WEAK NULLSTELLENSATZ
|
|
- Estella Austin
- 5 years ago
- Views:
Transcription
1 WEAK NULLSTELLENSATZ YIFAN WU, Abstract. We prove weak Nullstellensatz which states if a finitely generated k algebra is a field, then it is a finite algebraic field extension of k. We then use weak Nullstellensatz to prove Hilbert s Nullstellensatz. Contents. Finite Algebra versus Finite Generated Algebra 2. Weak Nullstellensatz 4 3. Hilbert s Nullstellensatz 5 References 7. Finite Algebra versus Finite Generated Algebra Let A and B be two commutative rings with. Let f : A B be a ring homomorphism that sends to. For a A and b B, define a product a b = f(a)b. This product defines a scalar multiplication in the ring B with scalars coming from A. It gives the ring B an A module structure. Definition. (A Algebra). Let A, B, f be as above. The ring B together with the A module structure as described above is said to be an A algebra. Let A, B, B 2 be commutative rings with, and let f : A B, f 2 : A B 2 be two ring homomorphisms that send s to s. B and B 2 are two A algebra as in the above definition. We now define an A algebra homomorphism between B and B 2. Definition.2 (A Algebra Homomorphism). A map h : B B 2 is said to an A algebra homomorphism if the following two conditions hold simutaneously: () h is a ring homomorphism between B and B 2. (2) h is an A module homomorphism between B and B 2. Date: First Version: May 6, 207. Last Update: May 0, 207.
2 Weak Nullstellensatz Lemma.3. Let A, B, B 2, f : A B, f 2 : A B 2 be as above. A map h : B B 2 is an A algebra homomorphism if and only if h is a ring homomorphism and h f = f 2. Proof. ( ) Assume h : B B 2 is an A algebra homomorphism, then for a A h(f (a)) f (a)=f (a) B in B h is A module homomorphism = h(f (a) B ) = h(a B ) = a h( B ) = f 2 (a)h( B ) = f 2 (a) B2 = f 2 (a). ( ) Suppose h f = f 2. We want to show that h : B B 2 is an A module homomorphism. Let a A and b B. h(a b) = h(f (a)b) h is ring homomorphism = h(f (a))h(b) = f 2 (a)h(b) = a h(b). Next we come up to two finite condition that is somehow confusing at first glance. The first one is called finite algebra. The second one is called finitely generated algebra. Definition.4 (Finite A Algebra). Let f : A B be a ring homomorphism and B be an A algebra as above. B is said to be a finite A algebra if B is finitely generated as an A module, i.e., there exists a finite set of elements b,, b n B such that B = A b + + A b n. Definition.5 (Finitely Generated A Algebra). Let f : A B be a ring homomorphism and B be an A algebra as above. B is said to be a finitely generated A algebra if B is finitely generated as a polynomial ring with coefficients from A, i.e., there exists a finite set of elements b,, b n B such that B = A[b,, b n ] = { a (e,,e n) b e b en n : a (e,,e n) A, e,, e n Z 0 }. finite sum Remark.6. Note A b + + A b n A[b,, b n ]. So if B is a finite A algebra then B is a finitely generated A algebra. The converse does not hold in general. Example.7. Let k be a field. The polynomial ring with n variables k[x,, X n ] is a finitely generately k algebra, but not a finite k algebra. Remark.8. Consider the case in which A = k is a field. Then every ring homomorphism from A = k to another ring B is always injective so we can view f : A = k B as an embedding and A = k as a subset of B. In this case, B is a finitely generated k algebra if and only if B = k[b,, b n ] with b,, b n B. We conclude this section with the following result which puts together our previous definitions on finite algebra and on finite generated algebra. 2
3 Yifan Wu, Proposition.9. Suppose A is a Noetherian commutative ring. Let A A 2 A 3 where A 2 and A 3 are also commutative rings. Suppose A 3 satisfies the following two condition simultaneously: () A 3 is a finitely generated A algebra. (2) A 3 is a fnite A 2 algebra. Then A 2 is a finitely generated A algebra. Proof. For this proof, we need some results regarding Noetherian rings and Noetherian modules. All these results can be found at [5]. Let s say c,, c m generate A 3 as A algebra, and d,, d n generate A 3 as A 2 module. More precisely, this means A 3 = A [c,, c m ] ( ) = A 2 d + + A 2 d n, c,, c m, d,, d n A 3. This relation ( ) tells us we can write ( ) c i = j e ij d j, e ij A 2, ( ) d i d j = k e ijk d k, e ijk A 2. Consider A e = A [e ij, e ijk ], i.e., the A algebra generated by all e ij, e ijk from above. We have A A e A 2. By [5, Corollary 3.3], A e is a Noetherian ring. We investigate what ( ) and ( ) tell us. The obveservation is c 2 i ( ) = ( j ( ) = j,k e ij d j ) 2 Binomial Theorem = ê jk d j d k, ê jk A e j,k ê jk e jkl d l = ê jkl d l, ê jkl A e. j,k,l l This is saying that a square c 2 i can be written in linear combination of d l s with coefficients in A e. Of course if we work hard any high power can also be written like this. So we just succesfully argue that every element in A 3 is a linear combination of d l s with coefficients from A e, so A 3 = A e d + + A e d n. By [5, Theorem 2.7], A 3 is a Noetherian A e module. Being a A e submodule, A 2 is finitely generated as A e module, A 2 = A e b + + A e b w, b,, b w A 2, A 2 = A [e ij, e ijk ] b + + A [e ij, e ijk ] b w = A [e ij, e ijk, b,, b w ] is a finitely generately A algebra. 3
4 Weak Nullstellensatz 2. Weak Nullstellensatz Theorem 2. (Weak Nullstellensatz). Let k be a field. Let B = k[b,, b n ] with b,, b n B be a finitely generated k algebra. Suppose B is a field. Then all b,, b n are algebraic over k. Hence B/k is an algebraic field extension. Before proving weak Nullstellensatz, we adapt Euclid s proof of infinitude of primes to obtain the following result: Proposition 2.2. Let k be a field, there are infinitely many irreducible polynomials in the polynomial ring with n variables k[x,, X n ]. Proof. k[x,, X n ] is a unique factorization domain. Suppose there is only finitely many irreducible polynomials f (X,, X n ),, f m (X,, X n ) k[x,, X n ]. Consider f f 2 f m + k[x,, X n ]. It cannot be factorized using f,, f m hence must be irreducible, contradicting that f,, f m are the only irreducible polynomials. Proof of 2.. Consider the field B = k[b,, b n ]. By re-indexing, we can assume b,, b s are algebraically independent over k, and b s+,, b n are algebraic over the subfield k(b,, b s ) B. Suppose on the contrary that B/k is not an algebraic extension. This means s. By hypothesis, A 3 = B is a finitely generated (A = k) algebra. Also note b s+,, b n are algebraic over A 2 so A 3 is a finite A 2 algebra. Hence we may apply.9 to A = k A 2 = k(b,, b s ) A 3 = B = k[b,, b n ], and conclude that A 2 = k(b,, b s ) is a finitely generated k algebra. By definition of finitely generated algebra, this means that This further means that k(b,, b s ) = k[c,, c t ], c,, c t k(b,, b s ). c = f (b,, b s ) g (b,, b s ),, c t = f t(b,, b s ) g t (b,, b s ), f,, f t, g,, g t k[x,, X s ]. Now by 2.2, there exists an irreducible polynomial h k[x,, X s ] which does not divide any of g,, g t k[x,, X s ]. Consider the element h(b,, b s ) k(b,, b s ) = k[c,, c t ]. 4
5 Yifan Wu, Since k[c h(b,,b s),, c t ], it must be the case that for certain C (e,,e t) k, e,, e t 0, = finite sum h(b,, b s ) = finite sum ( ) e f (b,, b s ) C (e,,e t) g (b,, b s ) C (e,,e t)c e c et t ( ) et ft (b,, b s ). g t (b,, b s ) Mutiply both side of this relation with g p g pt t for large enough p,, p t 0, in order to kill the denominators of right hand side, then it leads to a contradiction since h does not divide g,, g t. The conlusion is that it must be the case B/k is an algebraic extension, as desired. 3. Hilbert s Nullstellensatz As far as the author can see, there are a handful of verions of Hilbert s Nullstellensatz. Consider the polynomial ring of n variables k[x,, X n ] with coefficients from a field k. Consider an ideal I k[x,, X n ]. The elements in this ideal I are just polynomials f(x,, X n ). The common zeros of these polynomials form so called algebraic sets, or affine varieties. Definition 3. (Affine Varieties). The set k n is also called affine n space and denoted A n. Assign to each ideal I k[x,, X n ] the set of common zeros V(I) = {(a,, a n ) A n : f(a,, a n ) = 0 f I} A n. Subsets of A n of this form are called affine varieties. Remark 3.2. Hilbert Basis Theorem [5, Theorem 3.] states that every ideal I k[x,, X n ] is finitely generated, i.e., I(f,, f m ). So V(I) = V(f ) V(f m ). This is saying every affine variety is a finite intersection of the zero sets of single polynomial. Definition 3.3 (Ideal of a Set of Points of A n ). Conversely, given a subset X of A n. Assign to X an ideal in k[x,, X n ] I(X) = {f k[x,, X n ] : f vanishs on all of X}. Remark 3.4. It is easy to see for any ideal I k[x,, X n ], the relation I I(V(I)) always holds. What is less obvious at a first glance is that the radical I I(V(I)). The really non-trivial part is that equality holds, i.e., I = I(V(I)). 5
6 Weak Nullstellensatz Theorem 3.5 (Hilbert s Nullstellensatz). Let k be an algebraically closed field. Let I k[x,, X n ] be an ideal. Then I(V(I)) = I. Before proving this version, we might look at another version of Nullstellensatz. Theorem 3.6 (Hilbert s Nullstellensatz). Let k be an algebraically closed field. If V(I) = then I = k[x,, X n ]. Proof. We will deal with the contrapostitive, i.e., if I is a proper ideal inside k[x,, X n ], then the polynomials from I have at least one common zero. A standard result, which depends on Zorn s lemma, says the proper ideal I is contained in a maximal ideal of k[x,, X n ]. Since if the ideal becomes bigger, the set of common zeros only becomes smaller, so we might as well assume I is itself a maximal ideal. In this case, the residue ring B = k[x,, X n ]/I is a field. And it is a field extension of k. k embeds inside B = k[x,, X n ]/I as image of constant polynomials. Let X i denote the image of X i under the canonical homomorphism π : k[x,, X n ] B = k[x,, X n ]/I. Then B = k[x,, X n ] is a finitely generated k algebra. And it is a field. 2. precisely says that X,, X n are all algebraic over k. Since k is assume to be algebraically closed, it means all X,, X n k. So (X,, X n ) k n = A n. The next observation is that given a polynomial f(x,, X n ) I, plugging in X,, X n gives f(x,, X n ) = f(x,, X n ) = 0 B = k[x,, X n ]/I. So (X,, X n ) k n = A n is a common zero for all polynomials from I. We are done. Finally we come to the proof of 3.5. The proof uses so-called Rabinowitsch trick according to [4, Thereom.5, IX, ]. 6
7 Yifan Wu, Proof of 3.5. Let I k[x,, X n ] be an ideal. By 3.4, we know I I(V(I)) already. It remains to show the ( ) direction. By 3.2, assume I = (f,, f m ). Given f I(V(I)). This means f vanishes on common zeros of f,, f m. We want to show that there exists r Z >0 and g,, g m k[x,, X n ] such that ( ) f r (X,, X n ) = f (X,, X n )g (X,, X n )+ +f m (X,, X n )g m (X,, X n ). The so-called Rabinowitsch trick is to induce a new variable X n+ and then consider the following polynomials all as elements in k[x,, X n, X n+ ]: f (X,, X n ),, f m (X,, X n ), X n+ f(x,, X n ) This construction gurantees these m+ polynomials f (X,, X n ),, f m (X,, X n ), X n+ f(x,, X n ) have no common zero. Indeed if the first m of m+ polynomials vanish, then the last one evaluates to be 0. By 3.6, f (X,, X n ),, f m (X,, X n ), X n+ f(x,, X n ) generate the unit ideal k[x,, X n ]. We can find polynomials h,, h m, h k[x,, X n, X n+ ] such that = h (X,, X n, X n+ )f (X,, X n )+ +h m (X,, X n, X n+ )f m (X,, X n ) +h(x,, X n, X n+ ) ( X n+ f(x,, X n ) ) ( ). The second step of the so-called Rabinowitsch trick is to say goodbye to the new variable X n+. Replace X n+ by f(x,,x n) in ( ) throughout, we get = h (X,, X n, f(x,, X n ) )f (X,, X n ) + +h m (X,, X n, f(x,, X n ) )f m(x,, X n ) We can then multiply both side with a large enough power f(x,, X n ) r to kill denominator and conclude ( ) holds, as desired. References [] M. F. Atiyah, I. G. MacDonald, Introduction to Commutative Algebra, Westview, 206. [2] K. Hulek, Elementary Algebraic Geometry, American Mathematical Society, [3] J. Humphreys, Linear Algebraic Groups, Springer-Verlag, New York, 975. [4] S. Lang, Algebra Revised Third Edition, Springer, New York, [5] Y. Wu, Hilbert Basis Theorem, ExpositoryArticles/LinearAlgebraicGroups/Hilbert_Basis_Theorem.pdf. 3, 5 [6] O. Zariski, P. Samuel, Commutative Algebra Volume II, Van Nostrand, Princeton,
Algebraic Varieties. Chapter Algebraic Varieties
Chapter 12 Algebraic Varieties 12.1 Algebraic Varieties Let K be a field, n 1 a natural number, and let f 1,..., f m K[X 1,..., X n ] be polynomials with coefficients in K. Then V = {(a 1,..., a n ) :
More informationHilbert s Nullstellensatz
Hilbert s Nullstellensatz An Introduction to Algebraic Geometry Scott Sanderson Department of Mathematics Williams College April 6, 2013 Introduction My talk today is on Hilbert s Nullstellensatz, a foundational
More informationMath 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
More informationExtension theorems for homomorphisms
Algebraic Geometry Fall 2009 Extension theorems for homomorphisms In this note, we prove some extension theorems for homomorphisms from rings to algebraically closed fields. The prototype is the following
More informationALGEBRAIC 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
More informationReid 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
More informationRings 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...
More information4.4 Noetherian Rings
4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)
More informationDimension Theory. Mathematics 683, Fall 2013
Dimension Theory Mathematics 683, Fall 2013 In this note we prove some of the standard results of commutative ring theory that lead up to proofs of the main theorem of dimension theory and of the Nullstellensatz.
More informationAlgebraic function fields
Algebraic function fields 1 Places Definition An algebraic function field F/K of one variable over K is an extension field F K such that F is a finite algebraic extension of K(x) for some element x F which
More information10. Smooth Varieties. 82 Andreas Gathmann
82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It
More informationExtended Index. 89f depth (of a prime ideal) 121f Artin-Rees Lemma. 107f descending chain condition 74f Artinian module
Extended Index cokernel 19f for Atiyah and MacDonald's Introduction to Commutative Algebra colon operator 8f Key: comaximal ideals 7f - listings ending in f give the page where the term is defined commutative
More informationMATH 326: RINGS AND MODULES STEFAN GILLE
MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called
More informationALGEBRAIC 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.
More informationR S. with the property that for every s S, φ(s) is a unit in R S, which is universal amongst all such rings. That is given any morphism
8. Nullstellensatz We will need the notion of localisation, which is a straightforward generalisation of the notion of the field of fractions. Definition 8.1. Let R be a ring. We say that a subset S of
More informationCourse 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...............................
More informationYuriy Drozd. Intriduction to Algebraic Geometry. Kaiserslautern 1998/99
Yuriy Drozd Intriduction to Algebraic Geometry Kaiserslautern 1998/99 CHAPTER 1 Affine Varieties 1.1. Ideals and varieties. Hilbert s Basis Theorem Let K be an algebraically closed field. We denote by
More informationMATH 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......................................................................
More informationAlgebra Homework, Edition 2 9 September 2010
Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.
More informationSpring 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)
More information4. Noether normalisation
4. Noether normalisation We shall say that a ring R is an affine ring (or affine k-algebra) if R is isomorphic to a polynomial ring over a field k with finitely many indeterminates modulo an ideal, i.e.,
More informationAlgebraic 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
More informationCHAPTER 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
More information3. The Sheaf of Regular Functions
24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice
More informationExploring the Exotic Setting for Algebraic Geometry
Exploring the Exotic Setting for Algebraic Geometry Victor I. Piercey University of Arizona Integration Workshop Project August 6-10, 2010 1 Introduction In this project, we will describe the basic topology
More informationLecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman
Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman October 17, 2006 TALK SLOWLY AND WRITE NEATLY!! 1 0.1 Integral Domains and Fraction Fields 0.1.1 Theorems Now what we are going
More informationALGEBRAIC GEOMETRY (NMAG401) Contents. 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30
ALGEBRAIC GEOMETRY (NMAG401) JAN ŠŤOVÍČEK Contents 1. Affine varieties 1 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30 1. Affine varieties The basic objects
More informationarxiv:math/ v1 [math.ra] 9 Jun 2006
Noetherian algebras over algebraically closed fields arxiv:math/0606209v1 [math.ra] 9 Jun 2006 Jason P. Bell Department of Mathematics Simon Fraser University 8888 University Drive Burnaby, BC, V5A 1S6
More informationHILBERT FUNCTIONS. 1. Introduction
HILBERT FUCTIOS JORDA SCHETTLER 1. Introduction A Hilbert function (so far as we will discuss) is a map from the nonnegative integers to themselves which records the lengths of composition series of each
More informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationMATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 1 SOLUTIONS
MATH 63: ALGEBRAIC GEOMETRY: HOMEWORK SOLUTIONS Problem. (a.) The (t + ) (t + ) minors m (A),..., m k (A) of an n m matrix A are polynomials in the entries of A, and m i (A) = 0 for all i =,..., k if and
More informationALGEBRA EXERCISES, PhD EXAMINATION LEVEL
ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)
More informationLecture 4. Corollary 1.2. If the set of all nonunits is an ideal in A, then A is local and this ideal is the maximal one.
Lecture 4 1. General facts Proposition 1.1. Let A be a commutative ring, and m a maximal ideal. Then TFAE: (1) A has only one maximal ideal (i.e., A is local); (2) A \ m consists of units in A; (3) For
More information10. 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.
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GROUP ACTIONS ON POLYNOMIAL AND POWER SERIES RINGS Peter Symonds Volume 195 No. 1 September 2000 PACIFIC JOURNAL OF MATHEMATICS Vol. 195, No. 1, 2000 GROUP ACTIONS ON POLYNOMIAL
More informationMATH 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
More informationSummer Algebraic Geometry Seminar
Summer Algebraic Geometry Seminar Lectures by Bart Snapp About This Document These lectures are based on Chapters 1 and 2 of An Invitation to Algebraic Geometry by Karen Smith et al. 1 Affine Varieties
More informationThe 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
More informationABSTRACT 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
More informationCOMMUNICATIONS IN ALGEBRA, 15(3), (1987) A NOTE ON PRIME IDEALS WHICH TEST INJECTIVITY. John A. Beachy and William D.
COMMUNICATIONS IN ALGEBRA, 15(3), 471 478 (1987) A NOTE ON PRIME IDEALS WHICH TEST INJECTIVITY John A. Beachy and William D. Weakley Department of Mathematical Sciences Northern Illinois University DeKalb,
More informationInjective Modules and Matlis Duality
Appendix A Injective Modules and Matlis Duality Notes on 24 Hours of Local Cohomology William D. Taylor We take R to be a commutative ring, and will discuss the theory of injective R-modules. The following
More informationMath 40510, Algebraic Geometry
Math 40510, Algebraic Geometry Problem Set 1, due February 10, 2016 1. Let k = Z p, the field with p elements, where p is a prime. Find a polynomial f k[x, y] that vanishes at every point of k 2. [Hint:
More informationA MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ
A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In
More informationMath 145. Codimension
Math 145. Codimension 1. Main result and some interesting examples In class we have seen that the dimension theory of an affine variety (irreducible!) is linked to the structure of the function field in
More informationMath 203A - Solution Set 1
Math 203A - Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in
More information4.5 Hilbert s Nullstellensatz (Zeros Theorem)
4.5 Hilbert s Nullstellensatz (Zeros Theorem) We develop a deep result of Hilbert s, relating solutions of polynomial equations to ideals of polynomial rings in many variables. Notation: Put A = F[x 1,...,x
More information5 Dedekind extensions
18.785 Number theory I Fall 2016 Lecture #5 09/22/2016 5 Dedekind extensions In this lecture we prove that the integral closure of a Dedekind domain in a finite extension of its fraction field is also
More informationOn Invariants of Complex Filiform Leibniz Algebras ABSTRACT INTRODUCTION
Malaysian Journal of Mathematical Sciences (): 47-59 (009) Isamiddin S.Rakhimov Department of Mathematics Faculty of Science and Institute for Mathematical Research Universiti Putra Malaysia 4400 UPM Serdang
More information(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d
The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers
More informationMath 203A - Solution Set 1
Math 203A - Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in
More informationLECTURE Affine Space & the Zariski Topology. It is easy to check that Z(S)=Z((S)) with (S) denoting the ideal generated by elements of S.
LECTURE 10 1. Affine Space & the Zariski Topology Definition 1.1. Let k a field. Take S a set of polynomials in k[t 1,..., T n ]. Then Z(S) ={x k n f(x) =0, f S}. It is easy to check that Z(S)=Z((S)) with
More informationCHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998
CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic
More informationTHROUGH THE FIELDS AND FAR AWAY
THROUGH THE FIELDS AND FAR AWAY JONATHAN TAYLOR I d like to thank Prof. Stephen Donkin for helping me come up with the topic of my project and also guiding me through its various complications. Contents
More informationFormal 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
More information55 Separable Extensions
55 Separable Extensions In 54, we established the foundations of Galois theory, but we have no handy criterion for determining whether a given field extension is Galois or not. Even in the quite simple
More informationCommutative Algebra and Algebraic Geometry. Robert Friedman
Commutative Algebra and Algebraic Geometry Robert Friedman August 1, 2006 2 Disclaimer: These are rough notes for a course on commutative algebra and algebraic geometry. I would appreciate all suggestions
More informationChapter 1 Hilbert s Nullstellensatz
Chapter 1 Hilbert s Nullstellensatz Hilbert s Nullstellensatz may be seen as the starting point of algebraic geometry. It provides a bijective correspondence between affine varieties, which are geometric
More informationHomework 4 Solutions
Homework 4 Solutions November 11, 2016 You were asked to do problems 3,4,7,9,10 in Chapter 7 of Lang. Problem 3. Let A be an integral domain, integrally closed in its field of fractions K. Let L be a finite
More informationAN 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,
More informationMIT Algebraic techniques and semidefinite optimization February 16, Lecture 4
MIT 6.972 Algebraic techniques and semidefinite optimization February 16, 2006 Lecture 4 Lecturer: Pablo A. Parrilo Scribe: Pablo A. Parrilo In this lecture we will review some basic elements of abstract
More informationPure Math 764, Winter 2014
Compact course notes Pure Math 764, Winter 2014 Introduction to Algebraic Geometry Lecturer: R. Moraru transcribed by: J. Lazovskis University of Waterloo April 20, 2014 Contents 1 Basic geometric objects
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 1. Contents 1. Commutative algebra 2 2. Algebraic sets 2 3. Nullstellensatz (theorem of zeroes) 4
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 1 RAVI VAKIL Contents 1. Commutative algebra 2 2. Algebraic sets 2 3. Nullstellensatz (theorem of zeroes) 4 I m going to start by telling you about this course,
More informationLecture 2. (1) Every P L A (M) has a maximal element, (2) Every ascending chain of submodules stabilizes (ACC).
Lecture 2 1. Noetherian and Artinian rings and modules Let A be a commutative ring with identity, A M a module, and φ : M N an A-linear map. Then ker φ = {m M : φ(m) = 0} is a submodule of M and im φ is
More information12 Hilbert polynomials
12 Hilbert polynomials 12.1 Calibration Let X P n be a (not necessarily irreducible) closed algebraic subset. In this section, we ll look at a device which measures the way X sits inside P n. Throughout
More informationABSOLUTE VALUES AND VALUATIONS
ABSOLUTE VALUES AND VALUATIONS YIFAN WU, wuyifan@umich.edu Abstract. We introduce the basis notions, properties and results of absolute values, valuations, discrete valuation rings and higher unit groups.
More informationThe Topology and Algebraic Functions on Affine Algebraic Sets Over an Arbitrary Field
Georgia State University ScholarWorks @ Georgia State University Mathematics Theses Department of Mathematics and Statistics Fall 11-15-2012 The Topology and Algebraic Functions on Affine Algebraic Sets
More informationFinite Fields. [Parts from Chapter 16. Also applications of FTGT]
Finite Fields [Parts from Chapter 16. Also applications of FTGT] Lemma [Ch 16, 4.6] Assume F is a finite field. Then the multiplicative group F := F \ {0} is cyclic. Proof Recall from basic group theory
More informationTHE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS. K. R. Goodearl and E. S. Letzter
THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS K. R. Goodearl and E. S. Letzter Abstract. In previous work, the second author introduced a topology, for spaces of irreducible representations,
More informationMATH 131B: ALGEBRA II PART B: COMMUTATIVE ALGEBRA
MATH 131B: ALGEBRA II PART B: COMMUTATIVE ALGEBRA I want to cover Chapters VIII,IX,X,XII. But it is a lot of material. Here is a list of some of the particular topics that I will try to cover. Maybe I
More informationAlgebraic 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
More informationMA 252 notes: Commutative algebra
MA 252 notes: Commutative algebra (Distilled from [Atiyah-MacDonald]) Dan Abramovich Brown University February 4, 2017 Abramovich MA 252 notes: Commutative algebra 1 / 13 Rings of fractions Fractions Theorem
More informationPROBLEMS, MATH 214A. Affine and quasi-affine varieties
PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasi-affine varieties 1. Let X A 2 be defined by x 2 + y 2 = 1 and x = 1. Find the ideal I(X). 2. Prove that the subset
More informationAN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES
AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES DOMINIC L. WYNTER Abstract. We introduce the concepts of divisors on nonsingular irreducible projective algebraic curves, the genus of such a curve,
More informationFields and Galois Theory. Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory.
Fields and Galois Theory Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory. This should be a reasonably logical ordering, so that a result here should
More informationMath 203A - Solution Set 3
Math 03A - Solution Set 3 Problem 1 Which of the following algebraic sets are isomorphic: (i) A 1 (ii) Z(xy) A (iii) Z(x + y ) A (iv) Z(x y 5 ) A (v) Z(y x, z x 3 ) A Answer: We claim that (i) and (v)
More informationCommutative Algebra. Gabor Wiese. Winter Term 2013/2014. Université du Luxembourg.
Commutative Algebra Winter Term 2013/2014 Université du Luxembourg Gabor Wiese gabor.wiese@uni.lu Version of 16th December 2013 2 Preface In number theory one is naturally led to study more general numbers
More informationRings With Topologies Induced by Spaces of Functions
Rings With Topologies Induced by Spaces of Functions Răzvan Gelca April 7, 2006 Abstract: By considering topologies on Noetherian rings that carry the properties of those induced by spaces of functions,
More informationAlgebraic geometry of the ring of continuous functions
Algebraic geometry of the ring of continuous functions Nicolas Addington October 27 Abstract Maximal ideals of the ring of continuous functions on a compact space correspond to points of the space. For
More informationRing Theory Problems. A σ
Ring Theory Problems 1. Given the commutative diagram α A σ B β A σ B show that α: ker σ ker σ and that β : coker σ coker σ. Here coker σ = B/σ(A). 2. Let K be a field, let V be an infinite dimensional
More informationPowers and Products of Monomial Ideals
U.U.D.M. Project Report 6: Powers and Products of Monomial Ideals Melker Epstein Examensarbete i matematik, 5 hp Handledare: Veronica Crispin Quinonez Examinator: Jörgen Östensson Juni 6 Department of
More informationFactorization in Polynomial Rings
Factorization in Polynomial Rings Throughout these notes, F denotes a field. 1 Long division with remainder We begin with some basic definitions. Definition 1.1. Let f, g F [x]. We say that f divides g,
More informationMath 203A - Solution Set 1
Math 203A - Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in
More informationERRATA. Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on February 14, 2018)
ERRATA Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on February 14, 2018) These are errata for the Third Edition of the book. Errata from previous editions have been
More informationINJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA
INJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA These notes are intended to give the reader an idea what injective modules are, where they show up, and, to
More informationNOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22
NOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22 RAVI VAKIL Hi Dragos The class is in 381-T, 1:15 2:30. This is the very end of Galois theory; you ll also start commutative ring theory. Tell them: midterm
More informationInstitutionen för matematik, KTH.
Institutionen för matematik, KTH. Contents 7 Affine Varieties 1 7.1 The polynomial ring....................... 1 7.2 Hypersurfaces........................... 1 7.3 Ideals...............................
More informationTheorem 5.3. Let E/F, E = F (u), be a simple field extension. Then u is algebraic if and only if E/F is finite. In this case, [E : F ] = deg f u.
5. Fields 5.1. Field extensions. Let F E be a subfield of the field E. We also describe this situation by saying that E is an extension field of F, and we write E/F to express this fact. If E/F is a field
More informationSchemes via Noncommutative Localisation
Schemes via Noncommutative Localisation Daniel Murfet September 18, 2005 In this note we give an exposition of the well-known results of Gabriel, which show how to define affine schemes in terms of the
More informationABSTRACT. Department of Mathematics. interesting results. A graph on n vertices is represented by a polynomial in n
ABSTRACT Title of Thesis: GRÖBNER BASES WITH APPLICATIONS IN GRAPH THEORY Degree candidate: Angela M. Hennessy Degree and year: Master of Arts, 2006 Thesis directed by: Professor Lawrence C. Washington
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE ABOUT VARIETIES AND REGULAR FUNCTIONS.
ALGERAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE AOUT VARIETIES AND REGULAR FUNCTIONS. ANDREW SALCH. More about some claims from the last lecture. Perhaps you have noticed by now that the Zariski topology
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to
More informationIn Theorem 2.2.4, we generalized a result about field extensions to rings. Here is another variation.
Chapter 3 Valuation Rings The results of this chapter come into play when analyzing the behavior of a rational function defined in the neighborhood of a point on an algebraic curve. 3.1 Extension Theorems
More informationA course in. Algebraic Geometry. Taught by Prof. Xinwen Zhu. Fall 2011
A course in Algebraic Geometry Taught by Prof. Xinwen Zhu Fall 2011 1 Contents 1. September 1 3 2. September 6 6 3. September 8 11 4. September 20 16 5. September 22 21 6. September 27 25 7. September
More informationThis is a closed subset of X Y, by Proposition 6.5(b), since it is equal to the inverse image of the diagonal under the regular map:
Math 6130 Notes. Fall 2002. 7. Basic Maps. Recall from 3 that a regular map of affine varieties is the same as a homomorphism of coordinate rings (going the other way). Here, we look at how algebraic properties
More informationwhere 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
More information1 Absolute values and discrete valuations
18.785 Number theory I Lecture #1 Fall 2015 09/10/2015 1 Absolute values and discrete valuations 1.1 Introduction At its core, number theory is the study of the ring Z and its fraction field Q. Many questions
More informationHere is another way to understand what a scheme is 1.GivenaschemeX, and a commutative ring R, the set of R-valued points
Chapter 7 Schemes III 7.1 Functor of points Here is another way to understand what a scheme is 1.GivenaschemeX, and a commutative ring R, the set of R-valued points X(R) =Hom Schemes (Spec R, X) This is
More informationMATH32062 Notes. 1 Affine algebraic varieties. 1.1 Definition of affine algebraic varieties
MATH32062 Notes 1 Affine algebraic varieties 1.1 Definition of affine algebraic varieties We want to define an algebraic variety as the solution set of a collection of polynomial equations, or equivalently,
More information