The ring of global sections of a differential scheme
|
|
|
- Stephen Scott
- 5 years ago
- Views:
Transcription
1 The ring of global sections of a differential scheme Dmitry Trushin The Einstein Institute of Mathematics The Hebrew University of Jerusalem January 2012 Dmitry Trushin () The ring of global sections January, / 10
2 Affine schemes All rings are associative, commutative, and with an identity Dmitry Trushin () The ring of global sections January, / 10
3 Affine schemes All rings are associative, commutative, and with an identity Affine varieties over K Reduced finitely generated K-algebras X A n K K[X ] Max B B Dmitry Trushin () The ring of global sections January, / 10
4 Affine schemes All rings are associative, commutative, and with an identity Affine varieties over K Reduced finitely generated K-algebras X A n K K[X ] Max B B R a ring (Spec R, O R ) a locally ringed space Dmitry Trushin () The ring of global sections January, / 10
5 Affine schemes All rings are associative, commutative, and with an identity Affine varieties over K X A n K Reduced finitely generated K-algebras K[X ] Max B B R a ring (Spec R, O R ) a locally ringed space Definition ϕ: Spec R q R q and ϕ(q) R q ϕ is regular at p if for all q U ϕ(q) = a/b in R q O R (U) = regular functions in U X = Spec R Dmitry Trushin () The ring of global sections January, / 10
6 Affine differential scheme A differential ring is a ring R with = {δ 1,..., δ m }, δ i δ j = δ j δ i Dmitry Trushin () The ring of global sections January, / 10
7 Affine differential scheme A differential ring is a ring R with = {δ 1,..., δ m }, δ i δ j = δ j δ i Affine -varieties over K X A n K Max B Reduced -finitely generated K-algebras K{X } B Dmitry Trushin () The ring of global sections January, / 10
8 Affine differential scheme A differential ring is a ring R with = {δ 1,..., δ m }, δ i δ j = δ j δ i Affine -varieties over K X A n K Max B Reduced -finitely generated K-algebras K{X } B R a ring (Spec R, O R ) a locally ringed space Dmitry Trushin () The ring of global sections January, / 10
9 Affine differential scheme A differential ring is a ring R with = {δ 1,..., δ m }, δ i δ j = δ j δ i Affine -varieties over K X A n K Max B Reduced -finitely generated K-algebras K{X } B R a ring (Spec R, O R ) a locally ringed space (cannot use Spec R) Dmitry Trushin () The ring of global sections January, / 10
10 Affine differential scheme A differential ring is a ring R with = {δ 1,..., δ m }, δ i δ j = δ j δ i Affine -varieties over K X A n K Max B Reduced -finitely generated K-algebras K{X } B R a ring (Spec R, O R ) a locally ringed space (cannot use Spec R) Definition ϕ: Spec R q R q and ϕ(q) R q ϕ is regular at p if for all q U ϕ(q) = a/b in R q O R (U) = regular functions in U X = Spec R Dmitry Trushin () The ring of global sections January, / 10
11 The problem Commutative case Differential case Set R = O R (Spec R) Set R = O R (Spec R) ι: R R ι: R R Dmitry Trushin () The ring of global sections January, / 10
12 The problem Commutative case Differential case Set R = O R (Spec R) Set R = O R (Spec R) ι: R R ι: R R always isomorphism not necessarily injective not necessarily surjective Dmitry Trushin () The ring of global sections January, / 10
13 The problem Commutative case Differential case Set R = O R (Spec R) Set R = O R (Spec R) ι: R R ι: R R always isomorphism not necessarily injective not necessarily surjective Conjecture (Jerald Kovacic) The map ι : Spec R Spec R q ι 1 (q) is a homeomorphism. Dmitry Trushin () The ring of global sections January, / 10
14 Reduced schemes Commutative case Differential case ϕ: Spec R q k(q) ϕ: Spec R q k(q) ϕ(q) = a/b in k(q) and O R (U) = regular functions in U Dmitry Trushin () The ring of global sections January, / 10
15 Reduced schemes Commutative case Differential case ϕ: Spec R q k(q) ϕ: Spec R q k(q) ϕ(q) = a/b in k(q) and O R (U) = regular functions in U Set R = O R (Spec R) Set R = O R (Spec R) ι r : R R Dmitry Trushin () The ring of global sections January, / 10
16 Reduced schemes Commutative case Differential case ϕ: Spec R q k(q) ϕ: Spec R q k(q) ϕ(q) = a/b in k(q) and O R (U) = regular functions in U Set R = O R (Spec R) Set R = O R (Spec R) ι r : R R Theorem ι r : R D R. Then ι r : Spec D Spec R is a homeomorphism. Dmitry Trushin () The ring of global sections January, / 10
17 Reduced schemes Commutative case Differential case ϕ: Spec R q k(q) ϕ: Spec R q k(q) ϕ(q) = a/b in k(q) and O R (U) = regular functions in U Set R = O R (Spec R) Set R = O R (Spec R) ι r : R R Theorem ι r : R D R. Then ι r : Spec D Spec R is a homeomorphism. Corollary The mapping ι r : Spec R Spec R is a homeomorphism. Dmitry Trushin () The ring of global sections January, / 10
18 Keigher rings Definition (Keigher ring) I R is a -ideal r(i ) is a -ideal. Dmitry Trushin () The ring of global sections January, / 10
19 Keigher rings Definition (Keigher ring) I R is a -ideal r(i ) is a -ideal. R is a Ritt algebra (Q R) R is a Keigher ring. Dmitry Trushin () The ring of global sections January, / 10
20 Keigher rings Definition (Keigher ring) I R is a -ideal r(i ) is a -ideal. R is a Ritt algebra (Q R) R is a Keigher ring. Theorem R is a Keigher ring, and we are given ι: R D R. Then ι : Spec D Spec R is a homeomorphism. Dmitry Trushin () The ring of global sections January, / 10
21 Keigher rings Definition (Keigher ring) I R is a -ideal r(i ) is a -ideal. R is a Ritt algebra (Q R) R is a Keigher ring. Theorem R is a Keigher ring, and we are given ι: R D R. Then ι : Spec D Spec R is a homeomorphism. Corollary R is a Keigher ring. Then ι : Spec R Spec R is a homeomorphism. Dmitry Trushin () The ring of global sections January, / 10
22 Improvement Ehud Hrushovski: R to be a Keigher ring is a very strong condition Dmitry Trushin () The ring of global sections January, / 10
23 Improvement Ehud Hrushovski: R to be a Keigher ring is a very strong condition Theorem R is a differential ring such that nil R is differential and we are given ι: R D R. Then is a homeomorphism. ι : Spec D Spec R Dmitry Trushin () The ring of global sections January, / 10
24 Improvement Ehud Hrushovski: R to be a Keigher ring is a very strong condition Theorem R is a differential ring such that nil R is differential and we are given ι: R D R. Then is a homeomorphism. Corollary ι : Spec D Spec R R is a differential ring such that nil R is differential. Then is a homeomorphism. ι : Spec R Spec R Dmitry Trushin () The ring of global sections January, / 10
25 Iterative derivations Definition Let δ = {δ k } k 0, δ k : R R: 1) δ 0 (x) = x 3) δ k (ab) = µ+ν=k δµ (a)δ ν (b) 2) δ k (a + b) = δ k (a) + δ k (b) 4) δ k δ m = ( ) k+m k δ k+m Dmitry Trushin () The ring of global sections January, / 10
26 Iterative derivations Definition Let δ = {δ k } k 0, δ k : R R: 1) δ 0 (x) = x 3) δ k (ab) = µ+ν=k δµ (a)δ ν (b) 2) δ k (a + b) = δ k (a) + δ k (b) 4) δ k δ m = ( ) k+m k δ k+m A differential ring R is a ring with = {δ 1,..., δ m }, δ i are iterative derivations, and δ i δ j = δ j δ i. Dmitry Trushin () The ring of global sections January, / 10
27 Iterative derivations Definition Let δ = {δ k } k 0, δ k : R R: 1) δ 0 (x) = x 3) δ k (ab) = µ+ν=k δµ (a)δ ν (b) 2) δ k (a + b) = δ k (a) + δ k (b) 4) δ k δ m = ( ) k+m k δ k+m A differential ring R is a ring with = {δ 1,..., δ m }, δ i are iterative derivations, and δ i δ j = δ j δ i. Definition ϕ: Spec R q R q and ϕ(q) R q ϕ is regular at p if for all q U ϕ(q) = a/b in R q O R (U) = regular functions in U X = Spec R Dmitry Trushin () The ring of global sections January, / 10
28 Iterative case Set R = O R (Spec R). Dmitry Trushin () The ring of global sections January, / 10
29 Iterative case Set R = O R (Spec R). Theorem R is a differential ring with iterative derivations and ι: R D R. Then is a homeomorphism. ι : Spec D Spec R Dmitry Trushin () The ring of global sections January, / 10
30 Iterative case Set R = O R (Spec R). Theorem R is a differential ring with iterative derivations and ι: R D R. Then ι : Spec D Spec R is a homeomorphism. Corollary R is a differential ring with iterative derivations. Then ι : Spec R Spec R is a homeomorphism. Dmitry Trushin () The ring of global sections January, / 10
31 The structure sheaf R ι R X ι X p p Dmitry Trushin () The ring of global sections January, / 10
32 The structure sheaf R ι R X ι X p p Question Does O R coincide with O R? Dmitry Trushin () The ring of global sections January, / 10
33 The structure sheaf R ι R X ι X p p Question Does O R coincide with O R? O R, p could contain more nilpotent elements than O R,p Dmitry Trushin () The ring of global sections January, / 10
34 The structure sheaf R ι R X ι X p p Question Does O R coincide with O R? O R, p could contain more nilpotent elements than O R,p Fact If R is reduced. Then O R = O R. Dmitry Trushin () The ring of global sections January, / 10
Spectra of rings differentially finitely generated over a subring
Spectra of rings differentially finitely generated over a subring Dm. Trushin Department of Mechanics and Mathematics Moscow State University 15 April 2007 Dm. Trushin () -spectra of rings April 15, 2007
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
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
AN INTRODUCTION TO AFFINE SCHEMES
AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,
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
BEZOUT S THEOREM CHRISTIAN KLEVDAL
BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this
Constructions of Derived Equivalences of Finite Posets
Constructions of Derived Equivalences of Finite Posets Sefi Ladkani Einstein Institute of Mathematics The Hebrew University of Jerusalem http://www.ma.huji.ac.il/~sefil/ 1 Notions X Poset (finite partially
Rings and groups. Ya. Sysak
Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then
214A HOMEWORK KIM, SUNGJIN
214A HOMEWORK KIM, SUNGJIN 1.1 Let A = k[[t ]] be the ring of formal power series with coefficients in a field k. Determine SpecA. Proof. We begin with a claim that A = { a i T i A : a i k, and a 0 k }.
History: If A is primitive, Z(A) = k. J(A) is nil. In particular an affine algebra over an uncountable field has nil Jacobson radical.
Affinization 1 History: In the 50s, Amitsur proved Koethe s conjecture for algebras over an uncountable field by using a short and clever counting argument. This same argument could be used to show that
Representations of Quivers
MINGLE 2012 Simon Peacock 4th October, 2012 Outline 1 Quivers Representations 2 Path Algebra Modules 3 Modules Representations Quiver A quiver, Q, is a directed graph. Quiver A quiver, Q, is a directed
Spring 2016, lecture notes by Maksym Fedorchuk 51
Spring 2016, lecture notes by Maksym Fedorchuk 51 10.2. Problem Set 2 Solution Problem. Prove the following statements. (1) The nilradical of a ring R is the intersection of all prime ideals of R. (2)
The Elementary Theory of the Frobenius Automorphisms
The Elementary Theory of the Frobenius Automorphisms Ehud Hrushovski July 24, 2012 Abstract We lay down elements of a geometry based on difference equations. Various constructions of algebraic geometry
Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed.
![Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed.](/thumbs/82/86100847.jpg "Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed.")
Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed. Answer: Note that the first generator factors as (y
Synopsis of material from EGA Chapter II, 3
Synopsis of material from EGA Chapter II, 3 3. Homogeneous spectrum of a sheaf of graded algebras 3.1. Homogeneous spectrum of a graded quasi-coherent O Y algebra. (3.1.1). Let Y be a prescheme. A sheaf
Some remarks on Frobenius and Lefschetz in étale cohomology
Some remarks on obenius and Lefschetz in étale cohomology Gabriel Chênevert January 5, 2004 In this lecture I will discuss some more or less related issues revolving around the main idea relating (étale)
Classification of definable groupoids and Zariski geometries
and Zariski geometries Dmitry Sustretov Ben Gurion University sustreto@mathbguacil February 26, 2014 1 Motivation: Azumaya algebras An Azumaya algebra is a generalisation of a central simple algebra for
Di erential Algebraic Geometry, Part I
Di erential Algebraic Geometry, Part I Phyllis Joan Cassidy City College of CUNY Fall 2007 Phyllis Joan Cassidy (Institute) Di erential Algebraic Geometry, Part I Fall 2007 1 / 46 Abstract Di erential
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
Frobenius maps on injective hulls and their applications Moty Katzman, The University of Sheffield.
Frobenius maps on injective hulls and their applications Moty Katzman, The University of Sheffield. Consider a complete local ring (S, m) of characteristic p > 0 and its injective hull E S = E S (S/m).
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
NONSINGULAR CURVES BRIAN OSSERMAN
NONSINGULAR CURVES BRIAN OSSERMAN The primary goal of this note is to prove that every abstract nonsingular curve can be realized as an open subset of a (unique) nonsingular projective curve. Note that
A Primer on Homological Algebra
A Primer on Homological Algebra Henry Y Chan July 12, 213 1 Modules For people who have taken the algebra sequence, you can pretty much skip the first section Before telling you what a module is, you probably
1 Flat, Smooth, Unramified, and Étale Morphisms
1 Flat, Smooth, Unramified, and Étale Morphisms 1.1 Flat morphisms Definition 1.1. An A-module M is flat if the (right-exact) functor A M is exact. It is faithfully flat if a complex of A-modules P N Q
MATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
ALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS
ALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS Your Name: Conventions: all rings and algebras are assumed to be unital. Part I. True or false? If true provide a brief explanation, if false provide a counterexample
ABSTRACT NONSINGULAR CURVES
ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine n-space the collection A n k of points P = a 1, a,..., a
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.
11 Annihilators. Suppose that R, S, and T are rings, that R P S, S Q T, and R U T are bimodules, and finally, that
11 Annihilators. In this Section we take a brief look at the important notion of annihilators. Although we shall use these in only very limited contexts, we will give a fairly general initial treatment,
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
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
Homotopy-theory techniques in commutative algebra
Homotopy-theory techniques in commutative algebra Department of Mathematical Sciences Kent State University 09 January 2007 Departmental Colloquium Joint with Lars W. Christensen arxiv: math.ac/0612301
Rings and Fields Theorems
Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)
Lecture 7: Etale Fundamental Group - Examples
Lecture 7: Etale Fundamental Group - Examples October 15, 2014 In this lecture our only goal is to give lots of examples of etale fundamental groups so that the reader gets some feel for them. Some of
12. Hilbert Polynomials and Bézout s Theorem
12. Hilbert Polynomials and Bézout s Theorem 95 12. Hilbert Polynomials and Bézout s Theorem After our study of smooth cubic surfaces in the last chapter, let us now come back to the general theory of
Pseudo-Modularity and Iwasawa Theory
Pseudo-Modularity and Iwasawa Theory Preston Wake and Carl Wang Erickson April 11, 2015 Preston Wake and Carl Wang Erickson Pseudo-Modularity and Iwasawa Theory April 11, 2015 1 / 15 The Ordinary Eigencurve
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
(1) is an invertible sheaf on X, which is generated by the global sections
7. Linear systems First a word about the base scheme. We would lie to wor in enough generality to cover the general case. On the other hand, it taes some wor to state properly the general results if one
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. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].](/thumbs/85/92532834.jpg "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.
SCHEMES. David Harari. Tsinghua, February-March 2005
SCHEMES David Harari Tsinghua, February-March 2005 Contents 1. Basic notions on schemes 2 1.1. First definitions and examples.................. 2 1.2. Morphisms of schemes : first properties.............
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
The most important result in this section is undoubtedly the following theorem.
28 COMMUTATIVE ALGEBRA 6.4. Examples of Noetherian rings. So far the only rings we can easily prove are Noetherian are principal ideal domains, like Z and k[x], or finite. Our goal now is to develop theorems
Hyperbolic Ordinariness of Hyperelliptic Curves of Lower Genus in Characteristic Three
Hyperbolic Ordinariness of Hyperelliptic Curves of Lower Genus in Characteristic Three Yuichiro Hoshi November 2018 Abstract. In the present paper, we discuss the hyperbolic ordinariness of hyperelliptic
MA 252 notes: Commutative algebra
MA 252 notes: Commutative algebra (Distilled from [Atiyah-MacDonald]) Dan Abramovich Brown University February 11, 2017 Abramovich MA 252 notes: Commutative algebra 1 / 13 Primary ideals Primary decompositions
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)
FIRST ASSIGNMENT. (1) Let E X X be an equivalence relation on a set X. Construct the set of equivalence classes as colimit in the category Sets.
FIRST SSIGNMENT DUE MOND, SEPTEMER 19 (1) Let E be an equivalence relation on a set. onstruct the set of equivalence classes as colimit in the category Sets. Solution. Let = {[x] x } be the set of equivalence
10. Noether Normalization and Hilbert s Nullstellensatz
10. Noether Normalization and Hilbert s Nullstellensatz 91 10. Noether Normalization and Hilbert s Nullstellensatz In the last chapter we have gained much understanding for integral and finite ring extensions.
Section Projective Morphisms
Section 2.7 - Projective Morphisms Daniel Murfet October 5, 2006 In this section we gather together several topics concerned with morphisms of a given scheme to projective space. We will show how a morphism
Lyubeznik Numbers of Local Rings and Linear Strands of Graded Ideals
Lyubeznik Numbers of Local Rings and Linear Strands of Graded deals Josep Àlvarez Montaner Abstract We report recent work on the study of Lyubeznik numbers and their relation to invariants coming from
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar
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)
Systems of linear equations. We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K,
Systems of linear equations We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K, f 11 t 1 +... + f 1n t n = 0, f 21 t 1 +... + f 2n t n = 0,.
Johns Hopkins University, Department of Mathematics Abstract Algebra - Spring 2013 Midterm Exam Solution
Johns Hopkins University, Department of Mathematics 110.40 Abstract Algebra - Spring 013 Midterm Exam Solution Instructions: This exam has 6 pages. No calculators, books or notes allowed. You must answer
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
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
Algebraic varieties and schemes over any scheme. Non singular varieties
Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two
Math 121 Homework 4: Notes on Selected Problems
Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W
Piecewise Noetherian Rings
Northern Illinois University UNAM 25 May, 2017 Acknowledgments Results for commutative rings are from two joint papers with William D. Weakley,, Comm. Algebra (1984) and A note on prime ideals which test
Algebra Qualifying Exam Solutions January 18, 2008 Nick Gurski 0 A B C 0
1. Show that if B, C are flat and Algebra Qualifying Exam Solutions January 18, 2008 Nick Gurski 0 A B C 0 is exact, then A is flat as well. Show that the same holds for projectivity, but not for injectivity.
NORMALIZATION OF THE KRICHEVER DATA. Motohico Mulase
NORMALIZATION OF THE KRICHEVER DATA Motohico Mulase Institute of Theoretical Dynamics University of California Davis, CA 95616, U. S. A. and Max-Planck-Institut für Mathematik Gottfried-Claren-Strasse
Model Theory of Fields with Virtually Free Group Actions
Model Theory of Fields with Virtually Free Group Actions Özlem Beyarslan joint work with Piotr Kowalski Boğaziçi University Istanbul, Turkey 29 March 2018 Ö. Beyarslan, P. Kowalski Model Theory of V.F
Algebraic Topology Final
Instituto Superior Técnico Departamento de Matemática Secção de Álgebra e Análise Algebraic Topology Final Solutions 1. Let M be a simply connected manifold with the property that any map f : M M has a
Math 418 Algebraic Geometry Notes
Math 418 Algebraic Geometry Notes 1 Affine Schemes Let R be a commutative ring with 1. Definition 1.1. The prime spectrum of R, denoted Spec(R), is the set of prime ideals of the ring R. Spec(R) = {P R
Locality for qc-sheaves associated with tilting
Locality for qc-sheaves associated with tilting - ICART 2018, Faculty of Sciences, Mohammed V University in Rabat Jan Trlifaj Univerzita Karlova, Praha Jan Trlifaj (Univerzita Karlova, Praha) Locality
ON COMPLETENESS AND LINEAR DEPENDENCE FOR DIFFERENTIAL ALGEBRAIC VARIETIES
ON COMPLETENESS AND LINEAR DEPENDENCE FOR DIFFERENTIAL ALGEBRAIC VARIETIES JAMES FREITAG*, OMAR LEÓN SÁNCHEZ, AND WILLIAM SIMMONS** Abstract. In this paper we deal with two foundational questions on complete
NilBott Tower of Aspherical Manifolds and Torus Actions
NilBott Tower of Aspherical Manifolds and Torus Actions Tokyo Metropolitan University November 29, 2011 (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29,
Weierstrass preparation theorem and singularities in the space of non-degenerate arcs
Weierstrass preparation theorem and singularities in the space of non-degenerate arcs Ngô Bảo Châu gratefully dedicated to Prof. Lê Tuấn Hoa for his 60th birthday 1 Introduction It has been long expected
THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3
THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3 KAZUNORI NAKAMOTO AND TAKESHI TORII Abstract. There exist 26 equivalence classes of k-subalgebras of M 3 (k) for any algebraically closed field
Commutative Di erential Algebra, Part III
Commutative Di erential Algebra, Part III Phyllis Joan Cassidy, City College of CUNY October 26, 2007 hyllis Joan Cassidy, City College of CUNY () Comm Di Alg III October 26, 2007 1 / 39 Basic assumptions.
Lectures on Galois Theory. Some steps of generalizations
= Introduction Lectures on Galois Theory. Some steps of generalizations Journée Galois UNICAMP 2011bis, ter Ubatuba?=== Content: Introduction I want to present you Galois theory in the more general frame
Written Homework # 5 Solution
Math 516 Fall 2006 Radford Written Homework # 5 Solution 12/12/06 Throughout R is a ring with unity. Comment: It will become apparent that the module properties 0 m = 0, (r m) = ( r) m, and (r r ) m =
SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY
SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY MARC LEVINE Contents 0. Introduction 1 1. The category of schemes 2 1.1. The spectrum of a commutative ring 2 1.2. Ringed spaces 5 1.3. Schemes 10 1.4. Schemes
SEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.
NOTES IN COMMUTATIVE ALGEBRA: PART 2
NOTES IN COMMUTATIVE ALGEBRA: PART 2 KELLER VANDEBOGERT 1. Completion of a Ring/Module Here we shall consider two seemingly different constructions for the completion of a module and show that indeed they
On local properties of non- Archimedean analytic spaces.
On local properties of non- Archimedean analytic spaces. Michael Temkin (talk at Muenster Workshop, April 4, 2000) Plan: 1. Introduction. 2. Category bir k. 3. The reduction functor. 4. Main result. 5.
Algebraic Geometry I Lectures 14 and 15
Algebraic Geometry I Lectures 14 and 15 October 22, 2008 Recall from the last lecture the following correspondences {points on an affine variety Y } {maximal ideals of A(Y )} SpecA A P Z(a) maximal ideal
Graduate Preliminary Examination
Graduate Preliminary Examination Algebra II 18.2.2005: 3 hours Problem 1. Prove or give a counter-example to the following statement: If M/L and L/K are algebraic extensions of fields, then M/K is algebraic.
The geometrical semantics of algebraic quantum mechanics
The geometrical semantics of algebraic quantum mechanics B. Zilber University of Oxford November 29, 2016 B.Zilber, The semantics of the canonical commutation relations arxiv.org/abs/1604.07745 The geometrical
Examples of Semi-Invariants of Quivers
Examples of Semi-Invariants of Quivers June, 00 K is an algebraically closed field. Types of Quivers Quivers with finitely many isomorphism classes of indecomposable representations are of finite representation
MATH 221 NOTES BRENT HO. Date: January 3, 2009.
MATH 22 NOTES BRENT HO Date: January 3, 2009. 0 Table of Contents. Localizations......................................................................... 2 2. Zariski Topology......................................................................
GLUING SCHEMES AND A SCHEME WITHOUT CLOSED POINTS
GLUING SCHEMES AND A SCHEME WITHOUT CLOSED POINTS KARL SCHWEDE Abstract. We first construct and give basic properties of the fibered coproduct in the category of ringed spaces. We then look at some special
Homework 3 MTH 869 Algebraic Topology
Homework 3 MTH 869 Algebraic Topology Joshua Ruiter February 12, 2018 Proposition 0.1 (Exercise 1.1.10). Let (X, x 0 ) and (Y, y 0 ) be pointed, path-connected spaces. Let f : I X y 0 } and g : I x 0 }
(dim Z j dim Z j 1 ) 1 j i
Math 210B. Codimension 1. Main result and some interesting examples Let k be a field, and A a domain finitely generated k-algebra. In class we have seen that the dimension theory of A is linked to the
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: MIDTERM SOLUTIONS
Algebraic Geometry: MIDTERM SOLUTIONS C.P. Anil Kumar Abstract. Algebraic Geometry: MIDTERM 6 th March 2013. We give terse solutions to this Midterm Exam. 1. Problem 1: Problem 1 (Geometry 1). When is
7. Localization. (d 1, m 1 ) (d 2, m 2 ) d 3 D : d 3 d 1 m 2 = d 3 d 2 m 1. (ii) If (d 1, m 1 ) (d 1, m 1 ) and (d 2, m 2 ) (d 2, m 2 ) then
7. Localization To prove Theorem 6.1 it becomes necessary to be able to a enominators to rings (an to moules), even when the rings have zero-ivisors. It is a tool use all the time in commutative algebra,
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
Math 751 Week 6 Notes
Math 751 Week 6 Notes Joe Timmerman October 26, 2014 1 October 7 Definition 1.1. A map p: E B is called a covering if 1. P is continuous and onto. 2. For all b B, there exists an open neighborhood U of
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
Math 632 Notes Algebraic Geometry 2
Math 632 Notes Algebraic Geometry 2 Lectures by Karen Smith Notes by Daniel Hast Winter 2013 Contents 1 Affine schemes 4 1.1 Motivation and review of varieties........................ 4 1.2 First attempt
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.
On the vanishing of Tor of the absolute integral closure
On the vanishing of Tor of the absolute integral closure Hans Schoutens Department of Mathematics NYC College of Technology City University of New York NY, NY 11201 (USA) Abstract Let R be an excellent
Non-separable AF-algebras
Non-separable AF-algebras Takeshi Katsura Department of Mathematics, Hokkaido University, Kita 1, Nishi 8, Kita-Ku, Sapporo, 6-81, JAPAN katsura@math.sci.hokudai.ac.jp Summary. We give two pathological
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
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.
k k would be reducible. But the zero locus of f in A n+1
Math 145. Bezout s Theorem Let be an algebraically closed field. The purpose of this handout is to prove Bezout s Theorem and some related facts of general interest in projective geometry that arise along
arxiv: v1 [math.ra] 5 Feb 2015
![arxiv: v1 [math.ra] 5 Feb 2015](/thumbs/95/125554910.jpg "arxiv: v1 [math.ra] 5 Feb 2015")
Noncommutative ampleness from finite endomorphisms D. S. Keeler Dept. of Mathematics, Miami University, Oxford, OH 45056 arxiv:1502.01668v1 [math.ra] 5 Feb 2015 Abstract K. Retert Dept. of Mathematics,
THE MONODROMY-WEIGHT CONJECTURE
THE MONODROMY-WEIGHT CONJECTURE DONU ARAPURA Deligne [D1] formulated his conjecture in 1970, simultaneously in the l-adic and Hodge theoretic settings. The Hodge theoretic statement, amounted to the existence