F -purity of hypersurfaces by Daniel Jes´us Hern´andez A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2011 Doctoral Committee: Professor Karen E. Smith, Chair Professor Melvin Hochster Professor Mircea Immanuel Mustat¸˘a Professor Guy A. Meadows Assistant Professor Wenliang Zhang c Daniel Jes´us Hern´andez 2011 All Rights Reserved Dedication: To my friends and family; I love you all ii ACKNOWLEDGEMENTS I thank my parents, Luis and Mercedes, and my brothers, Alfonso, Carlos, and Manuel, for their unconditional love and support. They inspire everything I do, and without them I would never have made it to where I am now. I also thank my grandparents, Maria de Jes´us and Leopoldo Hern´andez, and Elisa and Roberto Mart´ınez, as well as my many aunts, uncles and cousins. I’m lucky to have such a wonderful family. The presence of the beautiful Emily Witt in my life further illustrates how lucky I am. Emily has supported me every day for almost six years, and I surely would never have finished my Ph.D. without her. The time we spent together has been the best part of my time in Michigan. Coming to Ann Arbor has been the best decision I’ve made in my life, and Karen Smith is the person most responsible for making it happen. I can’t imagine having had a better advisor and mentor than Karen, and I thank her for having the patience to take me as a student. Karen is one of my best friends, and I’m very happy to have her, Sanelma, Helena and Tapio in my life. We have been through a lot together, and I thank her for teaching me so much about life and math. I hold Mel Hochster in the highest esteem, and am very grateful for all that he has taught me. The dedication and generosity he shows towards his students is something that I deeply admire, and his example is one I will strive to follow in my career. I also especially thank him for being so good to Emily, and for introducing us iii to cryptic crossword puzzles, as well as to the Michigan Math and Science Scholars program, of which I have many fond memories. I am very proud to say that I’d run through a brick wall for Mel. My decision to pursue a graduate degree in mathematics is due almost entirely to David Fried and Jeehoon Park, both of whom I met in Boston. Then a graduate student, Jeehoon was my first mentor. I thank him for all the time and attention he gave me, and for being such a good friend. I also apologize to him for not studying number theory. I also thank David Fried for giving me my first reading course, for being such a positive influence on me, and for helping me find the confidence to study math. I’ll never forget either of them. Without a doubt, the best memories I have of Ann Arbor are of time spent with my friends. I’d like to especially thank Gerardo Hern´andez Due˜nas, Yadira Baeza Rodr´ıguez, Gerardito Hern´andez Baeza, Luis N´u˜nez Betancourt, Luis Serrano Herrera, Jos´eLuis Gonzalez Zapata and Erin Emerson, Hyosang Kang, Richard and La´ısVasques, Ben and Alison Weiss, and Greg McNulty for their friendship. Though many others deserve mention, Rackham requires that my acknowledgements be shorter than the actual thesis, and I apologize for any omissions. Additionally, I’d like to thank an important group of people, the Math Basketball All-Stars. I am grateful to the following (math-hoops) friends for coming out and sharing part of their weekend with me: Ryan Kinser (the founder), Kevin Tucker, Alan Stapledon, Yogesh More, Diane Vavrichek, Jasun Gong, Russell Golman, Kyle Hofmann, Kacey Walker, Gerardo Hern´andez Due˜nas, Luis N´u˜nez Betancourt, Jae Kyoung (JK) Kim, Yu-Jui (Jay) Huang, Ming Qin, Chong Hyun Park, David Benson- Putnins, Greg Simon, and Jonah Blasiak. The time spent playing with them has made every sprained ankle and loose tooth worthwhile. iv I am especially grateful to Karen Smith, Mel Hochster, Mircea Mustat¸˘a, Guy Meadows, and Wenliang Zhang for serving on my thesis committee, to the NSF for support through RTG grant number 0502170, and to Rackham for support through a Rackham Merit Fellowship. Last, but not least, I would like to thank Bennet Fauber for being a cool and helpful guy, Stephen DeBacker for teaching me some number theory and for some awesome MMSS celebrations, Igor Kriz for teaching me some topology and for hang- ing out at social hour, Karen Rhea for being a great friend and teaching mentor, Yvonne Lai for showing me how to run a course, Bob Cranson for trying to get me into shape, and Tracy Coburn for being the friendliest custodian in East Hall. v TABLE OF CONTENTS DEDICATION .................................. ii ACKNOWLEDGEMENTS .......................... iii LIST OF FIGURES ............................... viii CHAPTER 1. Introduction .............................. 1 1 Motivation ............................ 1 2 Outline .............................. 6 2. Singularities of hypersurfaces .................... 8 1 Singularities of hypersurfaces defined via L2 conditions .... 8 2 Some standard constructions in positive characteristic .... 13 3 Singularities of hypersurfaces defined via the Frobenius map . 14 4 F -singularities of hypersurfaces in F -finite regular rings ... 18 th 1 1 pe Frobenius powers ................ 18 2 Test Ideals and F -jumping numbers ......... 21 3. Some connections ........................... 24 1 Preliminaries ........................... 25 2 On reduction to positive characteristic ............. 28 3 On F -purity for pairs reduced from C ............. 30 4 A connection between singularities ............... 33 4. On base p expansions ......................... 37 5. F -pure thresholds of hypersurfaces ................ 43 1 Truncations of the F -pure threshold .............. 43 vi 2 The set of all F -pure thresholds ................ 46 3 Purity at the F -pure threshold ................. 48 6. Splitting polytopes .......................... 51 7. F -purity and log canonical singularities ............. 55 1 Some background on F -purity of monomial ideals ....... 55 2 F -pure thresholds for polynomials with good support ..... 58 3 Log canonical singularities and dense F -pure type ...... 60 8. F -singularities of diagonal hypersurfaces ............. 67 1 A detailed discussion of the main results ............ 67 1 The F -pure threshold of a diagonal hypersurface .. 67 2 Test ideals and higher jumping numbers ....... 69 2 F -pure thresholds of diagonal hypersurfaces .......... 71 3 Test ideals of diagonal hypersurfaces .............. 73 1 ProofofTheorem8.4 ................. 73 2 Some supporting lemmas ............... 75 4 On higher jumping numbers of Fermat hypersurfaces ..... 78 1 More supporting lemmas ............... 79 9. F -pure thresholds of binomial hypersurfaces .......... 85 1 The polytope P associated to a binomial ........... 85 Õ 2 Some lemmas on the computation of fptm Ôf ......... 88 3 The F -pure threshold of a binomial hypersurface ....... 91 1 An example ...................... 92 2 An algorithm ..................... 93 3 ProofofTheorem9.14 ................ 95 4 Proof of the main technical lemmas ......... 96 BIBLIOGRAPHY ................................ 104 vii LIST OF FIGURES Figure 5.14.1 Three intervals that do not intersect FPT2. ............. 48 3 a b c c Ö × Ø Ù 6.7.1 The rational polytope P 0, 1 associated to x ,y , x y . ... 53 ´ 3 d1 d2 d3 Ö × 6.9.1 The rational polytope P 0, 1 associated to x1 , x2 , x3 . .. 53 2 4 7 9 8 4 Ö × Ø Ù 9.3.1 The rational polytope P 0, 1 associated to xy z , x y z . 86 9.7.1 The decomposition of P from Example 9.3. .............. 87 3 Ô Õ Ô Õ m 9.15.1 p 43, ℓ L 1, fpt f 16 1 .8 base 43 . ......... 93 3 Ô Õ Ô Õ m 9.15.2 p 37, ℓ L 2, fpt f 16 2 ε .6 34 22 7 14 29 base 37 . 94 viii CHAPTER 1 Introduction 1 Motivation th Let R be a commutative ring of prime characteristic p 0. The e -iterated e F pe Frobenius map R R (defined by r r ) is a ring homomorphism whose image pe e th Ô Õ is the subring R R consisting of all p powers of elements of R. We call R F -finite whenever the Frobenius map is a finite map of rings, and in this thesis we will deal almost exclusively with F -finite rings. The Frobenius map has been an important tool in commutative algebra since Kunz characterized regular rings as those for which R is flat over Rp [Kun69]. In general, singular rings exhibit pathological behavior with respect to the Frobenius endomorphism, and by imposing conditions on the structure of R as an Rp-module, new classes of singularities can be defined. For example, we say that R is F -pure p p (or F -split) if the inclusion R R splits as a map of R -modules [HR76]. This condition is equivalent to the condition that R contain a free Rp-module summand of rank one. Note that a regular ring is F -pure, though the converse need not be 2 2 ×ßÔ Õ true: the ring Fp Öx, y x y is F -pure for p 2, yet always has an isolated singularity at the origin. The notion of F -purity is a critical ingredient in the proof of the well-known Hochster-Roberts Theorem on the Cohen-Macaulay property of 1 2 rings of invariants [HR74]. Recently, more subtle applications of Frobenius to define singularities has led to new classes of F -singularities, many of which are motivated by the theory of tight ¦ closure. The tight closure of an ideal I R is an ideal I R which contains I and is “tight” in the sense that it is contained in (and is often much smaller than) many of the more common closures of I (e.g., the radical and integral closure of ¦ I).