A surprising fact about D-modules in characteristic p>0

A SURPRISING FACT ABOUT D-MODULES IN CHARACTERISTIC p > 0 JOSEP ` ALVAREZ MONTANER AND GENNADY LYUBEZNIK Abstract. Let R=k[x1, . . . , xd] be the polynomial ring in dindependent variables, where kis a field of characteristic p > 0. Let DRbe the ring of k-linear differential operators of Rand let fbe a polynomial in R. In this work we prove that the localization R[1 f] obtained from Rby inverting f is generated as a DR-module by 1 f. This is an amazing fact considering that the corresponding characteristic zero statement is very false. 1. Introduction Let kbe a field and let R=k[x1, . . . , xd], or R=k[[x1, . . . , xd]] be either a ring of polynomials or formal power series in a finite number of variables over k. Let DRbe the ring of k-linear differential operators on R. For every f∈R, the natural action of DRon Rextends uniquely to an action on R[1 f] via the standard quotient rule. Hence R[1 f] acquires a natural structure of DR-module. It is a remarkable fact that R[1 f] has finite length in the category of DR-modules. This fact has been proven in characteristic 0 by Bernstein [1, Corollary 1.4] in the polynomial case and by Bj¨ork [2, Theorems 2.7.12, 3.3.2] in the formal power series case and in characteristic p > 0 by Bøgvad [3, Proposition 3.2] in the polynomial case and by Lyubeznik [5, Theorem 5.9] in the formal power series case. Thus the ascending chain of submodules DR·1 f⊆ DR·1 f2⊆ · · · ⊆ DR·1 fk⊆ · · · ⊆ R[1 f] stabilizes, i.e. R[1 f] is generated by 1 fifor some i. This paper is motivated by the following natural question: What is the smallest isuch that 1 figenerates R[1 f]as a DR-module? 2000 Mathematics Subject Classification. Primary 13N10, 13B30. Research of the first author partially supported by a Fulbright grant and the Secretar´ıa de Estado de Educaci´on y Universidades of Spain and the European Social Funding. The second author greatfully acknowledges NSF support. 2 JOSEP ` ALVAREZ MONTANER AND GENNADY LYUBEZNIK When kis a field of characteristic zero and f∈Ris a non-zero polynomial it has been proven in [1, Theorem 1’] that there exists of a monic polynomial bf(s)∈k[s] and a differential operator Q(s)∈ DR[s] such that Q(s)·fs+1 =bf(s)·fs for every s. The polynomial bf(s) is called the Bernstein-Sato polynomial of fand is always a multiple of (s+ 1). Let −ibe the least integer root of bf(s). Then, bf(s)6= 0 for any integer s < −iand therefore fs∈ DR·fs+1. In particular, R[1 f] is generated by 1 fiand, as is shown in [6, Lemma 1.3], it cannot be generated by 1 fjfor j < i. This gives a complete answer to our question in characteristic zero. For example, consider the polynomial f=x2 1+· · ·+x2 2n∈R=k[x1, . . . , x2n]. Then we have the functional equation 1 4(∂2 ∂x2 1 +· · · +∂2 ∂x2 2n )·fs+1 = (s+ 1)(s+n)·fs where the Bernstein-Sato polynomial is bf(s) = (s+ 1)(s+n). Hence R[1 f] is generated by 1 fnas a DR-module but it can not be generated by 1 fifor i < n. But in characteristic p > 0 a differential operator of a fixed order annihilates the powers 1 fpsfor slarge enough, so a functional equation such as above even if it exists does not imply that 1 fps+1 ∈ DR·1 fpsfor all s. The goal of this paper is to prove the following amazing result. Theorem 1.1. Let R=k[x1, . . . , xd]where kis a field of characteristic p > 0 and let f∈Rbe a non-zero polynomial. Then R[1 f]is generated by 1 fas DRmodule. Our proof does not extend to the case of formal power series; some new idea seems to be needed in this case (see Remark 3.6). Acknowledgement The first author would like to thank the Department of Mathematics at the University of Minnesota for the warm welcome he received during his postdoctoral stay. 2. Differential operators in positive characteristic Let Nbe the set of non-negative integers. Throughout, we will use multiindex notation in the polynomial ring R=k[x1, . . . , xd], where kis a field of characteristic p > 0. So, given α= (α1, . . . , αd)∈Ndwe will denote the sum of its components by |α|and xαwill stand for the monomial xα=xα1 1· · · xαd d. A SURPRISING FACT ABOUT D-MODULES IN CHARACTERISTIC p > 0 3 A pair of multi-indices αand βare ordered as usual: α < β if and only if αi< βifor i= 1, . . . , d. For a general description of the ring of differential operators we refer to [4, §16.8]. For the case we are considering in this work we refer to [4, Th´eor`eme 16.11.2]. The ring of differential operators DR=D(R, k) associated to the polynomial ring Ris the ring extension of Rgenerated by the set of differential operators {Dt,i =1 t! ∂t ∂xt i |t∈N, i = 1, . . . , d } Given β∈Nd,Dβwill denote the differential operator Dβ:= Dβ1,1· · · Dβd,d. We can extend the multi-index notation to DRconsidering the k-basis formed by the monomials xαDβ. In the sequel, a differential operator Q∈ DRwill be written in right normal form, i.e. Q=X α,β∈ N d aαβ xαDβ, where all but finitely many aαβ ∈kare zero. Let k[Rpn] be the k-algebra generated by pn-th powers of elements of R. Let D(n) Rbe the ring extension of Rgenerated by the set of differential operators up to order pn, i.e. {Dα|α < pn}where pn= (pn, . . . , pn)∈Nd. Then we have an increasing chain of finitely generated ring extensions of R D(0) R⊆ D(1) R⊆ D(2) R⊆ · · · ⊆ DR whose union is DR. Lema 2.1. Let Q∈ D(n) Rand f∈k[Rpn]. Then Qcommutes with f, i.e. Q(f·g) = f·Q(g)for all g∈R. Proof. Writing out Q,fand gas sums of monomials and considering that Dt,i commutes with Dt0,j and xs jfor j6=i, one sees that it is enough to prove the statement for Q=Dt,i,f=xpn iand g=xs i, where t < pn,sis an integer and i= 1, . . . , d. In this case we have Dt,i(xpn+s i) = xpn i·Dt,i(xs i) just comparing the coefficient at xpn+s−t ion both sides modulo p.¤ 3. Proof of Theorem 1.1 We notice first that it is enough to show that if f∈Ris a non-zero polynomial, then 1 fpbelongs to the DR-module generated by 1 f. Once this 4 JOSEP ` ALVAREZ MONTANER AND GENNADY LYUBEZNIK is proved we can apply this result to fps−1to get 1 fps∈ DR·1 fps−1for every s > 1. Since the set 1 f,1 fp,1 fp2, . . . generates R[1 f] as R-module, we are done. We can also reduce to the case of kbeing a perfect field. If kis not perfect, let Kbe the perfect closure of k. Assume there is a differential operator Q=PaαβxαDβwith coefficients aαβ ∈Ksuch that Q(1 f) = 1 fp. This is equivalent to the fact that a system of finitely many linear equations with coefficients in khas solutions in K, where the non-zero coefficients aαβ of Q are thought of as the unknowns of the system. For example, if f=x1, we may be looking for a solution in the form Q=aDp−1,1, so we get an equation Q(1 x) = 1 xp. Since Q(1 x) = a1 xp, the corresponding linear system is just one equation a= 1. The system has a solution in K, namely, the coefficients of Q. Hence it is consistent, so it must have a solution in kbecause the coefficients of the linear system are in k(in fact the coefficients are in the prime field Z/pZ). So there is a differential operator Q0with coefficients in ksuch that Q0(1 f) = 1 fp. Henceforth we will assume that the base field kof our polynomial ring is perfect. It is enough to show that under this assumption 1 fpbelongs to the DR-submodule generated by 1 f, for all f∈R. Given a polynomial f∈Rand an integer n≥1, we can write in a unique way f(x) = X 0≤α<pn fα(xpn)xα where fα(z)∈k[z] are polynomials in dvariables. Consider the ideal Jn(f) generated by the polynomials fα(xpn) in the decomposition of fwith respect to xpn. Lema 3.1. Let f, g, h ∈Rbe polynomials such that f=g·h. Then Jn(f)⊆ Jn(g). Proof. Consider the decomposition of gwith respect to xpn g(x) = X 0≤α<pn gα(xpn)xα Set h(x) = Paβxβ, then f(x) = g(x)·h(x) = X 0≤α<pn gα(xpn) (Xaβxα+β) A SURPRISING FACT ABOUT D-MODULES IN CHARACTERISTIC p > 0 5 Rewriting in the form f(x) = X 0≤γ<pn fγ(xpn)xγ we get fγ(xpn) = Xaβgα(xpn)xjpn where the sum is taken over the multi-indices αand βsuch that xα+β=xjpnxγ for a given j∈N. In particular fγ(xpn)∈Jn(g). ¤ Lema 3.2. Let f, g ∈Rbe polynomials such that f=gp. Then Jn(f) = Jn−1(g)[p] Proof. It is enough to raise to the p−th power the decomposition of gwith respect to xpn−1.¤ Notice that D(n) R·fis an ideal of R. This ideal can be also described as follows: Lema 3.3. Jn(f) = D(n) R·f. Proof. By Lemma 2.1 every Q∈ D(n) Rcommutes with every fα(xpn) in the decomposition of fwith respect to xpn. Hence Q(f) = X 0≤α<pn fα(xpn)Q(xα) In particular, Q(f)∈Jn(f), i.e. D(n) R·f⊆Jn(f). To prove the opposite containment it is enough to show that every fα(xpn) belongs to the ideal D(n) R·f. Consider the multi-index β= (pn−1, . . . , pn−1) ∈Nd. Then we have Dβ(f) = fβ(xpn)∈ D(n) R·f Now we proceed by induction on σβ= Σd i=1(pn−1−βi), the case σβ= 0 being just proved. Let β∈Ndbe a multi-index such that Dβ∈ D(n) R. Then we have Dβ(f) = fβ(xpn) + X β0>β fβ0(xpn) (aβ0βxβ0−β) where aβ0β=µβ0 1 β1¶· · · µβ0 d βd¶. By induction fβ0(xpn)∈ D(n) R·ffor any β0> β, so we are done. ¤ 6 JOSEP ` ALVAREZ MONTANER AND GENNADY LYUBEZNIK Since kis a perfect field, the coefficients of fα(xpn) in the decomposition of f with respect to xpnare pn-th powers, hence fα(xpn) = ( e fα(x))pn, where e fα(x) are polynomials in R. Consider the ideal In(f) generated by the polynomials e fα(x). Notice that Jn(f) is the n-th Frobenius powers of the ideal In(f), i.e. Jn(f) = In(f)[pn]. Lema 3.4. Let f∈Rbe a polynomial. For any integer n≥1there is an inclusion of ideals In(fpn−1)⊆In−1(fpn−1−1) Proof. The equality Jn(fpn−p) = Jn−1(fpn−1−1)[p]given by Lemma 3.2 translates to In(fpn−p)[pn]= (In−1(fpn−1−1)[pn−1])[p]=In−1(fpn−1−1)[pn] This implies In(fpn−p) = In−1(fpn−1−1) due to the fact that the polynomial ring Ris regular. On the other hand, the inclusion Jn(fpn−1) = Jn(fpn−p·fp−1)⊆Jn(fpn−p) given by Lemma 3.1 implies In(fpn−1)⊆In(fpn−p) again because Ris regular so we get the desired inclusion. ¤ Lema 3.5. The descending chain of ideals I1(fp−1)⊇I2(fp2−1)⊇ · · · ⊇ In−1(fpn−1−1)⊇In(fpn−1)⊇ · · · stabilizes. Proof. Assume that deg f=e. Let fpn−1(x) = X 0≤α<pn fα(xpn)xα be the decomposition of fpn−1with respect to xpn. Since deg fpn−1=e(pn−1), the polynomials in the decomposition satisfy deg fα(xpn)≤e(pn−1) < epn. On the other hand, deg fα(xpn) = pndeg e fα(x) implies deg e fα(x)< e. Let Webe the k-vector space of polynomials of degree strictly smaller than e. For every n, the ideal In(fpn−1) is generated by In(fpn−1)∩We. Thus we have a descending sequence of k-vector subspaces of We (I1(fp−1)∩We)⊇(I2(fp2−1)∩We)⊇ · · · ⊇ (In(fpn−1)∩We)⊇ · · · that must stabilize because Weis a finite-dimensional k-vector space. ¤ A SURPRISING FACT ABOUT D-MODULES IN CHARACTERISTIC p > 0 7 Now we can complete the proof of Theorem 1.1 as follows. Assume that the chain of ideals given in Lemma 3.5 stabilize at the level s−1, i.e. I:= Is−1(fps−1−1) = Is(fps−1). From the equalities Js(fps−1) = I[ps]and Js−1(fps−1−1) = I[ps−1]we deduce Js(fps−p) = Js−1(fps−1−1)[p]= (I[ps−1])[p]=I[ps]=Js(fps−1) Thus we have fps−p∈Js(fps−1). By Lemma 3.3 there is a differential operator Q∈ D(s) Rsuch that Q(fps−1) = fps−p. Since Qcommutes with fps, we see that Q(fps−1 fps) = fps−p fps so we get Q(1 f) = 1 fpas we desired. ¤ Remark 3.6.In the case of formal power series all parts of our proof go through except the proof of Lemma 3.5. Most likely, the statement of Lemma 3.5 is still true in the case of formal power series but a very different proof is needed. Example 3.7. Let R=k[x1, x2, x3, x4] where kis a field of characteristic p > 0. Consider the polynomial f=x2 1+x2 2+x2 3+x2 4. We are going to find a differential operator Q∈ DRsuch that Q(1 f) = 1 fpjust checking out the monomials in fp−1. Let S(f, p, n) be the set of terms aαxαin fp−1such that α < pn. The set S(f, p, 1) is non-empty. Namely we have: •If 4 divides p−1, then aαxα∈S(f, p, 1) where α= (p−1 2,p−1 2,p−1 2,p−1 2) aα=(p−1)! (p−1 4!)4 •If 4 does not divide p−1, then aαxα∈S(f, p, 1) where α= (p+1 2,p+1 2,p−3 2,p−3 2) aα=(p−1)! (p+1 4!)2(p−3 4!)2 Let aαxαbe a leading term of S(f, p, 1) with respect to the usual order. Notice that 1 aαDα(fp−1) = 1. The differential operator Q=1 aαDαcommutes with fpby Lemma 2.1 so we get the desired result. References [1] I. N. Bernˇste˘ın, Analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Priloˇzen., 6 (4) (1972) 26–40. 8 JOSEP ` ALVAREZ MONTANER AND GENNADY LYUBEZNIK [2] J. E. Bj¨ork, Rings of differential operators, North Holland Mathematics Library, Amsterdam, 1979. [3] R. Bøgvad, Some results on D-modules on Borel varieties in characteristic p > 0, J. Algebra, 173 (3) (1995) 638–667. [4] A. Grothendieck and J. Dieudonn´e, ´ El´ements de g´eom´etrie alg´ebrique IV. ´ Etude locale des sch´emas et des morphismes de sch´emas, Publications Math´ematiques I.H.E.S. 32 (1967). [5] G. Lyubeznik, F-modules: applications to local cohomology and D-modules in characteristic p > 0, J. Reine Angew. Math. 491 (1997), 65–130. [6] U. Walther, Bernstein-Sato polynomials versus cohomology of the Milnor fiber for generic hyperplane arrangements, to appear in Compositio Math. Departament de Matem` atica Aplicada I, Universitat Polit` ecnica de Catalunya, Avinguda Diagonal 647, Barcelona 08028, SPAIN E-mail address:[email protected] Department of Mathematics, University of Minnesota, 206 Church St. S.E., Minneapolis, MN 55455, USA E-mail address:[email protected]