The Affine Algebraic Connection

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

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

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

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

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 ∈ 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 ∈ 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 ∈ 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 ∈ k[T1,..., Tn], then D(F ) is deﬁned as the set of n points a ∈ 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.)

The Zariski Topology

Zariski Topology This is a topology. Regular Functions and Morphisms

Algebra

The Zariski Topology

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

Zariski Topology This is a topology. Regular Functions and Morphisms Every closed set is an aﬃne algebraic set (Hilbert’s Basis

Algebra Theorem.)

Parallels If F ∈ k[T1,..., Tn], then D(F ) is deﬁned as the set of n points a ∈ A (k) such that F (a) 6= 0. This is called a distinguished open set. The Zariski Topology

Zariski Topology This is a topology. Regular Functions and Morphisms Every closed set is an aﬃne algebraic set (Hilbert’s Basis

Algebra Theorem.)

Parallels If F ∈ k[T1,..., Tn], then D(F ) is defined as the set of n points a ∈ 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). 2 In A (k) non-empty closed sets are finite or of the solution of some equation F (T1, T2) = 0 with F ∈ k[T1, T2], or unions of those. n In A (k), non-empty closed sets are finite unions of finite intersections of level sets (of polynomials). n Every non-empty open subset of A (k) is dense! n2 GLn(k) is a distinguished open subset of A (k).

Examples: Zariski Topology

The Aﬃne Algebraic Connection

J. David Taylor 1 A (k) = k with the cofinite topology (finite sets are Affine closed). Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels n In A (k), non-empty closed sets are ﬁnite unions of ﬁnite intersections of level sets (of polynomials). n Every non-empty open subset of A (k) is dense! n2 GLn(k) is a distinguished open subset of A (k).

Examples: Zariski Topology

The Aﬃne Algebraic Connection

Algebra

Parallels n Every non-empty open subset of A (k) is dense! n2 GLn(k) is a distinguished open subset of A (k).

Examples: Zariski Topology

The Aﬃne Algebraic Connection

J. David Taylor 1 A (k) = k with the cofinite topology (finite sets are Affine closed). Algebraic Sets 2 Zariski In A (k) non-empty closed sets are finite or of the solution Topology of some equation F (T1, T2) = 0 with F ∈ k[T1, T2], or Regular Functions and unions of those. Morphisms In n(k), non-empty closed sets are finite unions of finite Algebra A intersections of level sets (of polynomials). Parallels n2 GLn(k) is a distinguished open subset of A (k).

Examples: Zariski Topology

The Aﬃne Algebraic Connection

The Aﬃne Algebraic For example, the union of a circle, the Witch of Agnesi, and Connection three lines is an example of a closed set in 2(k). J. David A Taylor

Aﬃne Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels Aﬃne sets as Topological Spaces

The Aﬃne Algebraic Connection

J. David Taylor

Aﬃne Algebraic Sets Zariski If X is an aﬃne algebraic set, then X ⊆ n(k). We give it the Topology A

Regular subspace topology. This topology is still called the Zariski Functions and topology. Morphisms

Algebra

Parallels (i.e. f is locally representable by rational functions)

Why? Rational functions are the most general functions you can express with +, −, ×, ÷, and we’re doing algebra!

Regular Functions

The Aﬃne Algebraic Connection

J. David Taylor Deﬁnition

Aﬃne Let X be an aﬃne algebraic set and let a ∈ X . A function Algebraic Sets f : X → k is regular at a, if a has an open neighborhood U Zariski Topology and there are polynomials P, Q ∈ k[T1,..., Tn] such that Regular Functions and 1 Q has no zeros on U, and Morphisms P(b) 2 f (b) = for all b ∈ U. Algebra Q(b)

Parallels Why? Rational functions are the most general functions you can express with +, −, ×, ÷, and we’re doing algebra!

Regular Functions

The Aﬃne Algebraic Connection

J. David Taylor Deﬁnition

Parallels (i.e. f is locally representable by rational functions) Regular Functions

The Aﬃne Algebraic Connection

J. David Taylor Deﬁnition

Parallels (i.e. f is locally representable by rational functions)

Why? Rational functions are the most general functions you can express with +, −, ×, ÷, and we’re doing algebra! Note the similarity with the deﬁnition of continuous, diﬀerentiable, or holomorphic functions on their respective manifolds.

Regular Functions

The Aﬃne Algebraic Connection

J. David Taylor

Aﬃne Algebraic Sets Deﬁnition Zariski We say a function f : X → k is regular if it is regular at every Topology point. Regular Functions and Morphisms

Algebra

Parallels Regular Functions

The Aﬃne Algebraic Connection

J. David Taylor

Affine Algebraic Sets Definition Zariski We say a function f : X → k is regular if it is regular at every Topology point. Regular Functions and Morphisms Note the similarity with the definition of continuous, Algebra differentiable, or holomorphic functions on their respective Parallels manifolds. 2 2 2 x On x + y − 1 in A (k), the function 1−y defines a regular function on the open set D(y − 1). By the Weak Nullstellensatz, every regular function on a closed set is given by a polynomial. 1 2 So the function xy−1 on xy − 2 in A (k) can be represented by a polynomial.

Examples of Regular Functions

The Aﬃne Algebraic Connection

J. David Taylor

Aﬃne Any polynomial function is regular. Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels By the Weak Nullstellensatz, every regular function on a closed set is given by a polynomial. 1 2 So the function xy−1 on xy − 2 in A (k) can be represented by a polynomial.

Examples of Regular Functions

The Aﬃne Algebraic Connection

J. David Taylor

Algebra

Parallels 1 2 So the function xy−1 on xy − 2 in A (k) can be represented by a polynomial.

Examples of Regular Functions

The Aﬃne Algebraic Connection

J. David Taylor

Parallels Examples of Regular Functions

The Aﬃne Algebraic Connection

J. David Taylor

Aﬃne Any polynomial function is regular. Algebraic Sets On x2 + y 2 − 1 in 2(k), the function x deﬁnes a Zariski A 1−y Topology regular function on the open set D(y − 1). Regular Functions and By the Weak Nullstellensatz, every regular function on a Morphisms closed set is given by a polynomial. Algebra So the function 1 on xy − 2 in 2(k) can be Parallels xy−1 A represented by a polynomial. A regular morphism of algebraic sets ψ : X → Y is a function which is given in coordinates by regular functions. More precisely, for every a ∈ X , there are open neighborhoods

ψ(a) ∈ V ⊆ Y and a ⊆ U ⊆ ψ−1(V ) ⊆ X

and rational functions R1,..., Rm ∈ k(T1,..., Tn) such that

For all i, the denominator of Ri does not vanish on U, and ψ : U → V is given by

(t1,..., tn) 7→ (R1(t1,..., tn),..., Rm(t1,..., tn)).

Regular Morphisms

The Affine Algebraic Connection Definition J. David n m Taylor Let X ⊆ A (k), Y ⊆ A (k) be affine algebraic sets.

Aﬃne Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels More precisely, for every a ∈ X , there are open neighborhoods

ψ(a) ∈ V ⊆ Y and a ⊆ U ⊆ ψ−1(V ) ⊆ X

and rational functions R1,..., Rm ∈ k(T1,..., Tn) such that

For all i, the denominator of Ri does not vanish on U, and ψ : U → V is given by

(t1,..., tn) 7→ (R1(t1,..., tn),..., Rm(t1,..., tn)).

Regular Morphisms

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels Regular Morphisms

The Affine Algebraic Connection Definition J. David n m Taylor Let X ⊆ A (k), Y ⊆ A (k) be affine algebraic sets. A regular morphism of algebraic sets ψ : X → Y is a function which is Affine Algebraic Sets given in coordinates by regular functions. Zariski More precisely, for every a ∈ X , there are open neighborhoods Topology Regular −1 Functions and ψ(a) ∈ V ⊆ Y and a ⊆ U ⊆ ψ (V ) ⊆ X Morphisms

Algebra and rational functions R1,..., Rm ∈ k(T1,..., Tn) such that Parallels For all i, the denominator of Ri does not vanish on U, and ψ : U → V is given by

(t1,..., tn) 7→ (R1(t1,..., tn),..., Rm(t1,..., tn)). Every regular function on X is a regular morphism 1 X → A (k).

Deﬁnition A regular isomorphism of aﬃne algebraic sets is a regular morphism ψ : X → Y such that there is a regular morphism φ : Y → X satisfying

φ ◦ ψ = idX , and

ψ ◦ φ = idY .

Regular Morphisms

The Aﬃne Algebraic Connection

J. David Taylor Regular morphisms of aﬃne algebraic sets are continuous.

Aﬃne Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels Deﬁnition A regular isomorphism of aﬃne algebraic sets is a regular morphism ψ : X → Y such that there is a regular morphism φ : Y → X satisfying

φ ◦ ψ = idX , and

ψ ◦ φ = idY .

Regular Morphisms

The Aﬃne Algebraic Connection

J. David Taylor Regular morphisms of aﬃne algebraic sets are continuous. Every regular function on X is a regular morphism Aﬃne 1 Algebraic Sets X → A (k). Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels Regular Morphisms

The Aﬃne Algebraic Connection

J. David Taylor Regular morphisms of aﬃne algebraic sets are continuous. Every regular function on X is a regular morphism Aﬃne 1 Algebraic Sets X → A (k). Zariski Topology

Regular Deﬁnition Functions and Morphisms A regular isomorphism of aﬃne algebraic sets is a regular Algebra morphism ψ : X → Y such that there is a regular morphism Parallels φ : Y → X satisfying

φ ◦ ψ = idX , and

ψ ◦ φ = idY . 1 2 2 The stereographic projection A → V (x + y − 1) is a regular morphism. 2 1 The projection V (y − x ) → A (k):(a, b) 7→ a is an isomorphism. 1 1 The map A (k) → A (k): a 7→ c1a + c2 for c1, c2 ∈ k and c1 6= 0 is an automorphism. (In fact, these are all the 1 automorphisms of A (k)).

Examples of Regular Morphisms

The Aﬃne Algebraic Connection J. David 1 1 p Taylor The map A (Fp) → A (Fp): a 7→ a is a regular morphism that is a homeomorphism but not an Aﬃne Algebraic Sets isomorphism. Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels 2 1 The projection V (y − x ) → A (k):(a, b) 7→ a is an isomorphism. 1 1 The map A (k) → A (k): a 7→ c1a + c2 for c1, c2 ∈ k and c1 6= 0 is an automorphism. (In fact, these are all the 1 automorphisms of A (k)).

Examples of Regular Morphisms

Algebra

Parallels 1 1 The map A (k) → A (k): a 7→ c1a + c2 for c1, c2 ∈ k and c1 6= 0 is an automorphism. (In fact, these are all the 1 automorphisms of A (k)).

Examples of Regular Morphisms

The Affine Algebraic Connection J. David 1 1 p Taylor The map A (Fp) → A (Fp): a 7→ a is a regular morphism that is a homeomorphism but not an Affine Algebraic Sets isomorphism. Zariski 1 2 2 Topology The stereographic projection A → V (x + y − 1) is a Regular regular morphism. Functions and 2 1 Morphisms The projection V (y − x ) → A (k):(a, b) 7→ a is an Algebra isomorphism. Parallels 1 1 The map A (k) → A (k): a 7→ c1a + c2 for c1, c2 ∈ k and c1 6= 0 is an automorphism. (In fact, these are all the 1 automorphisms of A (k)). That is, A is a ring that has k as a subring and A is a k-vector space. A homomorphism of k-algebras φ : A → B is a ring homomorphism that is k-linear. A ring R is reduced if it has no nilpotent elements. That is, if f ∈ R and f q = 0 for some q > 0, then f = 0. A k-algebra A is finitely generated (as a k-algebra) if there is a surjective k-algebra homomorphism

k[T1,..., Tn] A.

Equivalently, if there are elements g1,..., gn ∈ A such that every element of A can be represented as a polynomial in the gi ’s with k-coeﬃcients.

Some algebra

The Aﬃne Algebraic Connection A k-algebra A is a ring homomorphism k → A. J. David Taylor

Aﬃne Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels A homomorphism of k-algebras φ : A → B is a ring homomorphism that is k-linear. A ring R is reduced if it has no nilpotent elements. That is, if f ∈ R and f q = 0 for some q > 0, then f = 0. A k-algebra A is ﬁnitely generated (as a k-algebra) if there is a surjective k-algebra homomorphism

k[T1,..., Tn] A.

Equivalently, if there are elements g1,..., gn ∈ A such that every element of A can be represented as a polynomial in the gi ’s with k-coeﬃcients.

Some algebra

The Aﬃne Algebraic Connection A k-algebra A is a ring homomorphism k → A. That is, A J. David is a ring that has k as a subring and A is a k-vector space. Taylor

Aﬃne Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels A ring R is reduced if it has no nilpotent elements. That is, if f ∈ R and f q = 0 for some q > 0, then f = 0. A k-algebra A is ﬁnitely generated (as a k-algebra) if there is a surjective k-algebra homomorphism

k[T1,..., Tn] A.

Equivalently, if there are elements g1,..., gn ∈ A such that every element of A can be represented as a polynomial in the gi ’s with k-coeﬃcients.

Some algebra

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels A k-algebra A is ﬁnitely generated (as a k-algebra) if there is a surjective k-algebra homomorphism

k[T1,..., Tn] A.

Equivalently, if there are elements g1,..., gn ∈ A such that every element of A can be represented as a polynomial in the gi ’s with k-coeﬃcients.

Some algebra

The Aﬃne Algebraic Connection A k-algebra A is a ring homomorphism k → A. That is, A J. David is a ring that has k as a subring and A is a k-vector space. Taylor A homomorphism of k-algebras φ : A → B is a ring Aﬃne Algebraic Sets homomorphism that is k-linear. Zariski A ring R is reduced if it has no nilpotent elements. That Topology q Regular is, if f ∈ R and f = 0 for some q > 0, then f = 0. Functions and Morphisms

Algebra

Parallels Equivalently, if there are elements g1,..., gn ∈ A such that every element of A can be represented as a polynomial in the gi ’s with k-coeﬃcients.

Some algebra

Equivalently, if there are elements g1,..., gn ∈ A such that every element of A can be represented as a polynomial in the gi ’s with k-coeﬃcients. The ring map k[T1, T2, T3] → A : Ti 7→ Ti is a surjective k-algebra homomorphism. 5 k[T ]/(T ) is not reduced but k[T1, T2]/(T1 + T2 + 1) is.

Algebra Examples

The Aﬃne Algebraic Connection

J. David Taylor The ring Aﬃne Algebraic Sets k[T1, T2, T3] Zariski A := Topology 2 3 (T1T2, T3 + T2 , T1T3, T2T3) Regular Functions and Morphisms is a ﬁnitely generated k-algebra.

Algebra

Parallels 5 k[T ]/(T ) is not reduced but k[T1, T2]/(T1 + T2 + 1) is.

Algebra Examples

The Aﬃne Algebraic Connection

J. David Taylor The ring Aﬃne Algebraic Sets k[T1, T2, T3] Zariski A := Topology 2 3 (T1T2, T3 + T2 , T1T3, T2T3) Regular Functions and Morphisms is a ﬁnitely generated k-algebra.

Algebra The ring map k[T1, T2, T3] → A : Ti 7→ Ti is a surjective Parallels k-algebra homomorphism. Algebra Examples

The Aﬃne Algebraic Connection

J. David Taylor The ring Aﬃne Algebraic Sets k[T1, T2, T3] Zariski A := Topology 2 3 (T1T2, T3 + T2 , T1T3, T2T3) Regular Functions and Morphisms is a ﬁnitely generated k-algebra.

Algebra The ring map k[T1, T2, T3] → A : Ti 7→ Ti is a surjective Parallels k-algebra homomorphism. 5 k[T ]/(T ) is not reduced but k[T1, T2]/(T1 + T2 + 1) is. Let AAS be the category of aﬃne algebraic sets over k with regular morphisms. Let FGR be the category of ﬁnitely generated reduced k-algebras with k-algebra homomorphisms. There is an anti-equivalence of categories E : AAS → FGR. I’m going to spell it out in detail.

Algebraic Correspondence

The Aﬃne Algebraic Connection

J. David Taylor Theorem: Aﬃne Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels Let FGR be the category of ﬁnitely generated reduced k-algebras with k-algebra homomorphisms. There is an anti-equivalence of categories E : AAS → FGR. I’m going to spell it out in detail.

Algebraic Correspondence

The Aﬃne Algebraic Connection

J. David Taylor Theorem: Aﬃne Algebraic Sets Let AAS be the category of aﬃne algebraic sets over k

Zariski with regular morphisms. Topology

Regular Functions and Morphisms

Algebra

Parallels There is an anti-equivalence of categories E : AAS → FGR. I’m going to spell it out in detail.

Algebraic Correspondence

The Aﬃne Algebraic Connection

J. David Taylor Theorem: Aﬃne Algebraic Sets Let AAS be the category of aﬃne algebraic sets over k

Zariski with regular morphisms. Topology

Regular Let FGR be the category of ﬁnitely generated reduced Functions and k-algebras with k-algebra homomorphisms. Morphisms

Algebra

Parallels I’m going to spell it out in detail.

Algebraic Correspondence

The Aﬃne Algebraic Connection

J. David Taylor Theorem: Aﬃne Algebraic Sets Let AAS be the category of aﬃne algebraic sets over k

Zariski with regular morphisms. Topology

The Aﬃne Algebraic Connection

J. David Taylor Theorem: Aﬃne Algebraic Sets Let AAS be the category of aﬃne algebraic sets over k

Zariski with regular morphisms. Topology

Regular Let FGR be the category of finitely generated reduced Functions and k-algebras with k-algebra homomorphisms. Morphisms Algebra There is an anti-equivalence of categories Parallels E : AAS → FGR. I’m going to spell it out in detail. Define the coordinate ring of X as the ring of regular functions on X under pointwise addition and multiplication. It is written A[X ]. A[X ] is a reduced k-algebra. The Weak Nullstellensatz implies that A[X ] is finitely generated. E(X ) := A[X ]

Algebraic Correspondence: Sets7→Rings

The Aﬃne Algebraic Connection

J. David Taylor Let X be an aﬃne algebraic set. Aﬃne Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels A[X ] is a reduced k-algebra. The Weak Nullstellensatz implies that A[X ] is ﬁnitely generated. E(X ) := A[X ]

Algebraic Correspondence: Sets7→Rings

The Aﬃne Algebraic Connection

Algebra

Parallels The Weak Nullstellensatz implies that A[X ] is ﬁnitely generated. E(X ) := A[X ]

Algebraic Correspondence: Sets7→Rings

The Aﬃne Algebraic Connection

Algebra

Parallels E(X ) := A[X ]

Algebraic Correspondence: Sets7→Rings

The Aﬃne Algebraic Connection

J. David Taylor Let X be an affine algebraic set. Affine Algebraic Sets Define the coordinate ring of X as the ring of regular Zariski functions on X under pointwise addition and Topology multiplication. It is written A[X ]. Regular Functions and Morphisms A[X ] is a reduced k-algebra. Algebra The Weak Nullstellensatz implies that A[X ] is finitely Parallels generated. Algebraic Correspondence: Sets7→Rings

The Aﬃne Algebraic Connection

Algebraic Correspondence: Morphisms7→Homs

The Aﬃne Algebraic Connection

J. David Taylor

Aﬃne Algebraic Sets Let ψ : X → Y be a regular morphism. Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels In fact, ψ∗ : A[Y ] → A[X ] is a k-algebra homomorphism. E(ψ) := ψ∗

Algebraic Correspondence: Morphisms7→Homs

The Aﬃne Algebraic Connection

J. David Taylor

Aﬃne Algebraic Sets Let ψ : X → Y be a regular morphism. Zariski ∗ ψ f 1 Topology If f ∈ A[Y ], then ψ f : X −→ Y −→ A (k) is a regular ∗ Regular morphism, so ψ f ∈ A[X ]. Functions and Morphisms

Algebra

Parallels E(ψ) := ψ∗

Algebraic Correspondence: Morphisms7→Homs

The Aﬃne Algebraic Connection

J. David Taylor

Aﬃne Algebraic Sets Let ψ : X → Y be a regular morphism. Zariski ∗ ψ f 1 Topology If f ∈ A[Y ], then ψ f : X −→ Y −→ A (k) is a regular ∗ Regular morphism, so ψ f ∈ A[X ]. Functions and Morphisms In fact, ψ∗ : A[Y ] → A[X ] is a k-algebra homomorphism. Algebra

Parallels Algebraic Correspondence: Morphisms7→Homs

The Aﬃne Algebraic Connection

J. David Taylor

ker ρ is a ﬁnitely generated ideal of k[T1,..., Tn] by Hilbert’s Basis Theorem. n Let X ⊆ A (k) be the aﬃne algebraic set determined by a set of generators of ker ρ. E−1(A) := X .

Algebraic Correspondence:Rings7→Sets

The Aﬃne Algebraic Connection

J. David Taylor Let A be a ﬁnitely generated reduced k-algebra. Aﬃne Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels ker ρ is a ﬁnitely generated ideal of k[T1,..., Tn] by Hilbert’s Basis Theorem. n Let X ⊆ A (k) be the aﬃne algebraic set determined by a set of generators of ker ρ. E−1(A) := X .

Algebraic Correspondence:Rings7→Sets

The Aﬃne Algebraic Connection

J. David Taylor Let A be a ﬁnitely generated reduced k-algebra. Aﬃne Algebraic Sets Let ρ : k[T1,..., Tn] A be a surjective k-algebra Zariski homomorphism. Topology

Regular Functions and Morphisms

Algebra

Parallels n Let X ⊆ A (k) be the aﬃne algebraic set determined by a set of generators of ker ρ. E−1(A) := X .

Algebraic Correspondence:Rings7→Sets

The Aﬃne Algebraic Connection

J. David Taylor Let A be a ﬁnitely generated reduced k-algebra. Aﬃne Algebraic Sets Let ρ : k[T1,..., Tn] A be a surjective k-algebra Zariski homomorphism. Topology

Regular ker ρ is a ﬁnitely generated ideal of k[T1,..., Tn] by Functions and Morphisms Hilbert’s Basis Theorem.

Algebra

Parallels E−1(A) := X .

Algebraic Correspondence:Rings7→Sets

The Aﬃne Algebraic Connection

J. David Taylor Let A be a ﬁnitely generated reduced k-algebra. Aﬃne Algebraic Sets Let ρ : k[T1,..., Tn] A be a surjective k-algebra Zariski homomorphism. Topology

Regular ker ρ is a ﬁnitely generated ideal of k[T1,..., Tn] by Functions and Morphisms Hilbert’s Basis Theorem. n Algebra Let X ⊆ A (k) be the aﬃne algebraic set determined by a Parallels set of generators of ker ρ. Algebraic Correspondence:Rings7→Sets

The Aﬃne Algebraic Connection

J. David Taylor Let A be a ﬁnitely generated reduced k-algebra. Aﬃne Algebraic Sets Let ρ : k[T1,..., Tn] A be a surjective k-algebra Zariski homomorphism. Topology

h(Si ) = Pi (T1,..., Tn) mod I for all i and some polynomials P1,..., Pm ∈ k[T1,..., Tn].

Let ψ : X → Y be given by ψ(a) = (P1(a),..., Pm(a)). E−1(h) := ψ

Algebraic Correspondence:Homs7→Morphisms

The Aﬃne Algebraic Connection

J. David Taylor

Aﬃne Let h : B → A be a homomorphism of ﬁnitely generated Algebraic Sets reduced k-algebras. Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels h(Si ) = Pi (T1,..., Tn) mod I for all i and some polynomials P1,..., Pm ∈ k[T1,..., Tn].

Let ψ : X → Y be given by ψ(a) = (P1(a),..., Pm(a)). E−1(h) := ψ

Algebraic Correspondence:Homs7→Morphisms

The Aﬃne Algebraic Connection

J. David Taylor

Aﬃne Let h : B → A be a homomorphism of ﬁnitely generated Algebraic Sets reduced k-algebras. Zariski ∼ ∼ Topology A = k[T1,..., Tn]/I and B = k[S1,..., Sm]/J Regular Functions and Morphisms

Algebra

Parallels Let ψ : X → Y be given by ψ(a) = (P1(a),..., Pm(a)). E−1(h) := ψ

Algebraic Correspondence:Homs7→Morphisms

The Aﬃne Algebraic Connection

J. David Taylor

Parallels E−1(h) := ψ

Algebraic Correspondence:Homs7→Morphisms

The Aﬃne Algebraic Connection

J. David Taylor

Aﬃne Let h : B → A be a homomorphism of ﬁnitely generated Algebraic Sets reduced k-algebras. Zariski ∼ ∼ Topology A = k[T1,..., Tn]/I and B = k[S1,..., Sm]/J Regular Functions and h(Si ) = Pi (T1,..., Tn) mod I for all i and some Morphisms polynomials P1,..., Pm ∈ k[T1,..., Tn]. Algebra Let ψ : X → Y be given by ψ(a) = (P1(a),..., Pm(a)). Parallels Algebraic Correspondence:Homs7→Morphisms

The Aﬃne Algebraic Connection

J. David Taylor

Galois Connection

The Aﬃne Algebraic Connection

J. David Taylor

Aﬃne Algebraic Sets Let X be an aﬃne algebraic set with coordinate ring A. Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels Let S ⊆ A. Deﬁne V (S) as the solution set of S.

Galois Connection

The Aﬃne Algebraic Connection

J. David Taylor

Affine Algebraic Sets Let X be an affine algebraic set with coordinate ring A. Zariski Topology Let Y ⊆ X . Define I (Y ) as the ideal of functions that Regular Functions and vanish on Y . Morphisms

Algebra

Parallels Galois Connection

The Aﬃne Algebraic Connection

J. David Taylor

Parallels There is a pair of inclusion reversing functions

I : P(X ) → P(A) and V : P(A) → P(X )

such that for all Z ∈ P(X ) and J ∈ P(A): J ⊆ I(Z) iﬀ Z ⊆ V(J) (Galois Connection) I(V(J)) = p(J) (Weak Nullstellensatz) V(I(Z)) = Z

Galois Connection

The Aﬃne Algebraic Connection

J. David Taylor Let X be an aﬃne algebraic set with coordinate ring A. Aﬃne Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels J ⊆ I(Z) iﬀ Z ⊆ V(J) (Galois Connection) I(V(J)) = p(J) (Weak Nullstellensatz) V(I(Z)) = Z

Galois Connection

The Aﬃne Algebraic Connection

J. David Taylor Let X be an aﬃne algebraic set with coordinate ring A. Aﬃne Algebraic Sets There is a pair of inclusion reversing functions

Zariski Topology I : P(X ) → P(A) and V : P(A) → P(X ) Regular Functions and Morphisms such that for all Z ∈ P(X ) and J ∈ P(A): Algebra

Parallels I(V(J)) = p(J) (Weak Nullstellensatz) V(I(Z)) = Z

Galois Connection

The Aﬃne Algebraic Connection

J. David Taylor Let X be an aﬃne algebraic set with coordinate ring A. Aﬃne Algebraic Sets There is a pair of inclusion reversing functions

Zariski Topology I : P(X ) → P(A) and V : P(A) → P(X ) Regular Functions and Morphisms such that for all Z ∈ P(X ) and J ∈ P(A): Algebra J ⊆ I(Z) iﬀ Z ⊆ V(J) (Galois Connection) Parallels V(I(Z)) = Z

Galois Connection

The Aﬃne Algebraic Connection

J. David Taylor Let X be an aﬃne algebraic set with coordinate ring A. Aﬃne Algebraic Sets There is a pair of inclusion reversing functions

The Aﬃne Algebraic Connection

J. David Taylor Let X be an aﬃne algebraic set with coordinate ring A. Aﬃne Algebraic Sets There is a pair of inclusion reversing functions

Zariski Topology I : P(X ) → P(A) and V : P(A) → P(X ) Regular Functions and Morphisms such that for all Z ∈ P(X ) and J ∈ P(A): Algebra J ⊆ I(Z) iff Z ⊆ V(J) (Galois Connection) Parallels I(V(J)) = p(J) (Weak Nullstellensatz) V(I(Z)) = Z irreducible closed sets vs prime ideals points vs maximal ideals dimensions match varieties vs field extensions compact Riemann surfaces vs finite extensions of C(t) calculate tangent spaces with commutative algebra classify singularities with commutative algebra (e.g. k T1, T2 /(T1T2)) J K vector bundles vs finitely generated projective modules etc.

Consequences

The Aﬃne Algebraic Connection closed sets vs radical ideals J. David Taylor

Aﬃne Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels points vs maximal ideals dimensions match varieties vs field extensions compact Riemann surfaces vs finite extensions of C(t) calculate tangent spaces with commutative algebra classify singularities with commutative algebra (e.g. k T1, T2 /(T1T2)) J K vector bundles vs finitely generated projective modules etc.

Consequences

The Aﬃne Algebraic Connection closed sets vs radical ideals J. David Taylor irreducible closed sets vs prime ideals

Aﬃne Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels dimensions match varieties vs field extensions compact Riemann surfaces vs finite extensions of C(t) calculate tangent spaces with commutative algebra classify singularities with commutative algebra (e.g. k T1, T2 /(T1T2)) J K vector bundles vs finitely generated projective modules etc.

Consequences

The Aﬃne Algebraic Connection closed sets vs radical ideals J. David Taylor irreducible closed sets vs prime ideals Aﬃne points vs maximal ideals Algebraic Sets

Zariski Topology

Regular Functions and Morphisms

Algebra

Parallels varieties vs field extensions compact Riemann surfaces vs finite extensions of C(t) calculate tangent spaces with commutative algebra classify singularities with commutative algebra (e.g. k T1, T2 /(T1T2)) J K vector bundles vs finitely generated projective modules etc.

Consequences

The Aﬃne Algebraic Connection closed sets vs radical ideals J. David Taylor irreducible closed sets vs prime ideals Aﬃne points vs maximal ideals Algebraic Sets Zariski dimensions match Topology

Regular Functions and Morphisms

Algebra

Parallels compact Riemann surfaces vs ﬁnite extensions of C(t) calculate tangent spaces with commutative algebra classify singularities with commutative algebra (e.g. k T1, T2 /(T1T2)) J K vector bundles vs ﬁnitely generated projective modules etc.

Consequences

The Aﬃne Algebraic Connection closed sets vs radical ideals J. David Taylor irreducible closed sets vs prime ideals Aﬃne points vs maximal ideals Algebraic Sets Zariski dimensions match Topology

Regular varieties vs ﬁeld extensions Functions and Morphisms

Algebra

Parallels calculate tangent spaces with commutative algebra classify singularities with commutative algebra (e.g. k T1, T2 /(T1T2)) J K vector bundles vs ﬁnitely generated projective modules etc.

Consequences

The Aﬃne Algebraic Connection closed sets vs radical ideals J. David Taylor irreducible closed sets vs prime ideals Aﬃne points vs maximal ideals Algebraic Sets Zariski dimensions match Topology

Regular varieties vs ﬁeld extensions Functions and Morphisms compact Riemann surfaces vs ﬁnite extensions of C(t) Algebra

Parallels classify singularities with commutative algebra (e.g. k T1, T2 /(T1T2)) J K vector bundles vs ﬁnitely generated projective modules etc.

Consequences

The Aﬃne Algebraic Connection closed sets vs radical ideals J. David Taylor irreducible closed sets vs prime ideals Aﬃne points vs maximal ideals Algebraic Sets Zariski dimensions match Topology

Regular varieties vs field extensions Functions and Morphisms compact Riemann surfaces vs finite extensions of C(t) Algebra calculate tangent spaces with commutative algebra Parallels vector bundles vs finitely generated projective modules etc.

Consequences

The Aﬃne Algebraic Connection closed sets vs radical ideals J. David Taylor irreducible closed sets vs prime ideals Aﬃne points vs maximal ideals Algebraic Sets Zariski dimensions match Topology

Regular varieties vs ﬁeld extensions Functions and Morphisms compact Riemann surfaces vs ﬁnite extensions of C(t) Algebra calculate tangent spaces with commutative algebra Parallels classify singularities with commutative algebra (e.g. k T1, T2 /(T1T2)) J K etc.

Consequences

The Aﬃne Algebraic Connection closed sets vs radical ideals J. David Taylor irreducible closed sets vs prime ideals Aﬃne points vs maximal ideals Algebraic Sets Zariski dimensions match Topology

Regular varieties vs field extensions Functions and Morphisms compact Riemann surfaces vs finite extensions of C(t) Algebra calculate tangent spaces with commutative algebra Parallels classify singularities with commutative algebra (e.g. k T1, T2 /(T1T2)) J K vector bundles vs finitely generated projective modules Consequences

The Aﬃne Algebraic Connection closed sets vs radical ideals J. David Taylor irreducible closed sets vs prime ideals Aﬃne points vs maximal ideals Algebraic Sets Zariski dimensions match Topology

Regular varieties vs field extensions Functions and Morphisms compact Riemann surfaces vs finite extensions of C(t) Algebra calculate tangent spaces with commutative algebra Parallels classify singularities with commutative algebra (e.g. k T1, T2 /(T1T2)) J K vector bundles vs finitely generated projective modules etc.