Cohomology Theories and Commutative Rings

July 20, 2015 One of the main goals of algebraic topology is to study topological spaces by means of algebraic invariants.

Example To every space X we assign cohomology groups Hn(X ; A).

Let us (temporarily) ﬁx the group A and denote Hn(X ; A) by Hn(X ).

Cohomology Theories and Commutative Rings Example To every space X we assign cohomology groups Hn(X ; A).

Let us (temporarily) ﬁx the group A and denote Hn(X ; A) by Hn(X ).

Cohomology Theories and Commutative Rings

One of the main goals of algebraic topology is to study topological spaces by means of algebraic invariants. Let us (temporarily) ﬁx the group A and denote Hn(X ; A) by Hn(X ).

Cohomology Theories and Commutative Rings

One of the main goals of algebraic topology is to study topological spaces by means of algebraic invariants.

Example To every space X we assign cohomology groups Hn(X ; A). Cohomology Theories and Commutative Rings

One of the main goals of algebraic topology is to study topological spaces by means of algebraic invariants.

Example To every space X we assign cohomology groups Hn(X ; A).

Let us (temporarily) ﬁx the group A and denote Hn(X ; A) by Hn(X ). (A1) Functoriality: Every continuous map f : X → Y determines f ∗ :Hn(Y ) → Hn(X ). n n n In particular, if x ∈ X then H (X ) ' H ({x}) ⊕ Hred(X ). (A2) Homotopy Invariance: f ' g ⇒ f ∗ = g ∗. n Q n (A3) Multiplicativity: H (qXα) ' H (Xα) n n+1 (A4) Suspension isomorphisms: Hred(X ) ' Hred (ΣX ). n n n (A5) Excision: Hred(X /Y ) → H (X ) → H (Y ) ( A if n = 0 (A6) Dimension Axiom: Hn(∗) = 0 otherwise.

Cohomology Theories and Commutative Rings

Axioms for Cohomology n n n In particular, if x ∈ X then H (X ) ' H ({x}) ⊕ Hred(X ). (A2) Homotopy Invariance: f ' g ⇒ f ∗ = g ∗. n Q n (A3) Multiplicativity: H (qXα) ' H (Xα) n n+1 (A4) Suspension isomorphisms: Hred(X ) ' Hred (ΣX ). n n n (A5) Excision: Hred(X /Y ) → H (X ) → H (Y ) ( A if n = 0 (A6) Dimension Axiom: Hn(∗) = 0 otherwise.

Cohomology Theories and Commutative Rings

Axioms for Cohomology

(A1) Functoriality: Every continuous map f : X → Y determines f ∗ :Hn(Y ) → Hn(X ). (A2) Homotopy Invariance: f ' g ⇒ f ∗ = g ∗. n Q n (A3) Multiplicativity: H (qXα) ' H (Xα) n n+1 (A4) Suspension isomorphisms: Hred(X ) ' Hred (ΣX ). n n n (A5) Excision: Hred(X /Y ) → H (X ) → H (Y ) ( A if n = 0 (A6) Dimension Axiom: Hn(∗) = 0 otherwise.

Cohomology Theories and Commutative Rings

Axioms for Cohomology

(A1) Functoriality: Every continuous map f : X → Y determines f ∗ :Hn(Y ) → Hn(X ). n n n In particular, if x ∈ X then H (X ) ' H ({x}) ⊕ Hred(X ). n Q n (A3) Multiplicativity: H (qXα) ' H (Xα) n n+1 (A4) Suspension isomorphisms: Hred(X ) ' Hred (ΣX ). n n n (A5) Excision: Hred(X /Y ) → H (X ) → H (Y ) ( A if n = 0 (A6) Dimension Axiom: Hn(∗) = 0 otherwise.

Cohomology Theories and Commutative Rings

Axioms for Cohomology

(A1) Functoriality: Every continuous map f : X → Y determines f ∗ :Hn(Y ) → Hn(X ). n n n In particular, if x ∈ X then H (X ) ' H ({x}) ⊕ Hred(X ). (A2) Homotopy Invariance: f ' g ⇒ f ∗ = g ∗. n n+1 (A4) Suspension isomorphisms: Hred(X ) ' Hred (ΣX ). n n n (A5) Excision: Hred(X /Y ) → H (X ) → H (Y ) ( A if n = 0 (A6) Dimension Axiom: Hn(∗) = 0 otherwise.

Cohomology Theories and Commutative Rings

Axioms for Cohomology

Cohomology Theories and Commutative Rings

Axioms for Cohomology

Cohomology Theories and Commutative Rings

Axioms for Cohomology

(A1) Functoriality: Every continuous map f : X → Y determines f ∗ :Hn(Y ) → Hn(X ). n n n In particular, if x ∈ X then H (X ) ' H ({x}) ⊕ Hred(X ). (A2) Homotopy Invariance: f ' g ⇒ f ∗ = g ∗. n Q n (A3) Multiplicativity: H (qXα) ' H (Xα) n n+1 (A4) Suspension isomorphisms: Hred(X ) ' Hred (ΣX ). n n n (A5) Excision: Hred(X /Y ) → H (X ) → H (Y ) ( A if n = 0 (A6) Dimension Axiom: Hn(∗) = 0 otherwise. Theorem (Eilenberg-Steenrod) Any collection of invariants {Hn : spaces → abelian groups } satisfying (A1) through (A6) is given by cohomology with coeﬃcients in the abelian group A = H0(∗).

Deﬁnition A cohomology theory E is a sequence of invariants {E n : spaces → abelian groups} satisfying (A1) through (A5). (The n n+1 isomorphisms Ered(X ) ' Ered (ΣX ) are part of the data.)

Cohomology Theories and Commutative Rings

Cohomology Theories Deﬁnition A cohomology theory E is a sequence of invariants {E n : spaces → abelian groups} satisfying (A1) through (A5). (The n n+1 isomorphisms Ered(X ) ' Ered (ΣX ) are part of the data.)

Cohomology Theories and Commutative Rings

Cohomology Theories

Theorem (Eilenberg-Steenrod) Any collection of invariants {Hn : spaces → abelian groups } satisfying (A1) through (A6) is given by cohomology with coeﬃcients in the abelian group A = H0(∗). (The n n+1 isomorphisms Ered(X ) ' Ered (ΣX ) are part of the data.)

Cohomology Theories and Commutative Rings

Cohomology Theories

Theorem (Eilenberg-Steenrod) Any collection of invariants {Hn : spaces → abelian groups } satisfying (A1) through (A6) is given by cohomology with coeﬃcients in the abelian group A = H0(∗).

Deﬁnition A cohomology theory E is a sequence of invariants {E n : spaces → abelian groups} satisfying (A1) through (A5). Cohomology Theories and Commutative Rings

Cohomology Theories

Theorem (Eilenberg-Steenrod) Any collection of invariants {Hn : spaces → abelian groups } satisfying (A1) through (A6) is given by cohomology with coeﬃcients in the abelian group A = H0(∗).

Deﬁnition A cohomology theory E is a sequence of invariants {E n : spaces → abelian groups} satisfying (A1) through (A5). (The n n+1 isomorphisms Ered(X ) ' Ered (ΣX ) are part of the data.) Let X be a ﬁnite cell complex. Then the set

{ complex vector bundles on X }/ isomorphism

forms a commutative monoid under ⊕. The associated abelian group is denoted by K 0(X ), called the complex K-theory of X . Example K 0(∗) ' Z

Cohomology Theories and Commutative Rings

Example: Complex K-Theory The associated abelian group is denoted by K 0(X ), called the complex K-theory of X . Example K 0(∗) ' Z

Cohomology Theories and Commutative Rings

Example: Complex K-Theory

Let X be a ﬁnite cell complex. Then the set

{ complex vector bundles on X }/ isomorphism

forms a commutative monoid under ⊕. Example K 0(∗) ' Z

Cohomology Theories and Commutative Rings

Example: Complex K-Theory

Let X be a ﬁnite cell complex. Then the set

{ complex vector bundles on X }/ isomorphism

forms a commutative monoid under ⊕. The associated abelian group is denoted by K 0(X ), called the complex K-theory of X . Cohomology Theories and Commutative Rings

Example: Complex K-Theory

Let X be a ﬁnite cell complex. Then the set

{ complex vector bundles on X }/ isomorphism

forms a commutative monoid under ⊕. The associated abelian group is denoted by K 0(X ), called the complex K-theory of X . Example K 0(∗) ' Z One can extend to a deﬁnition of K n(X ) for all integers n and all spaces X . These invariants satisfy (A1) through (A5), but not the dimension axiom (A6). Instead we have ( if n is even K n(∗) = Z 0 if n is odd.

Cohomology Theories and Commutative Rings

Example: Complex K-Theory These invariants satisfy (A1) through (A5), but not the dimension axiom (A6). Instead we have ( if n is even K n(∗) = Z 0 if n is odd.

Cohomology Theories and Commutative Rings

Example: Complex K-Theory

One can extend to a deﬁnition of K n(X ) for all integers n and all spaces X . Instead we have ( if n is even K n(∗) = Z 0 if n is odd.

Cohomology Theories and Commutative Rings

Example: Complex K-Theory

One can extend to a deﬁnition of K n(X ) for all integers n and all spaces X . These invariants satisfy (A1) through (A5), but not the dimension axiom (A6). Cohomology Theories and Commutative Rings

Example: Complex K-Theory

One can extend to a deﬁnition of K n(X ) for all integers n and all spaces X . These invariants satisfy (A1) through (A5), but not the dimension axiom (A6). Instead we have ( if n is even K n(∗) = Z 0 if n is odd. Theorem (Brown) n n Let {E }n∈Z be a cohomology theory. Then each E is representable functor. That is, there exists a space Z(n) such that for every (nice) space X

E n(X ) '{ Maps from X into Z(n) }/ homotopy

n n+1 The suspension isomorphisms Ered(X ) ' Ered (ΣX ) determine homotopy equivalences Z(n) ' ΩZ(n + 1).

Cohomology Theories and Commutative Rings

Brown Representability Then each E n is representable functor. That is, there exists a space Z(n) such that for every (nice) space X

E n(X ) '{ Maps from X into Z(n) }/ homotopy

n n+1 The suspension isomorphisms Ered(X ) ' Ered (ΣX ) determine homotopy equivalences Z(n) ' ΩZ(n + 1).

Cohomology Theories and Commutative Rings

Brown Representability

Theorem (Brown) n Let {E }n∈Z be a cohomology theory. That is, there exists a space Z(n) such that for every (nice) space X

E n(X ) '{ Maps from X into Z(n) }/ homotopy

n n+1 The suspension isomorphisms Ered(X ) ' Ered (ΣX ) determine homotopy equivalences Z(n) ' ΩZ(n + 1).

Cohomology Theories and Commutative Rings

Brown Representability

Theorem (Brown) n n Let {E }n∈Z be a cohomology theory. Then each E is representable functor. n n+1 The suspension isomorphisms Ered(X ) ' Ered (ΣX ) determine homotopy equivalences Z(n) ' ΩZ(n + 1).

Cohomology Theories and Commutative Rings

Brown Representability

Theorem (Brown) n n Let {E }n∈Z be a cohomology theory. Then each E is representable functor. That is, there exists a space Z(n) such that for every (nice) space X

E n(X ) '{ Maps from X into Z(n) }/ homotopy Cohomology Theories and Commutative Rings

Brown Representability

Theorem (Brown) n n Let {E }n∈Z be a cohomology theory. Then each E is representable functor. That is, there exists a space Z(n) such that for every (nice) space X

E n(X ) '{ Maps from X into Z(n) }/ homotopy

n n+1 The suspension isomorphisms Ered(X ) ' Ered (ΣX ) determine homotopy equivalences Z(n) ' ΩZ(n + 1). Every spectrum {Z(n)} determine a cohomology theory {E n} via

E n(X ) = { continuous maps f : X → Z(n)}/ homotopy.

Using Brown’s theorem, one gets a bijection

{cohomology theories}/iso '{spectra}/homotopy equivalence.

Cohomology Theories and Commutative Rings

Spectra

Deﬁnition

A spectrum is a sequence of pointed topological spaces {Z(n)}n∈Z and homotopy equivalences Z(n) ' ΩZ(n + 1). Using Brown’s theorem, one gets a bijection

{cohomology theories}/iso '{spectra}/homotopy equivalence.

Cohomology Theories and Commutative Rings

Spectra

Deﬁnition

A spectrum is a sequence of pointed topological spaces {Z(n)}n∈Z and homotopy equivalences Z(n) ' ΩZ(n + 1).

Every spectrum {Z(n)} determine a cohomology theory {E n} via

E n(X ) = { continuous maps f : X → Z(n)}/ homotopy. Cohomology Theories and Commutative Rings

Spectra

Deﬁnition

A spectrum is a sequence of pointed topological spaces {Z(n)}n∈Z and homotopy equivalences Z(n) ' ΩZ(n + 1).

Every spectrum {Z(n)} determine a cohomology theory {E n} via

E n(X ) = { continuous maps f : X → Z(n)}/ homotopy.

Using Brown’s theorem, one gets a bijection

{cohomology theories}/iso '{spectra}/homotopy equivalence. Example Let A be an abelian group. Then

Hn(X ; A) '{ maps f : X → K(A, n)}/homotopy

Here K(A, n) is an Eilenberg-MacLane space, characterized by ( A if ∗ = n π∗K(A, n) = 0 otherwise.

The sequence {K(A, n)} determines a spectrum denoted by HA, the Eilenberg-MacLane spectrum of A.

Cohomology Theories and Commutative Rings

Example: Ordinary Cohomology Here K(A, n) is an Eilenberg-MacLane space, characterized by ( A if ∗ = n π∗K(A, n) = 0 otherwise.

The sequence {K(A, n)} determines a spectrum denoted by HA, the Eilenberg-MacLane spectrum of A.

Cohomology Theories and Commutative Rings

Example: Ordinary Cohomology

Example Let A be an abelian group. Then

Hn(X ; A) '{ maps f : X → K(A, n)}/homotopy The sequence {K(A, n)} determines a spectrum denoted by HA, the Eilenberg-MacLane spectrum of A.

Cohomology Theories and Commutative Rings

Example: Ordinary Cohomology

Example Let A be an abelian group. Then

Hn(X ; A) '{ maps f : X → K(A, n)}/homotopy

Here K(A, n) is an Eilenberg-MacLane space, characterized by ( A if ∗ = n π∗K(A, n) = 0 otherwise. Cohomology Theories and Commutative Rings

Example: Ordinary Cohomology

Example Let A be an abelian group. Then

Hn(X ; A) '{ maps f : X → K(A, n)}/homotopy

Here K(A, n) is an Eilenberg-MacLane space, characterized by ( A if ∗ = n π∗K(A, n) = 0 otherwise.

The sequence {K(A, n)} determines a spectrum denoted by HA, the Eilenberg-MacLane spectrum of A. Set Space

Abelian Group Spectrum

Tensor Product ⊗ Smash product ∧

Associative Ring Associative Ring Spectrum

Commutative Ring Commutative Ring Spectrum

Cohomology Theories and Commutative Rings

Some Analogies

Classical Algebra Homotopy-Theoretic Algebra Abelian Group Spectrum

Tensor Product ⊗ Smash product ∧

Associative Ring Associative Ring Spectrum

Commutative Ring Commutative Ring Spectrum

Cohomology Theories and Commutative Rings

Some Analogies

Classical Algebra Homotopy-Theoretic Algebra

Set Space Tensor Product ⊗ Smash product ∧

Associative Ring Associative Ring Spectrum

Commutative Ring Commutative Ring Spectrum

Cohomology Theories and Commutative Rings

Some Analogies

Classical Algebra Homotopy-Theoretic Algebra

Set Space

Abelian Group Spectrum Associative Ring Associative Ring Spectrum

Commutative Ring Commutative Ring Spectrum

Cohomology Theories and Commutative Rings

Some Analogies

Classical Algebra Homotopy-Theoretic Algebra

Set Space

Abelian Group Spectrum

Tensor Product ⊗ Smash product ∧ Commutative Ring Commutative Ring Spectrum

Cohomology Theories and Commutative Rings

Some Analogies

Classical Algebra Homotopy-Theoretic Algebra

Set Space

Abelian Group Spectrum

Tensor Product ⊗ Smash product ∧

Associative Ring Associative Ring Spectrum Cohomology Theories and Commutative Rings

Some Analogies

Classical Algebra Homotopy-Theoretic Algebra

Set Space

Abelian Group Spectrum

Tensor Product ⊗ Smash product ∧

Associative Ring Associative Ring Spectrum

Commutative Ring Commutative Ring Spectrum An associative (or A∞) ring spectrum is a spectrum E equipped with a multiplication E ∧ E → E satisfying a suitable associative law. (There are many diﬀerent approaches to making this precise.) If E is an associative ring spectrum, then E ∗(X ) is a graded ring for any space X . However, more is true: associativity holds at the cocycle level, not just at the level of cohomology.

There is a similar notion of commutative (or E∞) ring spectrum.

Cohomology Theories and Commutative Rings

Ring Spectra (There are many diﬀerent approaches to making this precise.) If E is an associative ring spectrum, then E ∗(X ) is a graded ring for any space X . However, more is true: associativity holds at the cocycle level, not just at the level of cohomology.

There is a similar notion of commutative (or E∞) ring spectrum.

Cohomology Theories and Commutative Rings

Ring Spectra

An associative (or A∞) ring spectrum is a spectrum E equipped with a multiplication E ∧ E → E satisfying a suitable associative law. If E is an associative ring spectrum, then E ∗(X ) is a graded ring for any space X . However, more is true: associativity holds at the cocycle level, not just at the level of cohomology.

There is a similar notion of commutative (or E∞) ring spectrum.

Cohomology Theories and Commutative Rings

Ring Spectra

There is a similar notion of commutative (or E∞) ring spectrum.

Cohomology Theories and Commutative Rings

Ring Spectra

An associative (or A∞) ring spectrum is a spectrum E equipped with a multiplication E ∧ E → E satisfying a suitable associative law. (There are many diﬀerent approaches to making this precise.) If E is an associative ring spectrum, then E ∗(X ) is a graded ring for any space X . There is a similar notion of commutative (or E∞) ring spectrum.

Cohomology Theories and Commutative Rings

Ring Spectra

An associative (or A∞) ring spectrum is a spectrum E equipped with a multiplication E ∧ E → E satisfying a suitable associative law. (There are many diﬀerent approaches to making this precise.) If E is an associative ring spectrum, then E ∗(X ) is a graded ring for any space X . However, more is true: associativity holds at the cocycle level, not just at the level of cohomology. Cohomology Theories and Commutative Rings

Ring Spectra

An associative (or A∞) ring spectrum is a spectrum E equipped with a multiplication E ∧ E → E satisfying a suitable associative law. (There are many diﬀerent approaches to making this precise.) If E is an associative ring spectrum, then E ∗(X ) is a graded ring for any space X . However, more is true: associativity holds at the cocycle level, not just at the level of cohomology.

There is a similar notion of commutative (or E∞) ring spectrum. Example If R is an associative ring, then HR is an associative ring spectrum.

Example If R is a commutative ring, then HR is a commutative ring spectrum.

Example Complex K-theory is represented by a commutative ring spectrum K.

Cohomology Theories and Commutative Rings

Examples of Ring Spectra Example If R is a commutative ring, then HR is a commutative ring spectrum.

Example Complex K-theory is represented by a commutative ring spectrum K.

Cohomology Theories and Commutative Rings

Examples of Ring Spectra

Example If R is an associative ring, then HR is an associative ring spectrum. Example Complex K-theory is represented by a commutative ring spectrum K.

Cohomology Theories and Commutative Rings

Examples of Ring Spectra

Example If R is an associative ring, then HR is an associative ring spectrum.

Example If R is a commutative ring, then HR is a commutative ring spectrum. Cohomology Theories and Commutative Rings

Examples of Ring Spectra

Example If R is an associative ring, then HR is an associative ring spectrum.

Example If R is a commutative ring, then HR is a commutative ring spectrum.

Example Complex K-theory is represented by a commutative ring spectrum K. Let k be an associative ring. The following conditions are equivalent:

(a) Every nonzero element of k is invertible. (b) Every module over k is free. If these conditions are satisﬁed, we say that k is a (skew) ﬁeld.

Cohomology Theories and Commutative Rings

Fields in Classical Algebra (a) Every nonzero element of k is invertible. (b) Every module over k is free. If these conditions are satisﬁed, we say that k is a (skew) ﬁeld.

Cohomology Theories and Commutative Rings

Fields in Classical Algebra

Let k be an associative ring. The following conditions are equivalent: (b) Every module over k is free. If these conditions are satisﬁed, we say that k is a (skew) ﬁeld.

Cohomology Theories and Commutative Rings

Fields in Classical Algebra

Let k be an associative ring. The following conditions are equivalent:

(a) Every nonzero element of k is invertible. If these conditions are satisﬁed, we say that k is a (skew) ﬁeld.

Cohomology Theories and Commutative Rings

Fields in Classical Algebra

Let k be an associative ring. The following conditions are equivalent:

(a) Every nonzero element of k is invertible. (b) Every module over k is free. Cohomology Theories and Commutative Rings

Fields in Classical Algebra

Let k be an associative ring. The following conditions are equivalent:

(a) Every nonzero element of k is invertible. (b) Every module over k is free. If these conditions are satisﬁed, we say that k is a (skew) ﬁeld. Let E be an associative ring spectrum. The following conditions are equivalent:

(a) Every nonzero homogeneous element of E ∗({x}) is invertible. (b) Every graded module over E ∗({x}) is free. It follows from (b) that: (c) Every E-module (in spectra) is free. If these conditions are satisﬁed, we say that E is a cohomological ﬁeld.

Cohomology Theories and Commutative Rings

Fields in Homotopy Theory (a) Every nonzero homogeneous element of E ∗({x}) is invertible. (b) Every graded module over E ∗({x}) is free. It follows from (b) that: (c) Every E-module (in spectra) is free. If these conditions are satisﬁed, we say that E is a cohomological ﬁeld.

Cohomology Theories and Commutative Rings

Fields in Homotopy Theory

Let E be an associative ring spectrum. The following conditions are equivalent: (b) Every graded module over E ∗({x}) is free. It follows from (b) that: (c) Every E-module (in spectra) is free. If these conditions are satisﬁed, we say that E is a cohomological ﬁeld.

Cohomology Theories and Commutative Rings

Fields in Homotopy Theory

Let E be an associative ring spectrum. The following conditions are equivalent:

(a) Every nonzero homogeneous element of E ∗({x}) is invertible. It follows from (b) that: (c) Every E-module (in spectra) is free. If these conditions are satisﬁed, we say that E is a cohomological ﬁeld.

Cohomology Theories and Commutative Rings

Fields in Homotopy Theory

Let E be an associative ring spectrum. The following conditions are equivalent:

(a) Every nonzero homogeneous element of E ∗({x}) is invertible. (b) Every graded module over E ∗({x}) is free. If these conditions are satisﬁed, we say that E is a cohomological ﬁeld.

Cohomology Theories and Commutative Rings

Fields in Homotopy Theory

Let E be an associative ring spectrum. The following conditions are equivalent:

(a) Every nonzero homogeneous element of E ∗({x}) is invertible. (b) Every graded module over E ∗({x}) is free. It follows from (b) that: (c) Every E-module (in spectra) is free. Cohomology Theories and Commutative Rings

Fields in Homotopy Theory

Let E be an associative ring spectrum. The following conditions are equivalent:

Example Let p be a prime number and let K/p denote complex K-theory modulo p. (The coﬁber of the map p : K → K.)

∗ ±1 (K/p) ({x}) ' (Z/pZ)[t ] So K/p is a cohomological ﬁeld.

Cohomology Theories and Commutative Rings

Examples of Cohomological Fields Example Let p be a prime number and let K/p denote complex K-theory modulo p. (The coﬁber of the map p : K → K.)

∗ ±1 (K/p) ({x}) ' (Z/pZ)[t ] So K/p is a cohomological ﬁeld.

Cohomology Theories and Commutative Rings

Examples of Cohomological Fields

Example Let k be a (skew) field. Then Hk is a cohomological field. (The cofiber of the map p : K → K.)

∗ ±1 (K/p) ({x}) ' (Z/pZ)[t ] So K/p is a cohomological ﬁeld.

Cohomology Theories and Commutative Rings

Examples of Cohomological Fields

Example Let k be a (skew) ﬁeld. Then Hk is a cohomological ﬁeld.

Example Let p be a prime number and let K/p denote complex K-theory modulo p. ∗ ±1 (K/p) ({x}) ' (Z/pZ)[t ] So K/p is a cohomological ﬁeld.

Cohomology Theories and Commutative Rings

Examples of Cohomological Fields

Example Let k be a (skew) ﬁeld. Then Hk is a cohomological ﬁeld.

Example Let p be a prime number and let K/p denote complex K-theory modulo p. (The coﬁber of the map p : K → K.) So K/p is a cohomological ﬁeld.

Cohomology Theories and Commutative Rings

Examples of Cohomological Fields

Example Let k be a (skew) ﬁeld. Then Hk is a cohomological ﬁeld.

Example Let p be a prime number and let K/p denote complex K-theory modulo p. (The coﬁber of the map p : K → K.)

∗ ±1 (K/p) ({x}) ' (Z/pZ)[t ] Cohomology Theories and Commutative Rings

Examples of Cohomological Fields

Example Let k be a (skew) ﬁeld. Then Hk is a cohomological ﬁeld.

Example Let p be a prime number and let K/p denote complex K-theory modulo p. (The coﬁber of the map p : K → K.)

∗ ±1 (K/p) ({x}) ' (Z/pZ)[t ] So K/p is a cohomological ﬁeld. Theorem Let k and k0 be (skew) ﬁelds. Then k and k0 have the same characteristic if and only if k ⊗ k0 6= 0.

Deﬁnition Let E and E 0 be cohomological ﬁelds. Then E and E 0 have the same characteristic if and only if E ∧ E 0 6= 0.

This is an equivalence relation ∼.

Cohomology Theories and Commutative Rings

The Characteristic of a Field Deﬁnition Let E and E 0 be cohomological ﬁelds. Then E and E 0 have the same characteristic if and only if E ∧ E 0 6= 0.

This is an equivalence relation ∼.

Cohomology Theories and Commutative Rings

The Characteristic of a Field

Theorem Let k and k0 be (skew) ﬁelds. Then k and k0 have the same characteristic if and only if k ⊗ k0 6= 0. This is an equivalence relation ∼.

Cohomology Theories and Commutative Rings

The Characteristic of a Field

Theorem Let k and k0 be (skew) ﬁelds. Then k and k0 have the same characteristic if and only if k ⊗ k0 6= 0.

Deﬁnition Let E and E 0 be cohomological ﬁelds. Then E and E 0 have the same characteristic if and only if E ∧ E 0 6= 0. Cohomology Theories and Commutative Rings

The Characteristic of a Field

Theorem Let k and k0 be (skew) ﬁelds. Then k and k0 have the same characteristic if and only if k ⊗ k0 6= 0.

Deﬁnition Let E and E 0 be cohomological ﬁelds. Then E and E 0 have the same characteristic if and only if E ∧ E 0 6= 0.

This is an equivalence relation ∼. Let p be a prime number and let X be the classifying space of the group G = Z/pZ. If k is a field of characteristic 6= p, then Hn(X ; k) ' 0 for n > 0. If k is a field of characteristic p, then Hn(X ; k) ' k for all n ≥ 0. ( Rep(G) ⊗ ( /p ) if n is even (K/p)n(X ) = Z Z 0 if n is odd These examples yield modules of ranks 1, ∞, and p over the cohomology of a point. Corollary K/p is not of the same characteristic as any ordinary field.

Cohomology Theories and Commutative Rings

Telling Fields Apart If k is a field of characteristic 6= p, then Hn(X ; k) ' 0 for n > 0. If k is a field of characteristic p, then Hn(X ; k) ' k for all n ≥ 0. ( Rep(G) ⊗ ( /p ) if n is even (K/p)n(X ) = Z Z 0 if n is odd These examples yield modules of ranks 1, ∞, and p over the cohomology of a point. Corollary K/p is not of the same characteristic as any ordinary field.

Cohomology Theories and Commutative Rings

Telling Fields Apart

Let p be a prime number and let X be the classifying space of the group G = Z/pZ. If k is a ﬁeld of characteristic p, then Hn(X ; k) ' k for all n ≥ 0. ( Rep(G) ⊗ ( /p ) if n is even (K/p)n(X ) = Z Z 0 if n is odd These examples yield modules of ranks 1, ∞, and p over the cohomology of a point. Corollary K/p is not of the same characteristic as any ordinary ﬁeld.

Cohomology Theories and Commutative Rings

Telling Fields Apart

Let p be a prime number and let X be the classifying space of the group G = Z/pZ. If k is a ﬁeld of characteristic 6= p, then Hn(X ; k) ' 0 for n > 0. ( Rep(G) ⊗ ( /p ) if n is even (K/p)n(X ) = Z Z 0 if n is odd These examples yield modules of ranks 1, ∞, and p over the cohomology of a point. Corollary K/p is not of the same characteristic as any ordinary ﬁeld.

Cohomology Theories and Commutative Rings

Telling Fields Apart

Cohomology Theories and Commutative Rings

Telling Fields Apart

Cohomology Theories and Commutative Rings

Telling Fields Apart

Let p be a prime number and let X be the classifying space of the group G = Z/pZ. If k is a field of characteristic 6= p, then Hn(X ; k) ' 0 for n > 0. If k is a field of characteristic p, then Hn(X ; k) ' k for all n ≥ 0. ( Rep(G) ⊗ ( /p ) if n is even (K/p)n(X ) = Z Z 0 if n is odd These examples yield modules of ranks 1, ∞, and p over the cohomology of a point. Corollary K/p is not of the same characteristic as any ordinary field. Then k = E 0({x}) is a classical field. Does k have characteristic zero? If so, then E ∼ HQ. If not, let p be the characteristic of k and let X be the classifying space of Z/pZ. ∗ ∗ Does E (X ) have infinite rank over E ({x})? If so, then E ∼ HFp. If not, one can show that the rank of E ∗(X ) over E ∗({x}) is pn for 0 < n < ∞. We will say that E has height n.

Cohomology Theories and Commutative Rings

A Field Guide to Fields

Let E be a cohomological ﬁeld. Does k have characteristic zero? If so, then E ∼ HQ. If not, let p be the characteristic of k and let X be the classifying space of Z/pZ. ∗ ∗ Does E (X ) have inﬁnite rank over E ({x})? If so, then E ∼ HFp. If not, one can show that the rank of E ∗(X ) over E ∗({x}) is pn for 0 < n < ∞. We will say that E has height n.

Cohomology Theories and Commutative Rings

A Field Guide to Fields

Let E be a cohomological field. Then k = E 0({x}) is a classical field. If so, then E ∼ HQ. If not, let p be the characteristic of k and let X be the classifying space of Z/pZ. ∗ ∗ Does E (X ) have infinite rank over E ({x})? If so, then E ∼ HFp. If not, one can show that the rank of E ∗(X ) over E ∗({x}) is pn for 0 < n < ∞. We will say that E has height n.

Cohomology Theories and Commutative Rings

A Field Guide to Fields

Let E be a cohomological field. Then k = E 0({x}) is a classical field. Does k have characteristic zero? If not, let p be the characteristic of k and let X be the classifying space of Z/pZ. ∗ ∗ Does E (X ) have infinite rank over E ({x})? If so, then E ∼ HFp. If not, one can show that the rank of E ∗(X ) over E ∗({x}) is pn for 0 < n < ∞. We will say that E has height n.

Cohomology Theories and Commutative Rings

A Field Guide to Fields

Let E be a cohomological field. Then k = E 0({x}) is a classical field. Does k have characteristic zero? If so, then E ∼ HQ. ∗ ∗ Does E (X ) have infinite rank over E ({x})? If so, then E ∼ HFp. If not, one can show that the rank of E ∗(X ) over E ∗({x}) is pn for 0 < n < ∞. We will say that E has height n.

Cohomology Theories and Commutative Rings

A Field Guide to Fields

Cohomology Theories and Commutative Rings

A Field Guide to Fields

Cohomology Theories and Commutative Rings

A Field Guide to Fields

Cohomology Theories and Commutative Rings

A Field Guide to Fields

Let E be a cohomological field. Then k = E 0({x}) is a classical field. Does k have characteristic zero? If so, then E ∼ HQ. If not, let p be the characteristic of k and let X be the classifying space of Z/pZ. ∗ ∗ Does E (X ) have infinite rank over E ({x})? If so, then E ∼ HFp. If not, one can show that the rank of E ∗(X ) over E ∗({x}) is pn for 0 < n < ∞. We will say that E has height n. Theorem For every prime number p and every positive integer n, there exists a cohomological field E of height n such that E 0({x}) has characteristic p. All such fields are of the same characteristic.

This cohomological ﬁeld is called the nth Morava K-theory and denoted by K(n).

Example We can take K(1) = K/p.

Cohomology Theories and Commutative Rings

Morava K-Theories All such ﬁelds are of the same characteristic.

This cohomological ﬁeld is called the nth Morava K-theory and denoted by K(n).

Example We can take K(1) = K/p.

Cohomology Theories and Commutative Rings

Morava K-Theories

Theorem For every prime number p and every positive integer n, there exists a cohomological ﬁeld E of height n such that E 0({x}) has characteristic p. This cohomological ﬁeld is called the nth Morava K-theory and denoted by K(n).

Example We can take K(1) = K/p.

Cohomology Theories and Commutative Rings

Morava K-Theories

Theorem For every prime number p and every positive integer n, there exists a cohomological ﬁeld E of height n such that E 0({x}) has characteristic p. All such ﬁelds are of the same characteristic. Example We can take K(1) = K/p.

Cohomology Theories and Commutative Rings

Morava K-Theories

This cohomological ﬁeld is called the nth Morava K-theory and denoted by K(n). Cohomology Theories and Commutative Rings

Morava K-Theories

This cohomological ﬁeld is called the nth Morava K-theory and denoted by K(n).

Example We can take K(1) = K/p. One can extend the deﬁnition of K(n) to 0 ≤ n ≤ ∞ by setting

K(0) = HQ K(∞) = HFp.

The Morava K-theories K(n) are associative ring spectra. They cannot be made commutative if 0 < n < ∞. Only the characteristic of K(n) is well-deﬁned. One can choose minimal representatives (“prime ﬁelds”), but these do not have unique ring structures.

Cohomology Theories and Commutative Rings

A Few Caveats The Morava K-theories K(n) are associative ring spectra. They cannot be made commutative if 0 < n < ∞. Only the characteristic of K(n) is well-deﬁned. One can choose minimal representatives (“prime ﬁelds”), but these do not have unique ring structures.

Cohomology Theories and Commutative Rings

A Few Caveats

One can extend the deﬁnition of K(n) to 0 ≤ n ≤ ∞ by setting

K(0) = HQ K(∞) = HFp. They cannot be made commutative if 0 < n < ∞. Only the characteristic of K(n) is well-deﬁned. One can choose minimal representatives (“prime ﬁelds”), but these do not have unique ring structures.

Cohomology Theories and Commutative Rings

A Few Caveats

One can extend the deﬁnition of K(n) to 0 ≤ n ≤ ∞ by setting

K(0) = HQ K(∞) = HFp.

The Morava K-theories K(n) are associative ring spectra. Only the characteristic of K(n) is well-deﬁned. One can choose minimal representatives (“prime ﬁelds”), but these do not have unique ring structures.

Cohomology Theories and Commutative Rings

A Few Caveats

One can extend the deﬁnition of K(n) to 0 ≤ n ≤ ∞ by setting

K(0) = HQ K(∞) = HFp.

The Morava K-theories K(n) are associative ring spectra. They cannot be made commutative if 0 < n < ∞. One can choose minimal representatives (“prime ﬁelds”), but these do not have unique ring structures.

Cohomology Theories and Commutative Rings

A Few Caveats

One can extend the deﬁnition of K(n) to 0 ≤ n ≤ ∞ by setting

K(0) = HQ K(∞) = HFp.

The Morava K-theories K(n) are associative ring spectra. They cannot be made commutative if 0 < n < ∞. Only the characteristic of K(n) is well-deﬁned. Cohomology Theories and Commutative Rings

A Few Caveats

One can extend the deﬁnition of K(n) to 0 ≤ n ≤ ∞ by setting

K(0) = HQ K(∞) = HFp.

A Few Caveats

One can extend the deﬁnition of K(n) to 0 ≤ n ≤ ∞ by setting

K(0) = HQ K(∞) = HFp.

These interpolate between fields of characteristic zero (K(0) ' HQ) and fields of characteristic p (K(∞) ' HFp). In Lecture 2, we’ll discuss the behavior of representation theory over these fields of “intermediate characteristic”. In Lecture 3, we’ll study some rudimentary algebraic geometry in these settings, focusing in particular on roots of unity.

Cohomology Theories and Commutative Rings

“Intermediate Characteristics” These interpolate between fields of characteristic zero (K(0) ' HQ) and fields of characteristic p (K(∞) ' HFp). In Lecture 2, we’ll discuss the behavior of representation theory over these fields of “intermediate characteristic”. In Lecture 3, we’ll study some rudimentary algebraic geometry in these settings, focusing in particular on roots of unity.

Cohomology Theories and Commutative Rings

“Intermediate Characteristics”

For a fixed prime number p, there is an infinite family of cohomological fields K(1) K(2) K(3) ··· In Lecture 2, we’ll discuss the behavior of representation theory over these fields of “intermediate characteristic”. In Lecture 3, we’ll study some rudimentary algebraic geometry in these settings, focusing in particular on roots of unity.

Cohomology Theories and Commutative Rings

“Intermediate Characteristics”

For a fixed prime number p, there is an infinite family of cohomological fields K(1) K(2) K(3) ···

These interpolate between ﬁelds of characteristic zero (K(0) ' HQ) and ﬁelds of characteristic p (K(∞) ' HFp). In Lecture 3, we’ll study some rudimentary algebraic geometry in these settings, focusing in particular on roots of unity.

Cohomology Theories and Commutative Rings

“Intermediate Characteristics”

For a fixed prime number p, there is an infinite family of cohomological fields K(1) K(2) K(3) ···

“Intermediate Characteristics”

For a fixed prime number p, there is an infinite family of cohomological fields K(1) K(2) K(3) ···