Characteristic p Techniques in Commutative Algebra and Algebraic Geometry Math 732 - Winter 2019 Karen E Smith1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109 E-mail address: [email protected] 1Acknowledgements: These notes loosely follow notes by Karl Schwede for a course on similar topics in Spring 2017 at the University of Utah. The arguments in Chapter 1, Section 3 draw heavily from notes of Mel Hochster. The section on Fedder's Criterion draws heavily from Karl's notes. Preface The goal of this course is to introduce the Frobenius morphism and its uses in commutative algebra and algebraic geometry. These characteristic p techniques have been used in commutative algebra, for example, to establish that certain rings are Cohen-Macaulay, as in the famous Hochster-Roberts theorem for rings of invariants (over fields of arbitrary character- istic). In more geometric settings, characteristic p techniques can help analyze or quantify how singular|that is, how far from being smooth|a particular variety may be, or to establish that the singularities are suitably mild. Vast bodies of research in birational algebraic geometry that had been developed for complex varieties using differential forms and L2-techniques have now been seen to have \characteristic p" analogs. These can be used to recover much of the theory in the complex setting, but even better, can serve as replacements to the analytic techniques to establish results for varieties of prime characteristic p. Examples of such analogs we will discuss in this course include the F-pure threshold and test ideals, which are \characteristic p analogs" of the log canonical threshold and multiplier ideals in complex geometry. In this course, we will introduce this machinery, while also introducing and/or re- viewing many of the sub-areas of commutative algebra and algebraic geometry where it has been useful. We begin with Kunz's characterization of regularity (smooth- ness) as the flatness of Frobenius. First, of course, we review regular local rings more broadly, including Serre's famous characterization in terms of global dimension. From there, we will consider classes of \mild singularities" for complex varieties, review- ing the definitions of rational, log canonical and log terminal singularities which are defined using resolutions of singularities. These all have \characteristic p analogs" defined using Frobenius, which we will explore in depth. We will then study some global implications of Frobenius splitting, including vanishing theorems for line bun- dles and characterizations of Fano (or nearly Fano) varieties. As time and student interest permit, we may study singularity invariants such as the log canonical thresh- old and its \characteristic p analog" called the F-pure threshold, as well as multiplier and test ideals, Hilbert-Kunz Multiplicity and/or F-signature. CHAPTER 1 Introduction 1. Overview Let X be a Noetherian scheme of prime characteristic p. The Frobenius mor- phism is the scheme map F : X ! X OX ! F∗OX which is the identity map on the underlying topological space X but the pth power map on sections. For example, when X = Spec R is affine, the Frobenius map is essentially just the ring homomorphism R ! R sending r 7! rp. An important special case to keep in mind is the case where X is a variety over an algebraically closed field K of characteristic p > 0. The Frobenius map is a powerful tool, both in commutative algebra and algebraic geometry. Some of its uses include (a) Detecting regularity of X: A famous theorem of Kunz says that X is regular if and only if the Frobenius morphism is flat. For varieties, this says that X is smooth if and only if F∗OX is a locally free sheaf (or vector bundle). (b) By relaxing the flatness condition for the Frobenius in various ways, we get a host of other \mild singularity" classes in prime characteristic, with names you may have heard before such as F-regular and F-pure. Many theorems that hold for smooth varieties over fields of characteristic p can be extended to these classes of \F-singularities." (c) Frobenius can be used to detect \mild" singularities of complex varieties as well. Singularities of importance in complex birational geometry such as Kawamata log terminal or canonical singularities can be understood by \reduction to prime characteristic," where surprisingly checkable criteria for these singularities can be given in terms of the Frobenius map. (d) In commutative algebra, the Frobenius can be used to proving certain rings are Cohen-Macaulay or have other nice properties. Again, by reduction to prime characteristic, such techniques can be applied to any ring containing a field. (e) The Frobenius map can be used to prove vanishing theorems for cohomology of line bundles on varieties of prime characteristic under certain conditions. Thus Frobenius can often be used when resolution of singularities, Kodaira Vanishing, or other tools fail (or are unknown) in characteristic p. 3 4 1. INTRODUCTION (f) Frobenius can be used to define numerical invariants of singularities. For example, the F-pure threshold is an invariant defined using the Frobenius which is an analog of the analytic index of singularities (or log canonical threshold) for complex varieties. A smooth point on a hypersurface always has F-pure threshold 1, but singular points have smaller (positive) thresholds| the smaller, the more singular. As one concrete example, consider the simple cusp defined by y2 = x3. This singularity has F-pure threshold equal to 1=2 in char 2 2=3 in char 3 5 1 6 − 6p if char p = 5 mod 6 5 6 if char p = 1 mod 6: This reflects the fact that the cusp is somehow most singular in characteristic 2, slightly less singular in characteristic 3, and increasing less singular for large p. For infinitely many p (namely, those p congruent to 5 mod 6), it it is not really anymore singular than it is over complex numbers, in the sense that the log canonical threshold of the cusp over C is also 5=6. The fact that the cusp is \more singular" in some characteristics, even infinitely many, than it is over C is deeply connected to arithmetic issues related to supersingularity for elliptic curves. Here is one specific long-standing conjecture: for a given polynomial with integer coefficients (say), the log canonical threshold of the corresponding hypersurface over C is equal to the F-pure threshold of the hypersurface considered over Fp for infinitely many p. Our goal will be to expand on all of these topics over the semester. 1.0.1. Local Properties. Let P be a property of schemes (or, say, of varieties or of topological spaces). We say that \P is a local property" if for any scheme X, X has property P if and only if for all x 2 X, there is an open neighborhood U where property P holds. Note the locus X0 of points of an arbitrary scheme X where the local property P holds will be an open set of X. For example, smoothness and normality are both local properties of complex varieties. Any property characterizing the `singularities' of a scheme ought to be a local property. Typically, a property of a scheme X may be defined in terms of ring properties for its stalks OX;x. That is, we may say that a point x has property P if and only if the stalk OX;x has property P. Reducedness and normality are such a properties of schemes, as are more exotic properties such as Cohen-Macaulayness and Gorenstein- ness. Property P holding for X if and only if it holds for all points x 2 X follows immediately if P is a local property, but this is weaker assumption in general. Note that if P is such a local property and some stalk OX;x has property P, then all further localizations OX;y must have property P. Indeed, to say that OX;y is a further localization of OX;x is to say that the point x is in the closure of y, so every open set containing x will also contain y. 1. OVERVIEW 5 Non-local properties include connectedness, or the property that the sheaf !X has a non-zero section. It is not always obvious whether or not a given property is local, even when we expect it to be. For example, the property of regularity for Noetherian schemes, introduced mid-last century as a scheme analog of smoothness for algebraic varieties, was long expected to be a local property. However, until the advent of homological algebra, it wasn't even known that a localization of a regular local ring is regular; this was settled in the 1950's by Serre (and independently Auslander-Buchsbaum) who gave new homological characterizations of regularity. Unfortunately, it turns out that regularity is not a local property [Hoc], though it is for large classes of rings with mild hypothesis. CHAPTER 2 Regularity and Kunz's theorem 1. Frobenius We begin with the \local" picture, studying Frobenius for rings. Setting 1.1. The word \ring" in these notes always means a commutative ring with unity (unless explicitly stated otherwise). Our rings are usually Noetherian. Recall that a Noetherian ring is called local if it has a unique maximal ideal. The notation (R; m; K) will always denote a Noetherian local ring with maximal ideal m and residue field K = R=m. We start with some examples of the types of rings we are interested in. Example 1.2. (a) The polynomial ring C[x1; : : : ; xn]; this is the ring of polyno- mial (or regular) functions on the variety Cn.