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Affine Varieties Affine Varieties By Asash Yohannes Advisor: Dr. Dawit Cherinet Department of Mathematics Arbaminch University OCT. 2018 A THESIS ON AFFINE VARIETIES By ASASH YOHANNES A THESIS SUBMITTED TO THE DEPARTMENT OF MATHEMATICS COLLEGE OF NATURAL SCIENCE SCHOOL OF GRADUATE STUDIES ARBAMINCH UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS WITH SPECIALIZATION IN ALGEBRA. OCT. 2018 ARBAMINCH ETHIOPIA Acknowledgment Next to GOD, I would like to thank my advisor Dr.Dawit Cherinet for all his help and support during the work of my thesis. I would also like to thank every one who read and give constractive and helpful comments on doing this thesis. ii Declaration I, Asash Yohannes with student number SMSc=118=07, hereby declare that this thesis has not previously been submitted for assessment or completion of any post graduate qualification to another university or for another qualification. Date Asash Yohannes iii Certificate I hereby certify that I have read this thesis prepared by Asash Yohannes under my supervision and recommended that, it be accepted as fulfilling the thesis requirement. Date Dr. Dawit Cherinet iv Examiners' Thesis Approval Sheet As members of Board of Examiners of the final MSc. Graduate Thesis open defense examination, we certify that we have read and evaluated this Graduate Thesis prepared by ASASH YOHANNES entitled ”Affine Varieties" and recommended that it be accepted as fulfilling the thesis requirement for the Degree of Master of Science in Mathematics(Algebra). Name of Chairperson Signature Date Name of Principal Advisor Signature Date Name of External Examiner Signature Date Name of Internal Examiner Signature Date SGS Approval Signature Date Name Of The Department Head Signature Date Final approval and acceptance of the thesis is contingent upon submission of the final copy of the Graduate Thesis to the Council of the Graduate Studies(CGS) through the Department Graduate Committed(DGC) of the candidate's department. v Contents 1 Introduction 1 1.1 Statement of the problem . .1 1.2 Objectives of the study . .2 1.2.1 General objectives of the study . .2 1.2.2 Specific objectives of the study . .2 2 Gr¨obnerBases 3 2.1 Polynomials in One Variable . .3 2.2 Polynomials in Several Variables Over a Field . .8 2.2.1 Ideals in K[x1; : : : ; xn]....................9 2.2.2 Monomial Ordering in K[x1; ··· ; xn]............9 2.2.3 Division Algorithm in K[x1; : : : ; xn]............. 11 2.3 Monomial Ideals and Dickson's Lemma . 12 2.4 Hilbert-Basis Theorem and Gr¨obnerBases . 15 3 Primary Decomposition Of Ideals In Noetherian Rings 27 3.1 Prime, Maximal, Radical & Primary Ideals . 27 3.2 Primary Decomposition . 30 4 Affine Varieties 32 4.1 Affine Spaces . 32 4.2 Affine Varieties . 34 4.2.1 Properties Of Affine Varieties . 38 4.3 Parametrization Of Affine Varieties . 38 4.4 Ideals and Affine Varieties . 40 4.4.1 Hilbert's Nullstellensatz . 45 vi 4.4.2 Radical Ideals and the Ideal-Variety Correspondence . 47 4.4.3 Sums, Products and Intersections Of Ideals . 50 4.4.4 Zariski Closure and Quotient Ideals . 54 4.5 Irreducible Varieties . 57 4.6 Decomposition of Affine Varieties in to Irreducibles . 61 Chapter 1 Introduction Algebraic geometry is the combination of linear algebra and abstract algebra. Linear algebra study the solutions of system of linear equations, whereas abstract algebra deals with polynomial equations in one variable. Algebraic geometry combines these two fields of Mathematics by studying solutions of system of polynomial equations in several variables fi(x1; : : : ; xn) = 0; i = 1; : : : n; where fi 2 K[x1; : : : ; xn] and K is a field. One difference between linear equations and polynomial equations is that; theorems on linear equations does not depend on the field K, but for polynomial equations depend on whether the field K is algebraically closed or not(to lesser extent) where K has characteristic zero.The space where system of polynomial equations in which algebraic geometry deals with is said to be algebraic-Variety (or Affine variety). 1.1 Statement of the problem In this thesis we construct a statement that relates the affine varieties in Kn and ideals in K[x1; : : : ; xn], where K is a field. 1 Chapter 1 : Introduction 1.2 Objectives of the study 1.2.1 General objectives of the study The general objectives of the study are: • To determine the properties of affine varieties. • To characterize affine varieties. • To decompose affine varieties in to irreducible varieties. 1.2.2 Specific objectives of the study The specific objectives of the study are: • To study the relation ship between ideals and affine varieties. • To see the correspondence between prime ideals and irreducible varieties. 1.2 Objectives of the study 2 Chapter 2 Gr¨obnerBases 2.1 Polynomials in One Variable Consider the polynomial ring K[x] in x over a field K which is the set of expressions of the form m K[x] = ff(x) = a0 + a1x + ··· + amx ; am 6= 0g: i where a0; : : : ; am 2 K, are called coefficients of x , i = 0; : : : ; m. m am called the leading coefficient of f, amx called leading term of f denoted by LT (f), m is called the degree of f denoted by deg(f). We say a polynomial f 2 K[x] is the zero polynomial if a0 = a1 = ··· = am = 0. |: We also introduce the following notation for polynomials in K[x]. The word polynomial is an expression of the form, 2 n a0 + a1x + a2x + ··· + anx : This expression corresponds to the sequence (a0; a1; : : : ; an;:::) of its coefficients. As any function, a sequence a is determined by its values; for each i 2 N, we write a(i) = ai 2 R, so that a = (a0; a1; a2; : : : ; an; 0; 0;:::): The entries ai 2 R are called the coefficients of the sequence. The term coefficient means "acting together to some single end". Here, coefficients combine 3 Chapter 2 : Gr¨obnerBases with powers of x to give the terms of the sequence. The sequence (0; 0;:::; 0; 0;:::) is a polynomial called the the-zero polynomial. The element x 2 K[x] is given by x = (0; 1; 0;:::): Thus, we have x2 = x:x = (0; 1; 0;:::)(0; 1; 0;:::): = (0; 0; 1; 0 :::) Similarly x3 = (0; 0; 0; 1; 0;:::): x4 = (0; 0; 0; 0; 1; 0;:::): . Thus, we have, (a0; a1; a2; : : : an; 0; 0;:::) = (a0;:::) + (0; a1; 0;:::) + ··· + (0; 0; : : : ; an;:::) = a0(1; 0;:::) + a1(0; 1; 0;:::) + ··· + an(0; 0;:::; 1; 0;:::) 2 n = a0 + a1x + a2x + ··· + anx : Definition 2.1.1. Let X be a non empty subset of a ring R, and let fAi : i 2 Ig be a family of ideals in R containing X; then \ Ai i2I is called an ideal generated by X denoted by hXi: If X = fx1; : : : ; xng, then hXi = hx1; : : : ; xni, in this case we say the ideal is ”finitely generated ". If X is a singleton element X=fxg, then the ideal is generated by a single element, such an ideal is called "principal ideal." Theorem 2.1.2. Let R be a ring. Then R is an integral domain if and only if R[x] is an integral domain. 2.1 Polynomials in One Variable 4 Chapter 2 : Gr¨obnerBases Proof. (=)) Let R be an integral domain ) R is a non-trivial ring with unity Let a = (a0; a1;::: ); b = (b0; b1;::: ) 2 R[x] be non-zero. Since a; b 6= 0, 9 n; m 2 Z≥0 such that an 6= 0; bm 6= 0 and ai = 0 for i>n and bj = 0 for j>m. Now consider the (m + n)th term in the product a:b. It is given by X arbs = anbm 6= 0; r+s=m+n Since R is an integral domain, an 6= 0; & bm 6= 0 Note that r + s = m + n ) r ≥ n; s ≥ m ) (r = n or ar = 0); or (s = m or bs = 0). Therefore, the (m + n)th term in the product a:b is non-zero. Hence a:b 6= 0: ) R[x] contains no divisors of zero.Thus R[x] is an integral domain. ((=) Conversely Let R[x] be an integral domain. Since R is isomorphic to a sub-ring of R[x]. It follows that R is an integral domain. Theorem 2.1.3. Let K be a field, then K[x] is an integral domain in which every ideal is principal ideal. Proof. By the above theorem K[x] is an integral domain, since K is so. Let I ⊆ K[x] be an ideal. If I = f0g, then there is nothing to proof. Since I = h0i = f0g. Suppose I 6= f0g. ) I contains at least one non-zero polynomial. Consider the set S = fdeg(f(x): 0 6= f(x) 2 Ig: Since S is non-empty set of non-negative integers, S has a least element ( by well ordering axiom ). Let n 2 Z≥0 be the least element of S, then n = deg(f) for some f(x) 2 I. Since f(x) 2 I, we have hf(x)i ⊆ I, n ≤ deg(g(x)) for all 0 6= g(x) 2 I. We shall prove that I is generated by f(x). 2.1 Polynomials in One Variable 5 Chapter 2 : Gr¨obnerBases On the other hand, suppose g(x) 2 I. By division algorithm 9 q(x); r(x) 2 K[x] such that g(x) = q(x):f(x) + r(x); where r(x) = 0 or deg(r(x)) ≤ deg(f(x)) = n. Now, r(x) = g(x) − q(x):f(x) 2 I (Since g(x); f(x) 2 I and I is an ideal).
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