F -pure threshold Fedder type criterion

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Rossana Salvatori
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1 1 F -pure threshold Fedder type criterion 2, 7, ,,.,,., F,. F -pure threshold,., log canonical threshold ( 0 ) -,Mustata- -. log canonical threshold, log resolution. F -pure threshold Frobenius. log canonical threshold, F -pure threshold. ( F -pure threshold ), F -pure threshold, F -pure threshold 2.6,. 2. Noether. p, q e N p e. R R R \ P Min R P. R p, R p R p. F : R R x x p Frobenius. R p, e R R R R R R (x, a) x q a m08023c@math.nagoya-u.ac.jp

2 2 R. e R, F e R R. e R R. R I, e R I e I. R x e x., R, e R R I x e I, e x. I I [q], I q. I {a i }, I [q] {a q i }, e R e (I [q] )=I e R R p. 1 R R, R F-finite. F-finite ( (excellent ring). [9]p.260, [13]p.385, F-finite [8] ).,, F-finite., R F-finite, R,, F-finite, F-finite R p. F : R 1 R, R F-split., R M F 1:R M 1 R M, R F-pure. split, pure F. F -split, ψ( e 1) = 1 ψ Hom R ( e R, R). puresubring. (R ) split pure, [5] Corollary 5.3 F -finite Noether F -pure F -split. 2.1 F-purity p, F-pure threshold. F-pure threshold 0 log canonical threshold. log canonical threshold. [11]. 0 p F-pure threshold R p F-finite, a a R R, t R 0. (R, a t ) F-pure, q = p e, a t(q 1) α, R split. R F e e R e α e R x e x q e (αx q ) 2.5. R p, a R a R. sup{s R 0 (R, a s ) F-pure} a F-pure threshold, fpt(r, a)., R fpt(a).

3 F -pure threshold 3, (R, m), fpt(m). fpt(m),. fpt(m) [11]. 2.6 ([11] Theorem 2.7(1)). (R, m,k) p F-finite F-pure, k dim R = d.. 1) R. 2) fpt(m) >d 1. 3) fpt(m) =d., F -pure threshold., Fedder. 2.7 (Fedder, [11] Lemma 1.8). (, m) p F-finite, I,a, R = /I., (R, (ar) t ) F-pure, q a t(q 1) (I [q] : I) m [q]. 2.2 F -pure threshold, F -pure threshold., k, X =(X ij 1 i r, 1 j s) k r s, I = I t (X) (1 t min{r, s}) X t = k[x]., /I. k p,. m. r = s = t, 2 R = k[[x, Y, Z, W]]/(f = XY ZW). f q : f =(f q 1 ), p 2, (XY ZW) q 1 2 f q 1 (X, Y, Z, W) [q], p =2, (XY ) q 2 1 (ZW) q 2 f q 1 (X, Y, Z, W) [q] 2.7 fpt(m) =2.. (r, s, t) =(2, 3, 2). 2.9 ([2] Proposition 4.7). k p, X =(X ij 1 i 2, 1 j 3) k 2 3, = k[[x]],i = I 2 (X) X 2, (I [p] : I)=I [p] + I 2p 2. fpt(m).

4 q 2 fpt(m) 2+ 4 p. m t(q 1) (I [q] : I) m [q] t. m. [q/p] := k[[x q/p ij 1 i 2, 1 j 3]] [q/p], J X q/p =(X q/p ij ) 2 [q/p], J 2.9, J [p] : [q/p] J = J [p] + J 2p+2. q p [q/p] [q/p],.,.,. m 2q+4q/p 5. (J [p] : [q/p] J) = J [p] : J = I [q] : I [q/p] I [q] : I (J [p] : [q/p] J) =(J [p] + J 2p 2 ) = I [q] +(I [q/p] ) (2p 2) m [q] + m4q 4q/p (I [q] : I) m 2q+4q/p 5 m 4q 4q/p = m 6(q 1)+1 m [q] 2q + 4q p fpt(m) lim 6 =2+ 4 q q 1 p 2.11 ([2] Lemma 4.2)., I ht I = d. q I [q] : I,. I [q] : I I [q] + I dq d ht I =2, 2.11 I [q] : I I [q] + I 2q 2. 1, 3 X 12 X 23 X 13 X 22,X 12 X 23 X 13 X 22, q 1 1 q 1 3 I 2q 2 \ m [q]., X (X 11 >X 12 >X 13 >X 21 >X 22 >X 23 ), 1, 3 X 12 X 23,X 11 X 22, q 1 1 q 1 3 X q 1 11 Xq 1 12 Xq 1 22 Xq X q 1 13 Xq 1 21 q 1 1 q 1 3 m [q], m2(q 1) (I [q] : I) m [q] 2 fpt(m)., t =2. [7], F -pure threshold, k p fpt(m) =2.

5 F -pure threshold 5,.,, 2.9, k p, X =(X ij ) k r s (r s), = k[[x]], m = X,I = I r (X),R= /I, m = m R. fpt(m) =r(r 1).. I [q] : I.. i (1 i s r + 1), X i i + r 1 r. hti = s r +1, 1,..., s r+1. (r, s) =(2, 3) X ij X i j i>i i = i j>j, i X 1i X r i+r. q 1 1 q 1 s r+1 q 1, r(r 1) m r(r 1)(q 1) q 1 1 q 1 s r 1 m[q] s r 1 (I [q] : I) \ I [q], q r(r 1) fpt(m). m r(r 1)(q 1) (I [q] : I) m [q]. hti = d, I [q] : I m [q] + m rdq rd Macaulay , 2.12 d = s r +1, 1, fpt(m) r(r 1). m r(r 1)(q 1)+1 (I [q] : I) m r(r 1)(q 1)+1 (m [q] + m rqd rd ) m [q] + m (rd+r(r 1))(q 1)+1 m [q] [1] M. Blickle, Multiplier ideals and modules on toric varieties, Math. Z. 248 (2004), no. 1, [2] R. Fedder, F-purity and rational singularity, Trans. Amer. Math. oc. 278 (1983), no. 2, [3] N. Hara, K. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. oc. 355 (2003), no. 8, [4] N. Hara, K.-i. Watanabe, F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom. 11 (2002), no. 2, [5] M. Hochster, J. L. Roberts, The purity of the Frobenius and local cohomology, Advances in Math. 21 (1976), no. 2,

6 6 [6] C. Huneke, The theory of d-sequences and powers of ideals, Adv. in Math. 46 (1982), no. 3, [7] D. Hirose, Formulas of F-thresholds and F-jumping coefficients on toric rings, arxiv: v2 [math.ac] 4 Aug 2008 [8] E. Kunz, Characterizations of regular local rings of characteristic p, Amer. J. Math. 91 (1969) [9] H. Matsumura, Commutative ring theory, Cambridge University Press 1986 [10]. Takagi, F-singularities of pairs and inversion of adjunction of arbitrary codimention, Invent. Math. 157 (2004), no. 1, [11]. Takagi, K.-i. Watanabe, On F-pure thresholds, J. Algebra 282 (2004), no. 1, [12] K.-i. Watanabe, F-regular and F-pure normal graded rings, J. Pure Appl. Algebra 71 (1991), no. 2-3, [13] W. Burns, J. Herzog, Cohen-Macaulay rings, Cambridge University Press 1993.

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