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Tight Closure in Equal Characteristic Zero TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO by Melvin Hochster and Craig Huneke Contents PREFACE CHAPTER 1. PRELIMINARIES (1.1) Introduction (1.2) Conventions of terminology and notation; alphabetical index of terms and notations (1.3) The main results (1.4) Tight closure theory and test elements in positive characteristic (1.5) Some new results on test elements (1.6) F -regularity in positive characteristic (1.7) Further tight closure theory in positive characteristic CHAPTER 2. AFFINE ALGEBRAS (2.1) Descent data and descent (2.2) Tight closure for affine algebras over fields of characteristic zero (2.3) Comparison of fibers (2.4) Universal test elements (2.5) Basic properties of tight closure over affine algebras Version of October 8, 2020. Both authors were supported in part by grants from the National Science Foundation. We wish to express our appreciation to Florian Enescu for his corrections to an earlier version of this manuscript. There has been a change of numbering from versions of this manuscript dated prior to August 20, 1997. Section (1.5) has been added, and some of the material previously in Section (2.4) is now in Section (1.5). The previous (1.5) and (1.6) have become (1.6) and (1.7), respectively. Sections (3.6) and (3.7) of early versions have become Sections (4.1) and (4.2), respectively. 1 2 MELVIN HOCHSTER AND CRAIG HUNEKE CHAPTER 3. ARBITRARY NOETHERIAN ALGEBRAS OVER A FIELD (3.1) More about descent: affine progenitors (3.2) Definition and basic properties of direct and formal K-tight closure (3.3) Artin approximation and the structure of formal power series rings (3.4) The locally excellent case (3.5) Height-preserving descent from complete local rings over K to affine K-algebras CHAPTER 4. FURTHER PROPERTIES OF TIGHT CLOSURE (4.1) Some major applications (4.2) Change of rings (4.3) F -regularity and F -rationality (4.4) Phantom homology and homological theorems (4.5) Iterated operations and constraints on parameters (4.6) Big equational tight closure APPENDIX. QUESTIONS TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO 3 PREFACE The main objective of this monograph is to lay the foundations of tight closure theory for Noetherian rings containing a field of characteristic 0. However, we intend and hope that it will be useful in several other ways. The first chapter contains, in effect, a survey of tight closure in characteristic p. Defi- nitions and theorems are given in full, although for most proofs we simply give references. This chapter provides a quick introduction to characteristic p tight closure theory for newcomers, and may be a useful guide even for expert readers in finding material in the existing literature, which has become rather formidable. Moreover, it contains all of the equicharacteristic p material that is needed for the construction of the equicharacteristic 0 theory. For the most part, the equal characteristic zero theory only requires knowledge of the characteristic p theory for finitely generated algebras over a field and, in fact, the field can generally be taken to be finite. When no great cost is involved, we have frequently included results in greater generality, but we have not given the most general results known when too many technicalities would be involved. A few of the results in this section are new. The equicharacteristic 0 theory rests heavily on the method of reduction to characteristic p. The results of the second chapter are aimed at supplying what is necessary in this direction: almost nothing here is new, but we had difficulty locating references for the facts that we needed in the right form and generality, and so we took this opportunity to give a self-contained treatment of what we required. It was our intention to make the method of reduction to characteristic p understandable and accessible to an audience with only a modest background in commutative algebra and algebraic geometry. While our 4 MELVIN HOCHSTER AND CRAIG HUNEKE treatment was certainly influenced by the need to provide the tools that are required to develop tight closure theory, it was constantly in our minds that this chapter should enable many mathematicians to acquire the technique of reduction to characteristic p. Many of the results are concerned with comparing the behavior of the generic fiber of a map from an integral domain A with the behavior of \almost all" closed fibers, i.e., of comparing what happens when one tensors with the fraction field of A with what happens when one tensors with A/µ for µ a maximal ideal in a suitably small Zariski dense open subset of Max Spec A. We have not made an effort to scour the literature to determine first sources for these results. Many can be extracted, in some form, from the massive Grothendieck- Diedonn´etreatise El´ementsde´ G´eom´etrieAlg´ebriques | see for example [EGA1], [EGA2] in the Bibliography. The third chapter develops the basic properties of tight closure in equal characteristic 0. We note here that there is a theory for Noetherian K-algebras for each fixed field K of characteristic 0. We do not know whether all these notions agree. The theory for K = Q gives the smallest tight closure, which we call equational tight closure. Another notion, big equational tight closure, is treated in the fourth chapter. The fourth chapter contains a deeper exploration of properties of tight closure, including many of the most important and useful properties: that every ideal is tightly closed in a regular ring, that tight closure captures colons of parameter ideals (and so can be used to measure the failure of the Cohen-Macaulay property), that a ring in which every ideal (or even every parameter ideal) is tightly closed is Cohen-Macaulay (and normal), that direct summands of regular rings are Cohen-Macaulay (as a corollary of the first three character- istics of tight closure), that there is a tight closure version of the Brian¸con-Skoda theorem, and that there is a theory of \phantom homology" analogous to the characteristic p the- ory. The fourth chapter also contains a treatment of change of rings, and a brief treatment of rings with the property that every ideal is tightly closed (weakly F -regular rings) and those with the property that parameter ideals are tightly closed (F -rational rings) and their connection with rational singularities. There is also a likewise brief discussion of yet another notion of tight closure, big equational tight closure, as mentioned above. It gives TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO 5 a larger tight closure than any of the other theories, but still has the crucial property that all ideals of regular rings are tightly closed. It was our original intention to give more detailed treatments of several of the topics in the fourth chapter, but we have decided that it is more important to make what is already written available at this time. The Appendix contains a list of open questions. For the convenience of the reader we have provided a complete glossary of notation and terminology in (1.2.4). We have also included a very extensive bibliography in the hope that it will prove helpful to readers in dealing with the now voluminous literature on tight closure. Mel Hochster and Craig Huneke Ann Arbor, Michigan and Lawrence, Kansas March, 2019 6 MELVIN HOCHSTER AND CRAIG HUNEKE CHAPTER 1. PRELIMINARIES In this chapter we first give an introduction that explains some of the motivations for studying tight closure, gives an overview of the entire monograph, and makes connections with the literature. The second section reviews many conventions concerning terminol- ogy and notation and also contains an alphabetical list showing where various terms and symbols are defined. The third section gives a summary of the main results of the entire monograph. The last three sections review the theory of tight closure and F -regularity in characteristic p. (1.1) INTRODUCTION In [HH4] (see also [HH1-3]) the authors introduced the notion of tight closure for Noe- therian rings of characteristic p. The original motivation for studying tight closure in characteristic p is that it gives very simple proofs of a host of results that, before the development of this method, did not even seem related. These include: (1) A new proof that direct summands of regular rings are Cohen-Macaulay (hence, that rings of invariants of linearly reductive linear algebraic groups acting on regular rings are Cohen-Macaulay). (2) A new proof of the Brian¸con-Skoda theorem. (3) New proofs of various local homological conjectures. Moreover, in every case, many new results are obtained, often very strong and quite unexpected generalizations of previous results. Already at the time that [HH4] was written the authors had worked out, in a preliminary form, a theory of tight closure in equal characteristic zero, provided that the base ring is a finitely generated algebra over a field. A definition for the tight closure of an ideal in this case is given in [HH4]. TIGHT CLOSURE IN EQUAL CHARACTERISTIC ZERO 7 Our objective in this paper is to present, in a greatly improved form, the theory antic- ipated in [HH4]. One improvement over what is described in [HH4] is that this theory of tight closure is defined for all rings containing a field of characteristic zero. Moreover, it behaves sufficiently well to give counterparts for all of the results of types (1), (2), and (3) described above. In the next section of this chapter we give some terminological and notational conven- tions, as well as an alphabetical index of terms and notations. In the following section we give a summary of some of the main results of the paper, and in the last three sections of this chapter we present a brief r´esum´eof what is needed repeatedly throughout the paper from tight closure theory in characteristic p.
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