irect limit of {H j (f pt ; )} t an H j (f; ) = F (H j (f p ; )), the composition H j (f; ) H j (f p ; ) = F (H j (f p ; )) is also a generating morphism of H j a(). Note that H j a() is generate by a finitely generate -moule Ext j (/ a, ) (or H j (f; )) via the generating morphism Ext j (/ a, ) F (Ext j (/ a, )) (or Hj (f; ) H j (f p ; )). This leas to: 7 f

8 Definition 3.6. An F -moule M is calle F -finite if M amits a generating morphism β : M F (M) with M a finitely generate -moule. If, in aition, β is injective, M (or β(m)) is calle a root of M an β is calle a root morphism. Theorem 3.7. Every F -finite moule has a root. Proof. Let M be an F -finite moule with a generating morphism β : M F (M) with M a finitely generate -moule. Let β i : M F i (M) enote the composition M β F (M) F (β) F i 1 (β) F i (M). It is clear that ker(β i) ker(β i+1 ) an hence ker(β i ) = ker(β i+1 ) = for i 0. We claim that Im(β i ) is a root of M. Let f enote the composition of Im(β i ) Im(β i+1 ) an Im(β i+1 ) F (Im(β i )). Since F is exact, we have Im(F t (β i)) = F t (Im(β i)). Hence the irect system M β F (M) F (β) inuces a irect system Im(β i ) f F (Im(β i )) F (f) F 2 (Im(β i )) F 2 (f) Then we have a commutative iagram of irect systems M β F (M) F (β) F 2 (M) F 2 (β) β i Im(β i ) F (β i ) F 2 (β i) f F (Im(β i )) F (f) F 2 (Im(β i)) F 2 (f) Since vertical maps are surjective, the inuce map on the irect limits is surjective. The map on the irect limits is also injective since the kernel of each vertical map eventually goes to 0 in the irect system. Hence the irect limit of the bottom system is also M. Since Im(β i ) is a finite -moule an f is injective, Im(β i ) is a root of M. Example 3.8. H j a() has a root since it is an F -finite moule. One esirable feature of F -moule is the following. Theorem 3.9 (Theorem 1.4 in [Lyu97]). inj. im (M ) im (Supp (M )) for each F -moule M. emark An immeiate consequence of Theorem 3.9 is that, if m is a maximal ieal of, then Hm(H i a()) j (or more generally, HmH i j 1 a 1 Ha js s ()) is a irect sum of some copies of E(/ m). This leaves the following question open: let = k[x 1,..., x n ] be the polynomial ring over a fiel k with the usual graing (i.e. eg(x i ) = 1) an a 1,..., a s be grae ieals of. Set m = (x 1,..., x n ). Is it true that HmH i j 1 a 1 Ha js s () = E(/ m) t for some integer t as grae moules? Or, equivalently, are all the socle elements of HmH i j 1 a 1 Ha js s () have egree n? We will come back to this question when we iscuss D-moules. The following is a generalization of Theorem 2.3. Theorem 3.11 (Theorem 2.11 in [Lyu97]). Let M be an F -finite moule an M be a root of M. Then µ j (p, M ) µ j (p, M) 8

9 for each prime ieal p an each integer j. Exercise Let p be a prime ieal of. Assume that M is an -moule such that H j p(m) p are injective for all j. Prove that Exercise Prove Theorem µ j (p, M) = µ 0 (p, H j p(m)). Surprisingly, local cohomology moules, though may not be finitely generate or artinian as -moules, will have finite length in the category of F -moules. Theorem If is a noetherian regular ring essentially of finite type over a regular local ring of characteristic p, then every F -finite moule has finite length in the category of F -moules. 4. D-moules Let be a commutative ring. Z-linear ifferential operators on are efine inuctively as follows: for each r, the multiplication by r map r : is a ifferential operator of orer 0; for each positive integer n, the ifferential operators of orer less than or equal to n are those aitive maps δ : for which the commutator [ r, δ] = r δ δ r is a ifferential operator of orer less than or equal to n 1. If δ an δ are ifferential operators of orer at most m an n respectively, then δ δ is a ifferential operator of orer at most m + n. Thus, the ifferential operators on form a subring D() of En Z (). When is an algebra over a commutative ring A, we efine D(, A) to be the subring of D() consisting of ifferential operators that are A-linear. Note that D(, Z) = D(). Example 4.1. Let = A[x 1,..., x ] or A[[x 1,..., x ]]. Then 1 t i! t i x t i i can be viewe as a ifferential operator on even if the integer t i! is not invertible. In either case, D(, A) is the free -moule with basis 1 t 1! t 1 x t t! t 1 t 1! x t 1 t x t for (t 1,..., t ) N, or equivalently D(, A) = 1 1 t! (t 1 x t 1,..., t ) N By the quotient rule, it is clear that f is a D(, A)-moule. It is also straightforwar to check that the maps in the Čech complex are D(, A)-linear an hence each local cohomology moule Ha() i is a D(, A)-moule. When specializing A = Z, we have that D()/p D() = D(/p). emark 4.2. Let be a noetherian commutative ring of characteristic p. Assume that is a finitely generate p -moule. Then by [Yek92, 1.4.9] an [SVB97, 2.5.1] we have t D() = D(, Z /p Z) = e Hom p e (, ). Proposition 4.3. Let be a regular ring of characteristic p such that is a finitely generate p -moule. Then each F -moule is a D()-moule. 9

10 Proof. Let M be an F -moule with the structure isomorphism θ : M F (M ). By emark 4.2, it suffices to specify the action of each element δ Hom p e (, ) on M. Let θ e enote the composition F e 1 (θ) θ : M F e (M ). Clearly δ acts on F e (M ) = F e M via δ i M. We let δ act on M via θe 1 (δ i M ) θ e. Since δ Hom p e (, ) Hom (, ) for p e e > e, we nee to check that the action of δ is inepenent of the choice of e, i.e. we nee to check that (1) θ 1 e (δ i M ) θ e = θ 1 e (δ i M ) θ e. Set θ e e e = F e (θ e e) = i F e θ e e. Then θ e = θ e e e θ e. To check (1) is equivalent to checking which is clear. δ i M = (θ e e e ) 1 (δ i M ) θ e e e = (i F e θ e e) 1 (δ i M ) (i F e θ e e) As an application of D-moule, now we turn our attention to answering the question raise in emark Let = F p [x 1,..., x n ] an let D enote the ring of ifferential operators over. We will use [t] j to enote 1 t t!. x t j Definition 4.4. The r-th Euler operator, enote by E r, is efine as E r := x i 1 1 x in n [i 1] 1 n [in]. i 1 +i 2 + +i n=r,i 1 0,...,i n 0 In particular E 1 is the usual Euler operator n i=1 x i i. A grae D-moule M is calle Eulerian, if each homogeneous element z M satisfies ( ) eg(z) (2) E r z = z r for every r 1. emark 4.5. An F -moule M with structure isomorphism θ : M F (M ) is calle a grae F -moule if M is a grae -moule an θ is egree-preserving. It turns out that each grae F -moule is an Eulerian grae D-moule [MZ14, Theorem 4.4]. Exercise 4.6. Prove that, if M is a grae Eulerian D-moule, so are D-submoules an D-quotients of M. Exercise 4.7. Prove that M f is an Eulerian grae D-moule for an Eulerian grae D-moule M an a grae element f an. (Hint: [j] i is p e -linear for p e > j an hence [j] i ( z f ) = 1 pe f pe [j] i (z) for each z M.) Combining Exercises 4.6 an 4.7, we have Proposition 4.8. Let a 1,..., a s be grae ieals of. Then each local cohomology moule HmH i j 1 a 1 Ha js s () is an Eulerian grae D-moule. For each grae -moule M an integer l, we will use M(l) to enote the grae -moule whose egree-j piece is M j+l for each integer j. Exercise 4.9. Let M be an Eulerian grae D-moule. Prove that M(l) is Eulerian if an only if l = 0. 10

11 emark The ring of ifferential operators D has a natural graing eg(x i ) = 1 an eg( [j] i ) = j. Uner this graing, the map D D m Hn m()( n) efine by is a egree-preserving isomorphism. [i 1] 1 n [in] ( 1) i 1+ +i n x i xn in 1 Theorem If M is an Eulerian grae D-moule an Supp (M) = {m}, then M is isomorphic (as grae -moules) to a irect sum of copies of H n m(). Proof. Let Soc(M) enote the socle of M, i.e the sub--moule of M that is annihilate by m. It is straightforwar to check that Soc(M) is generate by grae elements an a minimal set of grae generators of Soc(M) is a grae k-basis for Soc(M). Let {β j } be a grae k-basis of Soc(M) with eg(β j ) = n j. There is a egreepreserving homomorphism of D-moules D j D m (n j) M which sens 1 of the j- th copy to β j. This map is an isomorphism because it inuces an isomorphism on socles. By emark 4.10, we have D j D m (n j) = j Hn m()( n + n j ). Hence M = j Hn m()( n + n j ). Since M is Eulerian, so is each Hm()( n n + n j ) which implies that n j = n for each j by Exercise 4.9. This finishes the proof. 5. interactions between F -moules an D-moules We have seen in Proposition 4.3 that each F -moule is a D()-moule. It turns out that the theories of F -moules an D-moules are intertwine an their interactions have ha many applications. We begin with the following result ue to Lyubeznik [Lyu97, 5.7]. Theorem 5.1. Let be a regular ring of characteristic p that is a finite p -moule. Then each F -finite moule has finite length in the category of D()-moules. We will see an application of this theorem in a moment. Exercise 5.2. Let φ : S be a ring homomorphism between commutative rings such that locally S is a finitely generate free -moule. Then the category of - moules is equivalent to the one of En (S)-moules. The functors are S an Hom (S, ) En (S). emark 5.3. We like to apply the equivalence in Exercise 5.2 to Frobenius enomorphism. Let be a regular ring of characteristic p that is a finite p -moule. Consier the e-th Frobenius F e (here we enote the target ring by F e assuming no confusion will arise). Clearly En (F e ) = En p e () (keep in min that this is part of the ring D() accoring to emark 4.2). Hence the category of En p e ()-moules is equivalence to the one of -moules. One immeiate consequence is that the categories of En p e ()-moules are equivalent for all e 1. The functor that gives the equivalence between the category of En p e e e ()-moules an the one of En pe+e ()-moules is F ( ) = F. More specifically, let M be a En p e ()-moule. The En pe+e ()-moule structure on F e (M) can be seen as follows. Since the category of En pe ()-moules is equivalent to the one of -moules, we know that M = F e N for some -moule N. 11

12 Then F e M = F e+e e N. An hence En pe+e () acts on F (M) via its action on F e+e N which is given by α i N for each α En pe+e (). By our iscussion so far, we can see that, when is a regular ring of characteristic p that is a finite p -moule, the functor F e ( ) is an equivalence of the category of D()-moules to itself. Theorem 5.4. Let be a regular ring of characteristic p that is a finite p -moule. Assume that M is an F -finite moule an M is a root of M. Then the image of M in M generates M as a D()-moule. Proof. Since M is a root of M, we know that β : M F (M) is injective an hence the irect limit (which is M ) of M F (M) F 2 (M) is a irect union. Thus, we may assume that M F (M) an M = e F e (M). Let M be the D()-submoule of M that is generate by M. Since M F (M), we have M F (M). If M were strictly containe in F (M) (which is also a D()- submoule of M ), then we woul have a strictly increasing chain of D()-submoules of M : M F (M) F(M) 2 which is a contraiction to Theorem 5.1. Therefore M = F (M). Consequently M = e F e (M) e F e (M) = M. This finishes the proof. Corollary 5.5. Let be a regular ring of characteristic p that is a finite p -moule. Assume that M is an F -finite moule an β : M F (M) is a generating morphism of M. Then the image of M in M generates M as a D()-moule. As an application of the interactions between F -moules an D-moules, we prove the following result [BBL + 14, 3.1(1)]. Theorem 5.6. Let = Z[x 1,..., x n ] an a be an ieal of. If a prime number p is a nonzeroivisor on Ext i (/ a, ), then p is also a nonzeroivisor on Hi a(). Proof. Let p be a prime integer. The exact sequence 0 p /p 0 inuces an exact sequence of Ext an an exact sequence of local cohomology moules; they fit into a commutative iagram: Ext i 1 (/ a, ) ψ H i 1 a β () ϕ Ext i 1 (/ a, /p) H i 1 a γ (/p) δ Ext i p (/ a, ) H i a() p Ext i (/ a, ) H i a() The bottom row is a complex of D()-moules; in particular, ϕ ( Ha i 1 () ) is a D()- submoule of Ha i 1 (/p). As ϕ ( Ha i 1 () ) is annihilate by p, it has a natural structure as a moule over the ring D()/p D(), which equals D(/p). Similarly, (3) Ha i 1 (/p) δ Im(δ) is a map of D(/p)-moules. Suppose p is a nonzeroivisor on Ext i (/ a, ). Then the map ψ is surjective; we nee to prove that p is a nonzeroivisor on Ha(), i equivalently, that ϕ is surjective. 12

13 By Corollary 5.5, the image M of γ generates Ha i 1 (/p) as a D(/p)-moule. As ψ is surjective, M is also the image of γ ψ = ϕ β. It follows that But ϕ ( H i 1 a M ϕ ( H i 1 a () ). () ) is a D(/p)-submoule of H i 1 (/p) that contains M. Hence ϕ ( Ha i 1 () ) = Ha i 1 (/p), i.e., ϕ is surjective, as esire. Corollary 5.7. Let = Z[x 1,..., x n ] an a be an ieal of. Then there are only finitely many prime numbers p such that Ha() j contains a nonzero p-torsion element for all j. We have seen in Theorem 3.14 that F -finite moules have finite length in the category of F -moule. On the other han, [Lyu97, 5.7] says that an F -finite moule with the D-moule structure given in Proposition 4.3 also has finite length in the category of D-moules. So, a natural question is how oes one compare these two lengths. It turns out that they are the same when the unerlying fiel is algebraically close. Theorem 5.8 (Theorem 1.1 in [Bli03]). Let be a noetherian regular ring that is essentially of finite type over an algebraically close fiel of characteristic p. Then the length of an F -finite moule in the category of F -moules is the same as its length in the category of D-moules. Question 5.9. Let a be an ieal of. D-moule? a Can one compute the length of H i a() as a At the moment, very little is known about Question 5.9; even when a is a principal ieal an i = 1. We will come back to this question when we iscuss non-regular rings in characteristic p. 6. ing-theoretic properties characterize by Frobenius Theorem 0.5 exemplifies that the Frobenius enomorphism encoes properties of the ring itself. Definition 6.1. is calle F -pure if M F M is injective for all -moules M. When (, m) is local, is calle F -injective if the Frobenius action H i m() F H i m() is injective all i. More generally when may not be local, is calle F -injective if p is so for each prime ieal p. Exercise 6.2. Let (, m) be a noetherian local ring of characteristic p. Let E(/ m) enote the injective hull of / m. Prove that is F -pure if an only if E(/ m) F E(/ m) is injective. Exercise 6.3. Let (, m) be a noetherian local ring of characteristic p. Assume that is a finite p -moule. Prove that the following are equivalent (1) is F -pure; (2) for each e 1, there is an pe -moule homomorphism ϕ : pe such that the composition pe ϕ pe is ientity on pe ; (3) for some e 1, there is an pe -moule homomorphism ϕ : pe such that the composition pe ϕ pe is ientity on pe. 13

14 Example 6.4. Let = k[[x 1,..., x ]] an f m = (x 1,..., x ) be a formal power series. Since /f is a hypersurface an hence Gorenstein, E((/f)/ m) = Hm 1 (/f ). So, /f is F -pure if an only if the Frobenius action F : Hm 1 (/f) Hm 1 (/f ) is injective. Consier the following commutative iagram inuce by 0 f /f 0: (4) 0 H 1 m (/f) H m() f H m() 0 F f p 1 F 0 Hm 1 (/f) Hm() f F H m() 0 It is easy to see from this iagram that F : Hm 1 (/f) Hm 1 (/f) is injective if the map in the mile f p 1 F is injective. From Example 0.3, we can see that this is the case if an only if f p 1 / m [p]. More generally, the quotient ring /I is F -pure if an only if (I [p] : I) m [p] ; this is calle Feer s Criterion. Exercise 6.5. Let = Fp[[x,y,z]] (x 3 +y 3 +z 3 ) z2 an consier the element [ xy ] H2 (x,y,z)(). Fin all prime numbers p such that the image of [ z2 xy ] uner the Frobenius action on H2 (x,y,z) () is 0. Question 6.6. Assume that f is a polynomial of egree in variables x 1,..., x with integer coefficients an that its Jacobian ieal is primary to (x 1,..., x ). Let f p enote the image of f uner := Z[[x 1,..., x ]] p := F p [[x 1,..., x ]]. Are there infinitely many prime numbers p such that p /f p p is F -pure? emark 6.7. When f is a egree 3 homogeneous polynomial that efines an elliptic curve, then by [Sil92, 5.11 on page 145] there are infinitely many prime numbers p such that f p efines an orinary elliptic curve an hence p /f p p is F -pure. Exercise 6.8. Assume that (, m) is an F -injective local ring. Prove that is reuce. Question 6.9. Let f be a regular element of a local ring (, m). Assume that /f is F -injective. Is it true that is F -injective? Exercise Let (, m) be Cohen-Macaulay an f be a regular element. Assume that /f is F -injective. Prove that is also F -injective. Question 6.9 is still open in its full generality. For recent evelopments, see [HMS14]. Definition Let be a noetherian commutative ring of characteristic p. We say that is strongly F -regular if for every non-zero element r there exists e N an φ Hom p e (, pe ) such that φ(r) = 1. Exercise Prove that, if is a strongly F -regular local ring, then is a omain. Exercise Let S be a split ring extension (i.e. there is an -moule homomorphism S such that S is the ientity on ). Assume that S is strongly F -regular. Prove that is also strongly F -regular. Question Let be a noetherian commutative ring of characteristic p. An ieal I of i calle Frobenius close if r p I [p] implies r I for all elements r. Assume now that is local an consier the following two conitions 14

15 (1) is F -injective; (2) each parameter ieal I is Frobenius close. Is it true that (1) an (2) are equivalent? Question Does the conclusion of Exercise 1.1 still hol when assuming is strongly F -regular instea of regular? It turns out that one can use properties of D-moules to characterize strong F - regularity. Theorem 6.16 (Theorem 2.2(4) in [Smi95]). Let (, m) be a noetherian commutative local ring of characteristic p. Assume that is a finite p -moule. Then is strongly F -regular if an only if is F -pure an a simple D-moule. Proof. First assume that is strongly F -regular. Then it is clear that is F -pure. We will show that D()r = for each nonzero element r. Since is strongly F -regular, there is an integer e 1 an a ϕ Hom p e (, pe ) such that ϕ(r) = 1. The composition of ϕ with pe, still enote by ϕ, is clearly an element of En p e () D(). Since ϕ(r) = 1, we have D()r =. Conversely, assume that is F -pure an a simple D-moule. Let r be a nonzero element of. Since is D()-simple, there is δ D() such that δ(r) = 1. By emark 4.2, δ is pe -linear for some e 1. By Exercise 6.3, there is a ϕ Hom p e (, pe ) that splits the inclusion pe. Now ϕ δ : pe is pe -linear an ϕ δ(r) = 1. Question Let be an integral omain of characteristic p that is a finite p -moule. Assume that every moule-finite ring extension S splits. Is it true that is D()- simple? The following result ue to Blickle [Bli04, 4.10] shes some light on Question 5.9. Theorem Let (, m) be an F -finite regular local ring an f m be an element of. Assume that /f is F -injective (or equivalently, F -pure). then H(f) 1 () is a simple D()-moule if an only if /f is strongly F -regular. eferences [BBL + 14] B. Bhatt, M. Blickle, G. Lyubeznik, A. K. Singh, an W. Zhang: Local cohomology moules of a smooth Z-algebra have finitely many associate primes, Invent. Math. 197 (2014), no. 3, [Bli03] M. Blickle: The D-moule structure of [F ]-moules, Trans. Amer. Math. Soc. 355 (2003), no. 4, (2004m:13016) [Bli04] M. Blickle: The intersection homology D-moule in finite characteristic, Math. Ann. 328 (2004), no. 3, (2005a:14005) [Har68]. Hartshorne: Cohomological imension of algebraic varieties, Ann. of Math. (2) 88 (1968), (38 #1103) [HS77]. Hartshorne an. Speiser: Local cohomological imension in characteristic p, Ann. of Math. (2) 105 (1977), no. 1, M (56 #353) [HE71] M. Hochster an J. A. Eagon: Cohen-Macaulay rings, invariant theory, an the generic [HMS14] perfection of eterminantal loci, Amer. J. Math. 93 (1971), (46 #1787) J. Horiuchi, L. E. Miller, an K. Shimomoto: Deformation of F -injectivity an local cohomology, Iniana Univ. Math. J. 63 (2014), no. 4, [Hun92] C. Huneke: Problems on local cohomology, Free resolutions in commutative algebra an algebraic geometry (Sunance, UT, 1990), es. Notes Math., vol. 2, Jones an Bartlett, Boston, MA, 1992, pp (93f:13010) 15

16 [HK91] C. Huneke an J. Koh: Cofiniteness an vanishing of local cohomology moules, Math. Proc. Cambrige Philos. Soc. 110 (1991), no. 3, (92g:13021) [HS93] C. L. Huneke an. Y. Sharp: Bass numbers of local cohomology moules, Trans. Amer. Math. Soc. 339 (1993), no. 2, (93m:13008) [Kun69] E. Kunz: Characterizations of regular local rings for characteristic p, Amer. J. Math. 91 (1969), (40 #5609) [Lyu97] G. Lyubeznik: F -moules: applications to local cohomology an D-moules in characteristic p > 0, J. eine Angew. Math. 491 (1997), M (99c:13005) [Lyu06] G. Lyubeznik: On the vanishing of local cohomology in characteristic p > 0, Compos. Math. 142 (2006), no. 1, (2007b:13029) [MZ14] L. Ma an W. Zhang: Eulerian grae D-moules, Math. es. Lett. 21 (2014), no. 1, [PS73] C. Peskine an L. Szpiro: Dimension projective finie et cohomologie locale. Applications à la émonstration e conjectures e M. Auslaner, H. Bass et A. Grothenieck, Inst. Hautes Étues Sci. Publ. Math. (1973), no. 42, (51 #10330) [Sil92] J. H. Silverman: The arithmetic of elliptic curves, Grauate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1992, Correcte reprint of the 1986 original (95m:11054) [Sin00] A. K. Singh: p-torsion elements in local cohomology moules, Math. es. Lett. 7 (2000), no. 2-3, (2001g:13039) [Smi95] K. E. Smith: The D-moule structure of F -split rings, Math. es. Lett. 2 (1995), no. 4, (96j:13024) [SVB97] K. E. Smith an M. Van en Bergh: Simplicity of rings of ifferential operators in prime characteristic, Proc. Lonon Math. Soc. (3) 75 (1997), no. 1, (98:16039) [Yek92] A. Yekutieli: An explicit construction of the Grothenieck resiue complex, Astérisque (1992), no. 208, 127, With an appenix by Pramathanath Sastry (94e:14026) 16