The Affine Algebraic Connection J. David Taylor The Affine Algebraic Connection Affine Algebraic Sets Zariski Topology J. David Taylor Regular Functions and Morphisms Algebra October 3, 2018 Parallels (All rings are commutative and unital) ... C to get the classical geometric picture. ... the algebraic closure of C(T ), then we'll study relations between algebraic functions. ... Q, to get information on algebraic numbers. ... Qp for some prime number p, to get local, p-adic data. ... Fp for combinatorial, residual arithmetic information. Keep in mind whichever example(s) you like best. Pick an algebraically closed field The Affine Algebraic Connection J. David Taylor Throughout, let k be an algebraically closed field. We could let k be ... Affine Algebraic Sets Zariski Topology Regular Functions and Morphisms Algebra Parallels ... the algebraic closure of C(T ), then we'll study relations between algebraic functions. ... Q, to get information on algebraic numbers. ... Qp for some prime number p, to get local, p-adic data. ... Fp for combinatorial, residual arithmetic information. Keep in mind whichever example(s) you like best. Pick an algebraically closed field The Affine Algebraic Connection J. David Taylor Throughout, let k be an algebraically closed field. We could let k be ... Affine Algebraic Sets ... C to get the classical geometric picture. Zariski Topology Regular Functions and Morphisms Algebra Parallels ... Q, to get information on algebraic numbers. ... Qp for some prime number p, to get local, p-adic data. ... Fp for combinatorial, residual arithmetic information. Keep in mind whichever example(s) you like best. Pick an algebraically closed field The Affine Algebraic Connection J. David Taylor Throughout, let k be an algebraically closed field. We could let k be ... Affine Algebraic Sets ... C to get the classical geometric picture. Zariski Topology ... the algebraic closure of C(T ), then we'll study relations Regular Functions and between algebraic functions. Morphisms Algebra Parallels ... Qp for some prime number p, to get local, p-adic data. ... Fp for combinatorial, residual arithmetic information. Keep in mind whichever example(s) you like best. Pick an algebraically closed field The Affine Algebraic Connection J. David Taylor Throughout, let k be an algebraically closed field. We could let k be ... Affine Algebraic Sets ... C to get the classical geometric picture. Zariski Topology ... the algebraic closure of C(T ), then we'll study relations Regular Functions and between algebraic functions. Morphisms ... Q, to get information on algebraic numbers. Algebra Parallels ... Fp for combinatorial, residual arithmetic information. Keep in mind whichever example(s) you like best. Pick an algebraically closed field The Affine Algebraic Connection J. David Taylor Throughout, let k be an algebraically closed field. We could let k be ... Affine Algebraic Sets ... C to get the classical geometric picture. Zariski Topology ... the algebraic closure of C(T ), then we'll study relations Regular Functions and between algebraic functions. Morphisms ... Q, to get information on algebraic numbers. Algebra Parallels ... Qp for some prime number p, to get local, p-adic data. Keep in mind whichever example(s) you like best. Pick an algebraically closed field The Affine Algebraic Connection J. David Taylor Throughout, let k be an algebraically closed field. We could let k be ... Affine Algebraic Sets ... C to get the classical geometric picture. Zariski Topology ... the algebraic closure of C(T ), then we'll study relations Regular Functions and between algebraic functions. Morphisms ... Q, to get information on algebraic numbers. Algebra Parallels ... Qp for some prime number p, to get local, p-adic data. ... Fp for combinatorial, residual arithmetic information. Pick an algebraically closed field The Affine Algebraic Connection J. David Taylor Throughout, let k be an algebraically closed field. We could let k be ... Affine Algebraic Sets ... C to get the classical geometric picture. Zariski Topology ... the algebraic closure of C(T ), then we'll study relations Regular Functions and between algebraic functions. Morphisms ... Q, to get information on algebraic numbers. Algebra Parallels ... Qp for some prime number p, to get local, p-adic data. ... Fp for combinatorial, residual arithmetic information. Keep in mind whichever example(s) you like best. and let X ⊆ kn be the solution set of the system of equations F1(T1;:::; Tn) = 0 . Fr (T1;:::; Tn) = 0 X is called an affine algebraic set. Affine Algebraic Sets The Affine Algebraic Connection J. David Taylor Definition Affine Let F ;:::; F 2 k[T ;:::; T ] Algebraic Sets 1 r 1 n Zariski Topology Regular Functions and Morphisms Algebra Parallels X is called an affine algebraic set. Affine Algebraic Sets The Affine Algebraic Connection J. David Taylor Definition Affine Let F ;:::; F 2 k[T ;:::; T ] and let X ⊆ kn be the solution Algebraic Sets 1 r 1 n Zariski set of the system of equations Topology Regular F1(T1;:::; Tn) = 0 Functions and Morphisms . Algebra . Parallels Fr (T1;:::; Tn) = 0 Affine Algebraic Sets The Affine Algebraic Connection J. David Taylor Definition Affine Let F ;:::; F 2 k[T ;:::; T ] and let X ⊆ kn be the solution Algebraic Sets 1 r 1 n Zariski set of the system of equations Topology Regular F1(T1;:::; Tn) = 0 Functions and Morphisms . Algebra . Parallels Fr (T1;:::; Tn) = 0 X is called an affine algebraic set. The 0 polynomial has all of kn as its solution set. XY − 1 has a hyperbola in k2 as its solution set. In k3, the system x3 + y 3 + xy 2 + yx2 − x − y = 0 x3 − y 3 + xy 2 − yx2 − x + y = 0 is the union of a cylinder and the line through its center. Examples of an Affine Algebraic Set The Affine Algebraic Connection J. David Taylor Here are some examples: Affine Algebraic Sets Zariski Topology Regular Functions and Morphisms Algebra Parallels XY − 1 has a hyperbola in k2 as its solution set. In k3, the system x3 + y 3 + xy 2 + yx2 − x − y = 0 x3 − y 3 + xy 2 − yx2 − x + y = 0 is the union of a cylinder and the line through its center. Examples of an Affine Algebraic Set The Affine Algebraic Connection J. David Taylor Here are some examples: Affine The 0 polynomial has all of kn as its solution set. Algebraic Sets Zariski Topology Regular Functions and Morphisms Algebra Parallels In k3, the system x3 + y 3 + xy 2 + yx2 − x − y = 0 x3 − y 3 + xy 2 − yx2 − x + y = 0 is the union of a cylinder and the line through its center. Examples of an Affine Algebraic Set The Affine Algebraic Connection J. David Taylor Here are some examples: Affine The 0 polynomial has all of kn as its solution set. Algebraic Sets 2 Zariski XY − 1 has a hyperbola in k as its solution set. Topology Regular Functions and Morphisms Algebra Parallels Examples of an Affine Algebraic Set The Affine Algebraic Connection J. David Taylor Here are some examples: Affine The 0 polynomial has all of kn as its solution set. Algebraic Sets 2 Zariski XY − 1 has a hyperbola in k as its solution set. Topology In k3, the system Regular Functions and Morphisms x3 + y 3 + xy 2 + yx2 − x − y = 0 Algebra 3 3 2 2 Parallels x − y + xy − yx − x + y = 0 is the union of a cylinder and the line through its center. This is a topology. Every closed set is an affine algebraic set (Hilbert's Basis Theorem.) If F 2 k[T1;:::; Tn], then D(F ) is defined as the set of n points a 2 A (k) such that F (a) 6= 0. This is called a distinguished open set. The distinguished open sets form a basis for the Zariski n topology on A (k). The Zariski Topology The Affine Algebraic Connection Definition J. David n n Taylor Let A (k) := k with the topology where affine algebraic sets n are closed. This is called the Zariski topology and A (k) is Affine Algebraic Sets called affine n-space. Zariski Topology Regular Functions and Morphisms Algebra Parallels Every closed set is an affine algebraic set (Hilbert's Basis Theorem.) If F 2 k[T1;:::; Tn], then D(F ) is defined as the set of n points a 2 A (k) such that F (a) 6= 0. This is called a distinguished open set. The distinguished open sets form a basis for the Zariski n topology on A (k). The Zariski Topology The Affine Algebraic Connection Definition J. David n n Taylor Let A (k) := k with the topology where affine algebraic sets n are closed. This is called the Zariski topology and A (k) is Affine Algebraic Sets called affine n-space. Zariski Topology This is a topology. Regular Functions and Morphisms Algebra Parallels If F 2 k[T1;:::; Tn], then D(F ) is defined as the set of n points a 2 A (k) such that F (a) 6= 0. This is called a distinguished open set. The distinguished open sets form a basis for the Zariski n topology on A (k). The Zariski Topology The Affine Algebraic Connection Definition J. David n n Taylor Let A (k) := k with the topology where affine algebraic sets n are closed. This is called the Zariski topology and A (k) is Affine Algebraic Sets called affine n-space. Zariski Topology This is a topology. Regular Functions and Morphisms Every closed set is an affine algebraic set (Hilbert's Basis Algebra Theorem.) Parallels The distinguished open sets form a basis for the Zariski n topology on A (k). The Zariski Topology The Affine Algebraic Connection Definition J. David n n Taylor Let A (k) := k with the topology where affine algebraic sets n are closed. This is called the Zariski topology and A (k) is Affine Algebraic Sets called affine n-space.