When exist ? " - linear does p a map acknowledgment : I Land on the traditional work People 8 Fires territories of Three The of faith ) Ojibwe ( keepers The T urn ( keepers of ) trade Odawa The Ann the fire ) ' wadmi ( keepers of The Bode Frm the this talk the land of I giving on am Kiikaapoi . mum on joint Based with work ' 20 , Takumi Murayama ' 18 , Smith Karen & Takumi Murayama t Karen Smith in preparation .
⇒ Throughout R of noetherian domain char . is p > 0 prime a and K : = Frack ) . e [ Detects singularities of - e : R → t*R Frobenius F : rise ( e > o ) R T r to e F*R with R - mod Here but R structure is ring a as by given = rpex ( restriction of , × eF*eR reps v. × scalars ) AFearmapeipsanR-linearn F* → R y : . = Ifpcx , y ) free R Etmpe : R then F*R - mod is a , basis with - xiyj Of i. j - I e p . , F*R → R p : = { y ( xiyt ) the O =j basis if given by I i on - - otherwise J O . Frobenius epli Hing This is a
- e Why do of - linear maps ? about existence care we nonzero p a variety such maps global of X variants • on , satisfies Kodaira X especially splittings , imply - - Ramanathan ] [ Mehta Tanning Y Kodaira vanishing fails in general ( Raynaud ) char in p . the theory used test extensively of ideals in • a , of multiplier ideals char analogue prime . [ Hochstein , Huneke , Hara Smith , Yoshida , Takagi Watanabe , , , among others ] Lyubeznik , Schwed e , Blickle , Tucker , Sharp Aber back , Enescu , F - signature study used in the of , and • recently , more it non - local variant . [ Smith , Vanden Bergh , Leushke , Tucker , Huneke , Aber bach , Enescu , De Stefani , Polska others ] Yao , Singh among , " il sufficiently existence implies E of such • maps many Cohen - Macaulay - Huneke ] [ Hoch steer is rumrunner T F- regular rings strongly .
⇒ • If K = Fracas ) ( K : KP ) so satisfies then , of - ' - linear map implies existence a nonzero p [ Smith - D ] R excellent is Mmmm I Large that of behave under integral class well ⑧ rings of and completions other loci closures regular openness . , , Resolution of thins Deep such singularities conjectured * as hold for to this class . QuesetionibhendoesRhavenonzerop-e-linearmap.IE is finite , xamplel Exercise : R → F*R then If F : nonzero exist ! - e - linear p maps - : KP ) so CK - e If then - linear existence of non zero p a , is finite Frobenius [ Smith - D ] maps .
⇒ Above example its and examples converse gives many - e of - linear maps with excellent ¥0 non rings p nonzero - . " Folklore " for example If if R R excellent 8 is is nice , , , = - linear then does R - e admit - ? nonzero p maps Theorem A ( Murayama - D ] For integer each F : n > 0 , TT local excellent regular - - Hens elian - Krull dim that does of not admit ring R A¥y w - e - linear p map nonzero . F - split F - pure are NII that Thus , I excellent rings . also raised by Hochstein long-standing Answers others of question a , like , Brickle Smith etc Zhang Schroeder . , , class of answer for large has positive Folklore question excellent but non-F.fi#te rings . = - . # Theorem B [ Murayama finite - D ] R essentially If is of : - type complete local - ring , a over - t - linear has R then nonzero p maps . F - split , for - pure F suck R Furthermore , .
⇐ Open ( ? ? ) there Question : Are - excellent 101 R that admit non - linear non-trivial - e ? p maps I hypothesis drop local then construct If examples such can we ( forthcoming - D ) Murayama work construction from ThmAproofsketo.mg we use a rigid analytic geometry # . NA field a field ( k , 11 ) A equipped with is 11 k → IR > o : satisfying 1 × 1=0 ① O x = IN ly I Kyl ② = max { IN , lyl } ( ultra metric I - inequality ) lxtyl ③ E ( k , Il ) becomes assume k metric Ix - yl and space via a we is complete wrt this metric the Tatealgebra have . duck k For : = { ÷Ia ; Xi i → a } Tick ) lait → 0 C- KEW ) as : . T , Ck ) is " KEI local ) Rigid analytic of ( analogue regular not - . ( Kiehl ) excellent - Euclidean domain - .
For 42,11 ) - Doo has of Tick ) Murayama char p > o , a - e map ⇒ k - linear has p nonzero a - e - linear continuous p nonzero map . mmmm t , Gabbert Blaszczok ( now Rzepka ) - Kuhlmann : F NA fields k - linear that " admit D¥T continuous p maps . functional non - Archimedean This analysis uses . To get local Hensel ian localize counterexample you , Henselize T ( x ) then , for , Ck ) at the ideal and max field Gabbert Rzepka given by k Kuhlmann NA a - . BLACK LIVES MATTER to INDIGENOUS I LIVES MATTER 0 LGBTQ LIVES I MATTER 0
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