We show that for a Frobenius split variety, there are only finitely many
closed subvarieties which are compatibly-split.Comment: 3 page

Kumar, Shrawan

Mehta, Vikram B.

English

arXiv

Let k be an algebraically closed field of characteristic p &gt; 0 and let X be a scheme over k (always assumed to be separated of finite type over k). The following is the main theorem of this note and we give here its complete and self-contained proof

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arXiv:0901.2098v1  [math.AG]  14 Jan 2009Finiteness of the number of compatibly-splitsubvarietiesShrawan Kumar and Vikram B. Mehta∗January 14, 20091 IntroductionLet k be an algebraically closed field of characteristic p > 0 and let X be ascheme over k (always assumed to be separated of finite type over k). Thefollowing is the main theorem of this note and we give here its complete andself-contained proof.(1.1) Theorem. Assume that X is Frobenius split by a splitting σ ∈HomOX (F∗OX ,OX), where F is the absolute Frobenius morphism (cf. [BK,Section 1.1]). Then, there are only finitely many closed subschemes of Xwhich are compatibly split (under σ).2 Proof of Theorem 1.1We first prove the following proposition which is of independent interest. Bya variety we mean a reduced but not necessarily irreducible scheme over k.(2.1) Proposition. Let X be a nonsingular irreducible variety which isFrobenius split by σ ∈ HomOX (F∗OX ,OX) ≃ H0(X, F∗(ω1−pX )), where ωXis the dualizing sheaf of X (cf. [BK, Proposition 1.3.7]), and let Y ⊂6=X bea compatibly-split closed subscheme of X. Then,Y ⊂ Z(σ̄),∗Acknowledgements. We thank A. Knutson for the correspondences which led tothis work. The first author was partially supported by a FRG grant from NSF.1where Z(σ̄) denotes the set of zeroes of σ̄ and σ̄ is the section of F∗(ω1−pX )obtained from σ via the above identification.Proof. Since any irreducible component of a compatibly-split closed sub-scheme is compatibly split (cf. [BK, Proposition 1.2.1]), we can assumethat Y is irreducible. Assume, if possible, that Y ∩ (X\Z(σ̄)) 6= ∅. Then,Y reg ∩ (X\Z(σ̄)) 6= ∅, where Y reg is the nonsingular locus of Y .Take y ∈ Y reg ∩ (X\Z(σ̄)). Choose a system of local parameters {t1, . . . ,tm, tm+1, . . . , tn} at y ∈ X such that {t1, . . . , tm} is a system of local parame-ters at y ∈ Y and 〈tm+1, . . . , tn〉 is the completion of the ideal of Y in X at y.(This is possible since both X and Y are nonsingular at y.) By assumption,σ̄ is a unit in the local ring OX,y. Moreover, σ induces a splitting σ̂ of thepower series ring k[[t1, . . . , tn]] compatibly splitting the ideal 〈tm+1, . . . , tn〉.Now, since σ̄ does not vanish at y, σ̂((t1 · · · tn)p−1)is a unit in the ringk[[t1, . . . , tn]]. In particular, σ̂ does not keep the ideal 〈tm+1, . . . , tn〉 stable.This is a contradiction to the assumption. Hence, Y ⊂ Z(σ̄), proving theproposition.(2.2) Proof of Theorem 1.1. By [BK, Proposition 1.2.1], we can assumewithout loss of generality that X is irreducible. We prove Theorem 1.1 byinduction on the dimension of X . If dim X = 0, then the theorem is clear.So assume that dim X = n and the theorem is true for varieties of dimension< n. Let Y ⊂6=X be a compatibly-split irreducible closed subscheme. Then,either Y ⊂ Xsing (where Xsing is the singular locus of X) or Y ∩ Xreg 6= ∅.In the latter case, by Proposition 2.1,Y ∩ Xreg ⊂ Z(σ̄o),where Z(σ̄o) denotes the set of zeroes of the splitting σ̄o of the open subsetXreg of X viewed as a section of F∗(ω1−pXreg). Thus, in this case,Y ⊂ Z(σ̄o),Z(σ̄o) being the closure of Z(σ̄o) in X. Hence, in either case,(1) Y ⊂ Z(σ̄o) ∪ Xsing.Considering the irreducible components, the same inclusion (1) holds forany compatibly-split closed subscheme Y ⊂ X such that Y 6= X.2Let {Yi}i∈I be the collection of all the distinct compatibly-split closedsubschemes Yi ⊂6=X and let Y :=⋃i∈I Yi. Since the ideal sheaf IY =⋂i∈I IYiand each IYi is stable under the splitting σ of X, the closed subscheme Y iscompatibly split. In particular, by (1), for each i ∈ I,Yi ⊂ Y ⊂ Z(σ̄o) ∪ Xsing.Since dim(Z(σ̄o) ∪ Xsing)< dim X; in particular, one has dim Y < dim X.Thus, by the induction hypothesis (applying the theorem with X replacedby Y ), I is a finite set. This completes the proof of the theorem.(2.2) Remark. Karl Schwede has also obtained the above theorem via ‘F -purity’ in a recent preprint [S]. As pointed out by Schwede, when X is projec-tive, the theorem also follows from [EH, Corollary 3.2] again via ‘F -purity’.(See also, [Sh] for another proof via ‘tight closure’.)References[BK] Brion, M. and Kumar, S.: Frobenius splitting methods in geometry andrepresentation theory, Birkhäuser, Boston (2005).[EH] Enescu, F. and Hochster, M.: The Frobenius structure of local coho-mology, Preprint (2007) (To appear in J. Algebra and Number Theory).[S] Schwede, K.: F -adjunction, Preprint (2009).[Sh] Sharp, R.Y.: Graded annihilators of modules over the Frobenius skewpolynomial ring and tight closure, Trans. Amer. Math. Soc. 359 (2007),4237-4258.Addresses:S.K.: Department of Mathematics, University of North Carolina, ChapelHill, NC 27599-3250, USA(email: shrawan@email.unc.edu)V.M.: School of Mathematics, Tata Institute of Fundamental Research,Calaba, Mumbai 400005, India(email: vikram@math.tifr.res.in)3

Finiteness of the Number of Compatibly Split Subvarieties

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Finiteness of the number of compatibly-split subvarieties

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