# 18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Kiehl s finiteness theorems

Size: px
Start display at page:

Download "18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Kiehl s finiteness theorems"

Transcription

1 18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Kiehl s finiteness theorems References: [FvdP, Chapter 4]. Again, Kiehl s original papers (in German) are: Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedische Funktionentheorie, Inv. Math. 2 (1967), ; and Theorem A und B in der nichtarchimedische Funktionentheorie, Inv. Math. 2 (1967), Addendum: Generic fibres more generally Jay pointed out a reference for a more general construction of generic fibres: de Jong, Crystalline Dieudonné theory via formal and rigid geometry (see Chapter 7, where he attributes the construction to Berthelot). Let P = Spf(R), where R is a quotient of a ring of the form R x 1,..., x m y 1,..., y n. (Note that the order of the two sets of brackets is important here!) You can then take the generic fibre of P to be the appropriate subspace of the product of the closed m-dimensional unit ball with the open m-dimensional unit ball (and its points will be the formal subschemes of P which are integral and finite flat over o K ). Another way to say the same thing is that you take, for each 0 < ɛ < 1 in the divisible closure of K, the subring of R x 1,..., x m y 1,..., y n consisting of series convergent for x 1,..., x m 1 and y 1,..., y n ɛ, form the quotient, and nest these affinoid spaces to get your rigid space. Addendum: Analytification of an algebraic variety Here s the question Andre asked last time, with my commentary on it. Suppose X is a quasiprojective algebraic variety over K. Embed X into some projective variety X; we can then equip the closed points of X with a rigid analytic structure by picking a model of X over K and taking the rigid analytic generic fibre of the completion along the special fibre. (This is independent of the choice of the model because the rigid analytic generic fibre doesn t see blowups in the special fibre.) The closed points of X form a subspace of X; the question is, is this subspace independent of the choice of X? It suffices to check that if X 1 X 2 is a proper morphism between two compactifications, then it induces an isomorphism on X as a rigid analytic space. (To compare two general compactifications, you can then compare each to the fibre product.) I think this is okay, but I didn t check it. Separated and proper spaces The category of rigid spaces has fibre products: these are generated by completed tensor products of affinoid algebras. We may thus say that a rigid space X is separated if the diagonal : X X K X is a closed immersion (i.e., is defined by a coherent sheaf of ideals). As for schemes, one has the following criterion for separatedness. 1

2 Proposition 1. A rigid space X is separated if and only if it admits an admissible affinoid covering {U i } such that for i j with U i U j, the intersection U i U j is affinoid and the canonical map O(U i ) K O(U j ) O(U i U j ) is surjective. In particular, any affinoid space is separated. If U is an affinoid subset of an affinoid space X, then U is said to be an interior subspace of X if it is contained in a subspace of the form f 1 (x),..., f n (x) ɛ for some f 1,..., f n O(X) such that K f 1,..., f n surjects onto O(X) and some ɛ < 1. (Note that the interior of X is not a useful notion here.) A rigid space X is proper if it is separated and there exist two finite admissible affinoid coverings {U i } i=1,...,n and {U i} i=1,...,n such that U i is an interior subspace of U i for i = 1,..., n. For instance, P n K (with homogeneous coordinates t 0,..., t n ) is proper because you can cover it by the subspaces U i,ɛ = {x P n K : t j (x) ɛ t i (x) } (i = 0,..., n) for any ɛ, and U i,ɛ is an interior subspace of U i,ɛ if ɛ < ɛ. Likewise, any closed analytic subspace of a proper space is proper, so the analytic space associated to a projective variety is also projective. The relative version of this construction is as follows. A morphism X Y is proper if after restricting Y to each element of some admissible affinoid cover, I can find two finite admissible affinoid coverings {U i } and {U i} of X, such that U i is a relative interior subspace of U i. The latter means that U i belongs to a subspace of U i of the form f 1 (x),..., f n (x) ɛ for some f 1,..., f n O(X) such that O(Y ) f 1,..., f n surjects onto O(U i) and some ɛ < 1. (This is admittedly a lousy definition; Kiehl rigged it up precisely to make the next theorem work. Is there a universally closed version of this definition, or a valuative criterion?) A morphism f : X Y is finite if for some admissible affinoid covering {U i } of Y, each f 1 (U i ) is affinoid and O(U i ) O(f 1 (U i )) is a finite morphism of affinoid algebras. One can recover f from the pushforward sheaf f O X, which is coherent; in particular, one can show that some may be replaced by any (see [FvdP, Definition 4.5.7]). Any finite morphism is proper, because f 1 (U i ) is a relative interior subspace of itself! Theorem 2 (Kiehl). (a) Let X be a proper rigid space over K. Then the (Čech) cohomology spaces of any coherent sheaf on X are finite dimensional over K. (b) Let f : X Y be a proper morphism of rigid spaces, with Y separated, and let F be a coherent sheaf on X. Then the direct image f F and the higher direct images R i f F are coherent sheaves on Y. In particular, the image f(x) is a closed analytic subspace of Y. Warning: the higher direct images here are constructed using Čech cohomology; I think you should be able to make a Čech/sheaf comparison in this relative case, but I didn t check it. 2

3 Sketch of proof. This sketch is basically the sketch from [FvdP, Theorem ]. The idea is to take the two coverings you have and show that, on one hand, you get the same Čech cohomology groups (by acyclicity) from both coverings, and on the other hand, the map F(U) F(U ) is a compact operator (a uniform limit of operators of finite rank) whenever U is an interior subspace of U. Thus on each cohomology space, the identity map is compact, which can only happen on a finite dimensional vector space. (The relative argument proceeds basically the same way.) I don t know a reference for rigid GAGA, but I seem to think it was written down somewhere by Kiehl. Anyway, [FvdP] says you can just imitate Serre s proof; rather than presume you know how this goes, I ll give a brief version here. (Serre s paper is as good a reference as any, maybe even better, because it s from the dark ages when Serre s Faisceaux algébriques coherents, which introduced sheaves into abstract algebraic geometry, was hot off the presses. It is: Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 ( ), His Théorèmes 1,2,3 are our (a),(b),(c).) Theorem 3 (Rigid GAGA). Let X be a projective algebraic variety over K. (a) For any algebraic coherent sheaf F on X, the natural homomorphisms from algebraic (sheaf) to analytic (Čech) cohomologies are bijections. (b) The analytification functor from coherent sheaves on X to coherent sheaves on the analytification of X is fully faithful. (c) Every analytic coherent sheaf comes from an algebraic coherent sheaf (so the functor in (b) is an equivalence of categories). Sketch of proof. Note that it s enough to prove everything for X = P n, since we can extend coherent sheaves by zero in both the algebraic and analytic categories to get from sheaves on X to sheaves on P n. For (a), you first prove it for O, then for the sheaves O(m) by induction on the dimension n of the projective space: if H is a hyperplane, you have an exact sequence 0 O( 1) O O H 0 where the first map is multiplication by a linear form vanishing on H. Then clever use of the five lemma gives you what you need. Since every coherent sheaf on P n is the kernel of a map from some O(m) to some O(n) (another theorem of Serre!), you end up getting (a) for all coherent sheaves. For (b), you apply (a) to the sheaf Hom between any two given algebraic sheaves: homomorphisms between the sheaves correspond to elements of H 0 of the sheaf hom. For (c) (the hard part), you again induct on dimension. The crux of the argument is to show that given F and a point x P n, you can find an integer m such that the analytic H 0 of the twist F(m) generates the stalk of F(m); by a compactness argument, you can choose m uniformly for all x. That is, all stalks of F(m) are generated by the global sections; 3

4 proceeding as in the proof of Kiehl s theorem on coherent sheaves on affinoid spaces, we deduce that F can be written as a cokernel of a map between algebraic coherent sheaves (both of the form O( m) d ), and by (b) is algebraic. The crux lemma is a bit intricate; it s Lemme 8 in Serre s paper. You again pick a hyperplane H and write down the exact sequence which on the right you can tensor with F: 0 O(H) = O( 1) O O H 0, F( 1) F F H 0; let K be the kernel on the left. Both K and F H are supported on H, so are algebraic by the induction hypothesis; in particular, by Serre s theorem, they lose their higher cohomology upon twisting by O(m) for m sufficiently large. If you split the four-term exact sequence 0 K(m) F(m 1) F(m) F H (m) 0 by adding G in the middle, you end up with surjections H 1 (P n,an, F(m 1)) H 1 (P n,an, G), H 1 (P n,an, G) H 1 (P n,an, F(m)); the point is that dim H 1 (P n,an, F(m)) is nonincreasing as m grows. It must thus stabilize after some m, at which point one finds that the global sections of F(m) surject onto those of O H (m). The latter generate the stalk of F H (m) at x for m large enough (because this is true for the algebraic stalk, and the map between the algebraic and analytic local rings is faithfully flat they have the same completion), so you win. Quasi-Stein spaces I would be remiss in not mentioning Kiehl s other big cohomological theorem, his analogue of Cartan s Theorem A and Theorem B in complex analysis. A rigid space X is quasi-stein if it admits an admissible affinoid covering U 1 U 2 in which the image of O(U i+1 ) is dense in O(U i ) for each i. Besides affinoid spaces themselves, examples include open balls, one-dimensional annuli, and products of other quasi-stein spaces. Here s Kiehl s main theorem about quasi-stein spaces. Kiehl s argument is a bit fragmentary (see below), so I m looking for a better reference (preferably not in German); suggestions? Theorem 4 (Kiehl). Let F be a coherent sheaf on a quasi-stein space X, with a covering {U i } as above. (a) The image of F(X) in F(U i ) is dense for all i. 4

5 (b) The cohomology groups H i (X, F) vanish for all i ( Theorem B ). Note that van der Put s theorem applies, so sheaf = Čech here. (c) For each x X, F x is generated as an O x -module by global sections of F ( Theorem A ). Proof. Note that F(U i+1 ) is dense in F(U i ), from which apparently (a) is immediate ( unmittelbar ), but I don t see why offhand. (Do you?) To prove (b), it s enough to prove H 1 always vanishes (by a dimension shifting argument); this is done by an explicit calculation (see p. 271 of Kiehl s Theorem A und B paper). To prove (c), let G be the ideal sheaf of x; since H 1 (X, F G) vanishes, the sequence G(X) = H 0 (X, F) H 0 (X, F/(F G)) = F x /m x F x 0 is exact, and (c) follows by Nakayama s lemma. This almost means you can pretend quasi-stein spaces are just like affinoid spaces from the point of view of the cohomology of coherent sheaves. However, note that (c) does not imply that F is generated by finitely many global sections, because there is no compactness argument available. And indeed, it may not be: e.g., let P 1, P 2,... be points in the open unit disc such that P i 1 as i, and take the direct sum of the ideal sheaves of the P i. On the other hand, it sometimes happens that all locally free coherent sheaves are generated by finitely many global sections even though the underlying space is not affinoid, e.g., on an open annulus (see exercise from an earlier handout). 5

### Locally G-ringed spaces and rigid spaces

18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Rigid analytic spaces (at last!) We are now ready to talk about rigid analytic spaces in earnest. I ll give the

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### Factorization of birational maps for qe schemes in characteristic 0

Factorization of birational maps for qe schemes in characteristic 0 AMS special session on Algebraic Geometry joint work with M. Temkin (Hebrew University) Dan Abramovich Brown University October 24, 2014

### Factorization of birational maps on steroids

Factorization of birational maps on steroids IAS, April 14, 2015 Dan Abramovich Brown University April 14, 2015 This is work with Michael Temkin (Jerusalem) Abramovich (Brown) Factorization of birational

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### THREE APPROACHES TO CHOW S THEOREM

THREE APPROACHES TO CHOW S THEOREM EVAN WARNER 1. Statement and overview The focus of this talk will be on Chow s theorem, a simple but very useful result linking complex analytic geometry to algebraic

### Synopsis of material from EGA Chapter II, 5

Synopsis of material from EGA Chapter II, 5 5. Quasi-affine, quasi-projective, proper and projective morphisms 5.1. Quasi-affine morphisms. Definition (5.1.1). A scheme is quasi-affine if it is isomorphic

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### 1. Algebraic vector bundles. Affine Varieties

0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

### Introduction and preliminaries Wouter Zomervrucht, Februari 26, 2014

Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally

### Preliminary Exam Topics Sarah Mayes

Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition

### Algebraic varieties. Chapter A ne varieties

Chapter 4 Algebraic varieties 4.1 A ne varieties Let k be a field. A ne n-space A n = A n k = kn. It s coordinate ring is simply the ring R = k[x 1,...,x n ]. Any polynomial can be evaluated at a point

### Cohomology and Base Change

Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is half-exact if whenever 0 M M M 0 is an exact sequence of A-modules, the sequence T (M ) T (M)

### Smooth morphisms. Peter Bruin 21 February 2007

Smooth morphisms Peter Bruin 21 February 2007 Introduction The goal of this talk is to define smooth morphisms of schemes, which are one of the main ingredients in Néron s fundamental theorem [BLR, 1.3,

### DESCENT THEORY (JOE RABINOFF S EXPOSITION)

DESCENT THEORY (JOE RABINOFF S EXPOSITION) RAVI VAKIL 1. FEBRUARY 21 Background: EGA IV.2. Descent theory = notions that are local in the fpqc topology. (Remark: we aren t assuming finite presentation,

### 15 Lecture 15: Points and lft maps

15 Lecture 15: Points and lft maps 15.1 A noetherian property Let A be an affinoid algebraic over a non-archimedean field k and X = Spa(A, A 0 ). For any x X, the stalk O X,x is the limit of the directed

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 47 AND 48

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 47 AND 48 RAVI VAKIL CONTENTS 1. The local criterion for flatness 1 2. Base-point-free, ample, very ample 2 3. Every ample on a proper has a tensor power that

### 14 Lecture 14: Basic generallities on adic spaces

14 Lecture 14: Basic generallities on adic spaces 14.1 Introduction The aim of this lecture and the next two is to address general adic spaces and their connection to rigid geometry. 14.2 Two open questions

### AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES

AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are C-linear. 1.

### where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset

Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETR CLASS 24 RAVI VAKIL CONTENTS 1. Normalization, continued 1 2. Sheaf Spec 3 3. Sheaf Proj 4 Last day: Fibers of morphisms. Properties preserved by base change: open immersions,

### ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

### 0.1 Spec of a monoid

These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.

### Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### Curves on P 1 P 1. Peter Bruin 16 November 2005

Curves on P 1 P 1 Peter Bruin 16 November 2005 1. Introduction One of the exercises in last semester s Algebraic Geometry course went as follows: Exercise. Let be a field and Z = P 1 P 1. Show that the

### A Mini-Course on Formal and Rigid Geometry

A Mini-Course on Formal and Rigid Geometry Padova April/May 2010 Siegfried Bosch (Münster) http://wwwmath.uni-muenster.de/u/bosch/frg-padova.pdf 1 Outline 1 Introduction Naive approach to non-archimedean

### Lecture 3: Flat Morphisms

Lecture 3: Flat Morphisms September 29, 2014 1 A crash course on Properties of Schemes For more details on these properties, see [Hartshorne, II, 1-5]. 1.1 Open and Closed Subschemes If (X, O X ) is a

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### 12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n

12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Application of cohomology: Hilbert polynomials and functions, Riemann- Roch, degrees, and arithmetic genus 1 1. APPLICATION OF COHOMOLOGY:

### UNIVERSAL PROPERTY OF NON-ARCHIMEDEAN ANALYTIFICATION

UNIVERSAL PROPERTY OF NON-ARCHIMEDEAN ANALYTIFICATION BRIAN CONRAD 1. Introduction 1.1. Motivation. Over C and over non-archimedean fields, analytification of algebraic spaces is defined as the solution

### COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski

### NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY

NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc

### INTERSECTION THEORY CLASS 6

INTERSECTION THEORY CLASS 6 RAVI VAKIL CONTENTS 1. Divisors 2 1.1. Crash course in Cartier divisors and invertible sheaves (aka line bundles) 3 1.2. Pseudo-divisors 3 2. Intersecting with divisors 4 2.1.

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44 RAVI VAKIL CONTENTS 1. Flat implies constant Euler characteristic 1 2. Proof of Important Theorem on constancy of Euler characteristic in flat families

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves 1 2. Quasicoherent sheaves form an abelian category 5 We began by recalling the distinguished affine base. Definition.

### NON-ARCHIMEDEAN ANALYTIC SPACES. Vladimir G. BERKOVICH. Department of Mathematics, The Weizmann Institute of Science P.O.B. 26, Rehovot, ISRAEL

NON-ARCHIMEDEAN ANALYTIC SPACES Vladimir G. BERKOVICH Department of Mathematics, The Weizmann Institute of Science P.O.B. 26, 76100 Rehovot, ISRAEL The following text is a short account of lectures at

### Peter Scholze Notes by Tony Feng. This is proved by real analysis, and the main step is to represent de Rham cohomology classes by harmonic forms.

p-adic Hodge Theory Peter Scholze Notes by Tony Feng 1 Classical Hodge Theory Let X be a compact complex manifold. We discuss three properties of classical Hodge theory. Hodge decomposition. Hodge s theorem

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### MATH 233B, FLATNESS AND SMOOTHNESS.

MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)

### DERIVED CATEGORIES OF COHERENT SHEAVES

DERIVED CATEGORIES OF COHERENT SHEAVES OLIVER E. ANDERSON Abstract. We give an overview of derived categories of coherent sheaves. [Huy06]. Our main reference is 1. For the participants without bacground

### We can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle

### CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES

CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES BRIAN OSSERMAN The purpose of this cheat sheet is to provide an easy reference for definitions of various properties of morphisms of schemes, and basic results

### APPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP

APPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP In this appendix we review some basic facts about étale cohomology, give the definition of the (cohomological) Brauer group, and discuss

### A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1

A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection

### mirko mauri, valerio proietti

O K A - C A R TA N F U N D A M E N TA L T H E O R E M O N S T E I N M A N I F O L D S mirko mauri, valerio proietti contents 1 Preparation 2 1.1 Coherent sheaves 2 1.2 Holomorphic convexity 4 1.3 Sheaf

### MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

p-adic UNIFORMIZATION OF CURVES EVAN WARNER 1. Introduction: why bother? Let us presume that we have at our disposal the fully-formed theory of rigidanalytic spaces., as setched in my last tal. Why would

### INTERSECTION THEORY CLASS 19

INTERSECTION THEORY CLASS 19 RAVI VAKIL CONTENTS 1. Recap of Last day 1 1.1. New facts 2 2. Statement of the theorem 3 2.1. GRR for a special case of closed immersions f : X Y = P(N 1) 4 2.2. GRR for closed

### DERIVED CATEGORIES OF STACKS. Contents 1. Introduction 1 2. Conventions, notation, and abuse of language The lisse-étale and the flat-fppf sites

DERIVED CATEGORIES OF STACKS Contents 1. Introduction 1 2. Conventions, notation, and abuse of language 1 3. The lisse-étale and the flat-fppf sites 1 4. Derived categories of quasi-coherent modules 5

### Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

### Lecture 9 - Faithfully Flat Descent

Lecture 9 - Faithfully Flat Descent October 15, 2014 1 Descent of morphisms In this lecture we study the concept of faithfully flat descent, which is the notion that to obtain an object on a scheme X,

### Chern classes à la Grothendieck

Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces

### DELIGNE S THEOREMS ON DEGENERATION OF SPECTRAL SEQUENCES

DELIGNE S THEOREMS ON DEGENERATION OF SPECTRAL SEQUENCES YIFEI ZHAO Abstract. We present the proofs of Deligne s theorems on degeneration of the Leray spectral sequence, and the algebraic Hodge-de Rham

### Discussion Session on p-divisible Groups

Discussion Session on p-divisible Groups Notes by Tony Feng April 7, 2016 These are notes from a discussion session of p-divisible groups. Some questions were posed by Dennis Gaitsgory, and then their

### ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES N. MOHAN KUMAR, A. P. RAO, AND G. V. RAVINDRA Abstract. We prove that any rank two arithmetically Cohen- Macaulay vector bundle on a general hypersurface

### RELATIVE AMPLENESS IN RIGID-ANALYTIC GEOMETRY

RELATIVE AMPLENESS IN RIGID-ANALYTIC GEOMETRY BRIAN CONRAD 1. Introduction 1.1. Motivation. The aim of this paper is to develop a rigid-analytic theory of relative ampleness for line bundles, and to record

### h M (T ). The natural isomorphism η : M h M determines an element U = η 1

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli

### Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

### BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,

CONNECTIONS, CURVATURE, AND p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26 RAVI VAKIL CONTENTS 1. Proper morphisms 1 Last day: separatedness, definition of variety. Today: proper morphisms. I said a little more about separatedness of

### ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0.

ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0. ANDREW SALCH During the last lecture, we found that it is natural (even just for doing undergraduatelevel complex analysis!)

### Realizing Families of Landweber Exact Theories

Realizing Families of Landweber Exact Theories Paul Goerss Department of Mathematics Northwestern University Summary The purpose of this talk is to give a precise statement of 1 The Hopkins-Miller Theorem

### ÉTALE π 1 OF A SMOOTH CURVE

ÉTALE π 1 OF A SMOOTH CURVE AKHIL MATHEW 1. Introduction One of the early achievements of Grothendieck s theory of schemes was the (partial) computation of the étale fundamental group of a smooth projective

### FORMAL GLUEING OF MODULE CATEGORIES

FORMAL GLUEING OF MODULE CATEGORIES BHARGAV BHATT Fix a noetherian scheme X, and a closed subscheme Z with complement U. Our goal is to explain a result of Artin that describes how coherent sheaves on

### where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism

8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the

### HARTSHORNE EXERCISES

HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

### UNIFORM STRUCTURES AND BERKOVICH SPACES

UNIFORM STRUCTURES AND BERKOVICH SPACES MATTHEW BAKER Abstract. A uniform space is a topological space together with some additional structure which allows one to make sense of uniform properties such

### Notes on p-divisible Groups

Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Some constructions using universal properties, old and new 1 2. Adjoint functors 3 3. Sheaves 6 Last day: What is algebraic geometry? Crash

### ALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!

ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.

### Math 210B. Artin Rees and completions

Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show

### Concentrated Schemes

Concentrated Schemes Daniel Murfet July 9, 2006 The original reference for quasi-compact and quasi-separated morphisms is EGA IV.1. Definition 1. A morphism of schemes f : X Y is quasi-compact if there

### Zariski s Main Theorem and some applications

Zariski s Main Theorem and some applications Akhil Mathew January 18, 2011 Abstract We give an exposition of the various forms of Zariski s Main Theorem, following EGA. Most of the basic machinery (e.g.

### Math 797W Homework 4

Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

### Université Paris Sud XI Orsay THE LOCAL FLATTENING THEOREM. Master 2 Memoire by Christopher M. Evans. Advisor: Prof. Joël Merker

Université Paris Sud XI Orsay THE LOCAL FLATTENING THEOREM Master 2 Memoire by Christopher M. Evans Advisor: Prof. Joël Merker August 2013 Table of Contents Introduction iii Chapter 1 Complex Spaces 1

### Geometry 9: Serre-Swan theorem

Geometry 9: Serre-Swan theorem Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have

### Half a Century of Rigid Analytic Spaces

Pure and Applied Mathematics Quarterly Volume 5, Number 4 (Special Issue: In honor of John Tate, Part 1 of 2 ) 1435 1467, 2009 Half a Century of Rigid Analytic Spaces Siegfried Bosch Dedicated to John

### ALGEBRAIC GROUPS JEROEN SIJSLING

ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined

### MOISHEZON SPACES IN RIGID GEOMETRY

MOISHEZON SPACES IN RIGID GEOMETRY BRIAN CONRAD Abstract. We prove that all proper rigid-analytic spaces with enough algebraically independent meromorphic functions are algebraic (in the sense of proper

### LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES

LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EAMPLES VIVEK SHENDE Historically, sheaves come from topology and analysis; subsequently they have played a fundamental role in algebraic geometry and certain

### ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset

### CHAPTER 1. AFFINE ALGEBRAIC VARIETIES

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the

### Tamagawa Numbers in the Function Field Case (Lecture 2)

Tamagawa Numbers in the Function Field Case (Lecture 2) February 5, 204 In the previous lecture, we defined the Tamagawa measure associated to a connected semisimple algebraic group G over the field Q

### AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES

AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES MATTHEW H. BAKER AND JÁNOS A. CSIRIK This paper was written in conjunction with R. Hartshorne s Spring 1996 Algebraic Geometry course at

### COMPONANTES IRRÉDUCTIBLE D ESPACES RIGIDES

IRREDUCIBLE COMPONENTS OF RIGID SPACES BRIAN CONRAD This paper lays the foundations for the global theory of irreducible components of rigid analytic spaces over a complete field k. We prove the excellence

### Section Higher Direct Images of Sheaves

Section 3.8 - Higher Direct Images of Sheaves Daniel Murfet October 5, 2006 In this note we study the higher direct image functors R i f ( ) and the higher coinverse image functors R i f! ( ) which will

### Arithmetic Algebraic Geometry

Arithmetic Algebraic Geometry 2 Arithmetic Algebraic Geometry Travis Dirle December 4, 2016 2 Contents 1 Preliminaries 1 1.1 Affine Varieties.......................... 1 1.2 Projective Varieties........................

### Fourier Mukai transforms II Orlov s criterion

Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and

### Lecture 6: Etale Fundamental Group

Lecture 6: Etale Fundamental Group October 5, 2014 1 Review of the topological fundamental group and covering spaces 1.1 Topological fundamental group Suppose X is a path-connected topological space, and

The Adjoint Functor Theorem. Kevin Buzzard February 7, 2012 Last modified 17/06/2002. 1 Introduction. The existence of free groups is immediate from the Adjoint Functor Theorem. Whilst I ve long believed

### MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23

MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23 6.4. Homotopy uniqueness of projective resolutions. Here I proved that the projective resolution of any R-module (or any object of an abelian category

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5 RAVI VAKIL CONTENTS 1. The inverse image sheaf 1 2. Recovering sheaves from a sheaf on a base 3 3. Toward schemes 5 4. The underlying set of affine schemes 6 Last

### Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely