A Dissertation entitled

The Steenrod Algebra is a Prime Ring and the Krull Dimensions of the Steenrod Algebra

by Robert P. Stephens

Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mathematics

Dr. Charles J. Odenthal, Committee Chair

Dr. Paul R. Hewitt, Committee Member

Dr. John H. Palmieri, Committee Member

Dr. Martin R. Pettet, Committee Member

Dr. Patricia R. Komuniecki, Dean College of Graduate Studies

The University of Toledo August 2011 Copyright 2011, Robert P. Stephens

This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of The Steenrod Algebra is a Prime Ring and the Krull Dimensions of the Steenrod Algebra by Robert P. Stephens

Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mathematics The University of Toledo August 2011

Kashkarev has shown that the mod 2 Steenrod algebra is a prime ring. For any odd prime p, we prove that the mod p Steenrod algebra is also a prime ring. In sequel, for any prime p, we show that the mod p Steenrod algebra (a local ring with nil maximal ideal) has infinite little Krull dimension. This contrasts sharply with the case of a commutative (or noetherian) local ring with nil maximal ideal which must have little Krull dimension equal to 0. Also, we show that the Steenrod algebra has no Krull dimension, classical Krull dimension, or Gabriel dimension.

iii Acknowledgments

I would like to thank my advisor Charles Odenthal for his guidance and encour- agement. His advice helped me focus my thoughts and ideas into sentences that were more precise and easier to understand.

I also thank Rose Kneen for her friendship and help throughout my time here at the University of Toledo.

I thank the anonymous reviewer for all his or her suggestions and comments.

Lots of appreciation goes to my Mom and Dad, Mary Ann and Joe Stephens, and my sister Katie and her family, Dylan and Kenneth. My family helped kept me afloat through tough times.

Finally, I’d like to think Jeffery Davis. His sense of humor could always cut through any bad mood and make me smile, which in turn helped keep me sane.

iv Contents

Abstract iii

Acknowledgments iv

Contents v

1 History and Conventions 2

1.1 Background ...... 2

1.1.1 Cohomology ...... 2

1.2 The Steenrod Squares and Powers ...... 4

1.2.1 Axiomatic Description of the Steenrod Squares and Powers . .5

1.2.2 Algebraic Description of the Steenrod Algebras ...... 8

1.2.3 Early Uses of the Steenrod Algebra ...... 10

1.3 Prime Ideals and Krull Dimensions ...... 11

1.4 Definitions and Conventions ...... 17

2 The Steenrod algebra is a Prime Ring 22

2.1 Preliminary Lemmas ...... 22

2.2 Main Primality Theorem ...... 26

2.3 Miscellaneous ...... 33

3 The Krull Dimensions of the Steenrod Algebra 36

v 3.1 Preliminary Results ...... 37

3.2 The Little Krull Dimension ...... 38

3.2.1 The Ideals Dj ...... 38 3.2.2 The map ψ ...... 51

3.2.3 The Hopf Algebra Approach ...... 54

3.3 Other Dimensions ...... 55

3.4 Future Work ...... 57

3.5 The case p =2 ...... 58

References 59

A Construction of the Steenrod Squares and Powers 64

B Hopf Algebras 69

vi Introduction

The main topic of this work is the Steenrod algebra, which plays an important role in the study of algebraic topology, but this treatise mostly concentrates on some of the algebraic properties of the Steenrod algebra. Namely, we show that, for an odd prime p, then Steenrod algebra is a prime ring. This work is an extension of the work by Kashkarev in [25], who showed that the mod 2 Steenrod algebra was a prime ring. In sequel, we show, for any prime p, that the mod p Steenrod algebra has infinite little Krull dimension and that the Steenrod algebra has no Krull dimension, classical Krull dimension, or Gabriel dimension. We also show that a semiprime ring with a nonzero nil ideal has no Gabriel dimension.

The current work is broken into 3 chapters. The first chapter presents some of the history and general knowledge of the Steenrod algebra, followed by some of the notational conventions we will use. Chapter 2 concentrates on showing the primality of the Steenrod algebra for odd primes, and Chapter 3 discusses the various Krull dimensions of the Steenrod algebra.

1 Chapter 1

History and Conventions

1.1 Background

Before we can introduce the Steenrod algebra, we will very briefly explain what is cohomology. This will allow us to better describe the Steenrod squares and Steenrod powers and how they are used.1 Some axioms and properties of these squares and powers will then be presented, followed by the algebraic description of the Steenrod algebra. We will also discuss some of the early uses of the Steenrod algebra.

We will then give an introduction to prime ideals and prime rings, followed by the various Krull dimensions and Gabriel dimension. Then we will set the notation that we will use for the rest of paper.

1.1.1 Cohomology

Our exposition won’t be the most general, but only what we need. An interested reader who is unfamiliar with cohomology can refer to [21], [36], and [41], for a topological viewpoint of cohomology, or [6], [7], [8], and [13], for group cohomology.

Details about constructing the chains below can be found in those references.

1For another general overview of the Steenrod algebra, see [49].

2 A chain complex over a commutative ring R is a sequence of R-modules and

R-maps C = (Cn, ∂n):

∂n ∂n+1 · · · ←− Cn−1 ←− Cn ←−−− Cn+1 ←−· · ·

with ∂n∂n+1 = 0 for all n. Since img ∂n+1 ⊆ ker ∂n, we can form the quotient module

n th H (C) = ker ∂n/img ∂n+1, which we call the n homology of C.

∗ Taking duals and writing C = HomR (Cn,R), we have the cochain complex over R ∗ ∂∗ ∗ ∂n ∗ n+1 ∗ · · · −→ Cn−1 −→ Cn −−−→ Cn+1 −→· · ·

∗ ∗ ∗ ∗ with ∂n+1∂n = 0 for all n. This time, img ∂n ⊆ ker ∂n+1, and we form the quotient

n ∗ ∗ th module H (C) = ker ∂n+1/img ∂n, the n cohomology of C. In topology, starting with a space X and a commutative ring R, there are several

ways to construct associated chain complexes of R-modules. Conveniently, these all

produce the same homology and cohomology, which are denoted

M ∗ M n H∗ (X; R) = Hn (X; R) and H (X; R) = H (X; R) , n n

∗ respectively, both H∗(X; R) and H (X; R) are graded R-modules. Given a map of

spaces f : X → Y , there are induced homomorphisms f∗ : Hn(X; R) → Hn(Y ; R) and f ∗ : Hn(Y ; R) → Hn(X; R). As an example, the diagonal map d : X → X × X, defined by d(x) = (x, x), induces a homomorphism of graded R-modules

∗ H∗ (X × X; R) −→d H∗ (X; R) .

We also have the cross product (see [21, Page 210])

Hk (X; R) × H` (Y ; R) −→c Hk+` (X × Y ; R) .

3 When Y = X, the composition d∗ ◦ c helps define a multiplicative structure on

H∗ (X; R) called the cup product (see [21] and [41] for details): for α ∈ Hk (X; R)

and β ∈ H` (X; R), the cup product is written α ^ β ∈ Hk+` (X; R). We note

here that the cup product is anticommutative; that is, α ^ β = (−1)k`β ^ α for

α ∈ Hk (X; R) and β ∈ H` (X; R). With this multiplicative structure, H∗ (X; R) is a

graded R-algebra

1.2 The Steenrod Squares and Powers

Before introducing the Steenrod squares and powers, we give definitions to the

properties that the Steenrod squares and powers have.

m A cohomology operation (see [21, 44]) is a transformation Θ = ΘX : H (X; R) → Hn (X; T ) defined for all spaces X, with fixed choice of m, n, R, and T , and satisfying

the naturality property f ∗Θ = Θf ∗ for all maps f : X → Y ; that is, Θ satisfies the

commuting diagram

ΘY Hm (Y ; R) / Hn (Y ; T )

f ∗ f ∗

 ΘX  Hm (X; R) / Hn (X; T ) for all maps f : X → Y . These operations “are used to decide questions about the existence of continuous mappings which cannot be settled by examining cohomology groups alone” [44]. We define the suspension of a space X, denoted SX, as the quotient of X × [0, 1] obtained by collapsing X × {0} to a point and X × {1} to

another point. A cohomology operation Θ is said to be stable if it commutes with the suspension operator; notationally, SΘ = ΘS. Though we will not make use of this property, it is useful in the study of algebraic topology. For instance, Hatcher

[21, Theorem 4.L2] uses stability to show that the stable homotopy groups of certain spheres are nontrivial. See [21, 44] for more information about suspension.

4 Two continuous maps f, g : X → Y are said to be homotopic, written f ∼ g, if

f can be continuously deformed into g; that is, f ∼ g if there exists a continuous family of maps ft : X → Y (0 ≤ t ≤ 1), such that f0 = f and f1 = g. Two spaces X and Y are homotopy-equivalent (or of the same homotopy type), written X ∼ Y ,

if there exists maps f : X → Y and g : Y → X, such that gf ∼ 1X and fg ∼ 1Y ,

where 1X and 1Y are the identity maps of X and Y respectively. The relation ∼ on spaces is an equivalence relation, and one can try to enumerate the homotopy classes

of continuous maps f : X → Y . This is called the homotopy classification problem, and the enumeration should be in terms of algebraic invariants of X and Y (see [12]).

We introduce the Bockstein homomorphism (see [21, 44]), which is the coboundary

n n+1 operation δ : H (X; Zp) → H (X; Zp) associated with the coefficient sequence

p 0 → Zp → Zp2 → Zp → 0.

The Bockstein homomorphism has the properties

δδ = 0 and δ (α ^ β) = δα ^ β + (−1)|α| α ^ δβ,

as well as naturality and stability.

1.2.1 Axiomatic Description of the Steenrod Squares and

Powers

In 1947, Steenrod [42] introduced a set of stable cohomology operators, later

named the Steenrod squares, as a generalization of the cup product, which he used

to solve the homotopy classification problem for maps of an (n + 1)-complex K in Sn.

5 These cohomology operators are denoted

i n n+i Sq : H (X; Z2) → H (X; Z2) , for i ≥ 0, (1.1)

∗ where the cohomology groups H (X; Z2) of some complex X have Z2 coefficients. These are the unique operations that satisfy the following axioms (see [44]):

1. Naturality: Sqi f ∗ = f ∗ Sqi for f : X → Y , a map of complexes.

2. Linearity: Sqi (α + β) = Sqi (α) + Sqi (β).

3. Sq0 = 1.

4. If i = |α|, then Sqi (α) = α2.

5. If i > |α|, then Sqi (α) = 0.

i P j k 6. The Cartan Formula [9]: Sq (α ^ β) = j+k=i Sq (α) ^ Sq (β).

These 6 axioms imply the following properties:

1. The operations Sqi are stable operations.

2. Sq1 = δ.

3. The Adem Relations [5]: If a < 2b, then

[a/2] X b − 1 − j SqaSqb = Sqa+b−jSqj, a − 2j j=1

with the binomial coefficients taken modulo 2.

Later on in 1953, Steenrod [43] introduced what were later called the Steenrod powers, which are an extension of the Steenrod squares to any odd prime p. These are denoted

i n n+2i(p−1) P : H (X; Zp) → H (X; Zp) , for i ≥ 0, (1.2)

6 ∗ where the cohomology groups H (X; Zp) of some complex X have Zp coefficients with p an odd prime. These are the unique operations that satisfy the following axioms

(see [44]):

1. Naturality: Pif ∗ = f ∗Pi for f : X → Y , a map of complexes.

2. Linearity: Pi (α + β) = Pi (α) + Pi (β).

3. P0 = 1.

4. If 2i = |α|, then Pi (α) = αp.

5. If 2i > |α|, then Pi (α) = 0.

i P j k 6. The Cartan Formula: P (α ^ β) = j+k=i P (α) ^ P (β).

These 6 axioms imply the following properties:

1. The operations Pi are stable operations.

2. The Adem Relations [5]: If a < pb, then

[a/p]   X a+t (p − 1) (b − t) − 1 PaPb = (−1) Pa+b−tPt, a − pt t=0

and if a ≤ pb, then

[a/p]   X a+t (p − 1) (b − t) PaδPb = (−1) δPa+b−tPt a − pt t=0 [(a−1)/p]   X a+t−1 (p − 1) (b − t) − 1 + (−1) Pa+b−tδPt, a − pt − 1 t=0

with the binomial coefficients taken modulo p.

For the interested reader, the construction of the Steenrod squares and powers can be found in Appendix A.

7 1.2.2 Algebraic Description of the Steenrod Algebras

An iterated p-power is a composition of two or more of the operations Sqi, when

p = 2, or δ and Pi, when p > 2. As examples of iterated powers, we have PiδPk and

Sqi Sqj, for i, j, k ≥ 0. It is clear that any iterated power is also a cohomology operator

∗ i on H (X; Zp). So, one could form an algebra, over Zp, involving the operations Sq from (1.1), when p = 2, and δ and Pi from (1.2), when p > 2. This is the so-called

Steenrod algebra.

Following Adem [5] in 1957, the Steenrod algebra Ap can be defined as follows:

0 1 Let R denote the free associative algebra over Zp generated by Sq , Sq ,..., if p = 2 and δ, P0, P1,..., if p > 2. Let I be the ideal of R consisting of elements f with

∗ the property that fα = 0 for all complexes K and all α ∈ H (K; Zp). Elements

of I are called relations. The Steenrod Algebra is then the quotient Ap = R/I. Adem showed in [5] that the relations, now called the Adem relations, given on Pages

6 and 7 generate the ideal I. Thus, the Steenrod algebra can be defined purely

algebraically by modding out these Adem relations from the algebra R. Adem also

n 2i o showed that the set Sq : i = 0, 1, 2,... generates the algebra A2, and the set

n pi o P : i = 0, 1, 2,... ∪ {δ} generates the algebra Ap for p > 2. We note here that the Steenrod algebra consists of all the stable cohomology operations.

∗ For any space X, the cohomology ring H (X; Zp) is an Ap-module. Given prop- erty 5 above for both p = 2 and p > 2, we have a certain property called instability:

∗ i • if x ∈ H (X; Zp) and i > |x|, then Sq x = 0, for p = 2;

∗ e i • if x ∈ H (X; Zp) and e + 2i > |x|, e = 0, 1, then δ P x = 0, for p > 2.

2 We say an Ap-module is unstable if it satisfies the preceding property. See [39] for details about unstable modules.

2The use of the words “unstable” and “instability” is not related to the use of the word “stable.”

8 The Steenrod algebra is a graded ring of finite type (that is the grading consists of finite dimensional vector spaces) with grading as follows: |Sqr| = r, for p = 2, and

|Pr| = 2r (p − 1) and |δ| = 1, for p > 2. Here, for example, the notation |δ| denotes the degree of the element δ.

In 1958, Milnor [33] showed that the Steenrod algebra is a cocommutative Hopf

3 algebra. The coalgebra structure of the Steenrod algebra Ap is given by

  α if |α| = 0 n P i j ε (α) = ∆ (Sq ) = i+j=n Sq ⊗ Sq  0 if |α| > 0. for p = 2 and

 ∆ (Pn) = P Pi ⊗ Pj  α if |α| = 0 i+j=n ε (α) = ∆ (δ) = δ ⊗ 1 + 1 ⊗ δ  0 if |α| > 0 for p > 2. The formulas for ∆ come from the Cartan formulas presented on Pages

6 and 7. We see that Ap is cocommutative. Since Ap is graded, it has an antipode, which Milnor discusses in [33, Section 7].

∗ Since Ap is of finite type, the dual of Ap, written Ap, is a commutative Hopf

∗ algebra. Milnor (see [33, Theorem 2] for details) showed that Ap is the tensor product

k of an exterior algebra generated by τ0, τ1,..., where |τk| = 2p − 1, and a polynomial

k algebra generated by ξ1, ξ2,..., where |ξk| = 2p − 2, when p is odd. When p = 2,

∗ k A2 is just a polynomial algebra generated by ξ1, ξ2,..., where |ξk| = 2 − 1. The

∗ coalgebra structure of Ap (see [33, Theorem 3]) for p > 2 is given by

 ∗ Pk pi m (ξk) = ξ ⊗ ξi  α if |α| = 0 i=0 k−i u∗ (α) = ∗ Pk pi m (τk) = τk ⊗ 1 + i=0 ξk−i ⊗ τi  0 if |α| > 0.

3See Appendix B for an introduction to Hopf algebras.

9 ∗ ∗ Pk 2i The comultiplication for A2 is simply m (ζk) = i=0 ζk−i ⊗ ζi. We see, as expected,

∗ that Ap is not cocommutative. Milnor used the simpler commutative multiplicative structure of the dual to form a new basis (defined in §1.4 below) for the Steenrod algebra, now called the Milnor basis, that allows one to calculate the composition, or multiplication, of the Steenrod operations in terms of his basis. Furthermore, Milnor showed that the Steenrod algebra has a filtration by finite dimensional subalgebras (see §1.4 below), which showed that every element of positive degree in the Steenrod algebra is nilpotent. We also note that the Steenrod algebra is a local ring where the maximal ideal consists of all the elements of positive degree.

1.2.3 Early Uses of the Steenrod Algebra

The Steenrod algebra has become an important tool in the subject of algebraic topology. Some of the early applications of the Steenrod algebra applied to unsolved problems were the Hopf invariant 1 problem and vector fields on spheres. We will briefly describe these below, and the reader may refer to [21] for more details.

For each n, let Sn−1 denote the unit sphere in Euclidian n-space Rn. When given a

m n 4 m+1 n map f : S → S , one can form a complex Cf by attaching a cell e to S by using

∗ f. When m = 2n − 1, cup products in H (Cf ) have a chance of being nontrivial.

n 2n Given generators α ∈ H (Cf ) and β ∈ H (Cf ), the multiplicative structure of

∗ 2 H (Cf ) is determined by the equation α = H(f)β, where H(f) is an integer called the Hopf invariant of f. The Hopf invariant 1 problem [23] asks for which values of n does H(f) = 1. Using the Steenrod algebra, Adams [2] showed that the map f : S2n−1 → Sn has Hopf invariant 1 only when n = 2, 4, 8.

On the subject of vector fields on spheres, there was a question about the maxi- mum number of orthogonal tangent vector fields on a sphere. Write n = (2a + 1) 2b

4The reader may refer to [21, Page 5] for more information about cells.

10 with b = c + 4d, where a, b, c, d ∈ Z and 0 ≤ c ≤ 3. Define ρ(n) = 2c + 8b. Following Eckmann [11], there exists ρ(n)−1 linearly independent vector fields on Sn−1. Adams

[3] showed that there does not exists ρ(n) linearly independent vector fields on Sn−1.

As Adams noted, Steenrod and Whitehead [45] showed this to be true for b ≤ 3; their proof uses the Steenrod squares. Instead of using the Steenrod squares, Adams refor- mulates the problem into K-theory, and chooses an appropriate cohomology operator to suit his needs. An interested reader may refer to Hatcher [21] for a proof for all n 6≡ −1 mod 16; his proof rephrases the stated problem in terms of Stiefel manifolds and uses the Steenrod squares.

1.3 Prime Ideals and Krull Dimensions

Since this paper discusses the topics of prime rings and Krull dimensions, we will briefly discuss the history of these topics. The first attempt to study prime ideals in a general ring – not necessarily a commutative ring – was by McCoy [32] in 1949. In a commutative ring R, an ideal P is called a prime ideal if ab ∈ P implies a ∈ P or b ∈ P , were a and b are elements of R. Fitting [15] used this definition for an arbitrary ring, calling the ideal a completely prime ideal. McCoy defines a prime ideal in weaker terms: In a ring R, an ideal P is called a prime ideal if AB ⊆ P implies A ⊆ P or

B ⊆ P , where A and B are ideals of R. McCoy derives several equivalent statements in [32] for his definition of prime ideal, including the following definition: In a ring R, an ideal P is called a prime ideal if aRb ⊆ P implies a ∈ P or b ∈ P , where a and b are elements of R. McCoy shows that there is a characterization for prime ideals: an ideal P is prime if and only if the complement of P is an m-system. An m-system M is a subset M of R such that for all a, b ∈ M, there exists x ∈ R such that axb ∈ M.

A prime ring R is a ring where the zero ideal is prime; an equivalent definition

is, for a, b ∈ R, it is true that aRb = 0 implies a = 0 or b = 0. This allows us to

11 present another description of prime ideals, which is what we use in this paper: an ideal P of a ring R is a prime ideal if R/P is a prime ring. The idea of prime rings in noncommutative ring theory is a generalization of the idea of integral domains in commutative ring theory. We note here a generalization of prime ideals (see [30] for more information): An ideal C of a ring R is called a semiprime ideal if for any ideal

A of R, it is true that A2 ⊆ C implies A ⊆ C. Equivalently, and ideal C is semiprime if aRa ⊆ C implies a ∈ C for all a ∈ R. Likewise, a semiprime ring is a ring where the zero ideal is semiprime. A prime ring is also semiprime.

There is a relationship between algebraic varieties and chains of prime ideals, studied by Emmy Noether. For a field F , recall that an affine algebraic set is a subset V of F n which is the set of common zeros of some collection I of polynomials in F [x1, x2, . . . , xn]. The set V is called an algebraic variety if V is irreducible; that is, V cannot be written as V1 ∪ V2 for proper algebraic sets in V . Letting

F [V ] = F [x1, x2, . . . , xn] /I denote the coordinate ring of the variety, the dimension of V is defined to be the transcendence degree of the field of fractions of F [V ] over

F (see [10]). If P is a prime ideal in F [x1, x2, . . . , xn] consisting of those polynomials that vanish on V , then this dimension of V equals the maximum length of a chain of prime ideals ascending from P (see [17]).

In 1928, Krull’s work (see [28]) on the principle ideal theorem used chains of prime ideals and the height of ascending chains of prime ideals. We define the height of a prime ideal P in a commutative ring R, denoted ht(P ), to be the largest integer n such that there exists a chain P = P0 > P1 > ··· > Pn. If the chain is unbounded, then we write ht(P ) = ∞. This definition led to the definition of the dimension of a commutative ring, later called the Krull dimension: the dimension of a commutative ring R is

sup {ht(P ): P ∈ Spec R} , where Spec R is the set of all proper prime ideals of R.

12 Not all noncommutative rings have enough prime ideals to make the definition above interesting or useful. We define here various types of Krull dimensions and the

Gabriel dimension. The rings mentioned below in these definitions are not necessarily commutative. These definitions can all be found in [18].

Little Krull Dimension

One attempt to adapt the above definition to arbitrary rings and ordinal numbers is the following definition from [18, Page 47].

A limit ordinal is an ordinal number that is neither zero nor a successor of an ordinal. Let λ (W ) denote the length of a well-ordered set W defined by

  ord W if the order type of W is a limit ordinal, λ (W ) =  ord W − 1 otherwise.

The little classical Krull dimension or, for short, little Krull dimension, which we denote by k dim R, for an arbitrary ring R is defined by

k dim R = sup {λ (W ): W ⊂ Spec R, well-ordered under reverse inclusion} .

For example, given any ordinal α, there is a principle right ideal domain with little Krull dimension α, and the dimension is given by the length of a single chain, see [18]. Also, for a countable field F , the polynomial ring F [x1, x2,...] has little

Krull dimension of ω1, the first uncountable ordinal (see Proposition 3.4.1 below). As another example, consider the nth Weyl algebra over a ring R, which is defined as

An(R) = R [x1, θ1, x2, θ2, . . . , xn, θn] ,

where the xi commute with themselves, the θi commute with themselves, and θxj −

13 5 xjθi = δij, the Kronecker delta. If F has characteristic zero, then An(F ) is a simple algebra, from which it follows that k dim An(F ) = 0. In fact, all simple algebras have little Krull dimension equal to zero. In this case, the little Krull dimension is does not supply useful information.

It is worth noting that in some cases, the little Krull dimension is the most in- teresting case of all the Krull dimensions. The Steenrod algebra is an example; the little Krull dimension of the Steenrod algebra is infinite while the other two Krull dimensions, defined below, do not exist.

Classical Krull Dimension

There is a more general definition for a Krull dimension that also uses ordinal numbers, which first was introduced by Krause in [26, Definition 11]. The following definition is from [18].

Let P0 denote the set of all maximal ideals of an arbitrary ring R. Then, if α > 0 is an ordinal, denote by Pα the set of prime ideals P of R such that each prime

Q properly containing P belongs to Pβ for some β < α. Then the classical Krull dimension of R, denoted by cl K dim R, is defined to be the first ordinal α such that

Pα = Spec R.

As simple examples, we have cl K dim Z = 1, where Z denotes the ring of integers, and cl K dim F [x1, x2, . . . , xn] = n, where F is a field. As with the little Krull dimension, all simple algebras have a classical Krull dimension equal to zero. For example, when F is a field with characteristic zero, we have cl K dim An(F ) = 0 since the nth Weyl algebra is simple. In this case, the classical Krull dimension does not yield much information about such algebras.

5 We use the notation An(R) for the Weyl algebra and hope it is not confused with the notation Ap for the Steenrod algebra or the notation for Ap(n), defined on Page 20.

14 Krull Dimension

The Krull dimension for a module was defined in [16] for finite ordinals. This

definition was extended to arbitrary ordinals in [18], and we present this definition

below.

Let R be a ring and M a right R-module. The Krull dimension of M, denoted

K dim M, is defined by transfinite recursion as follows: if M = 0, then K dim M =

−1. If α is an ordinal and K dim M 6< α, then K dim M = α provided there is no infinite descending chain M = M0 ⊃ M1 ⊃ · · · of submodules Mi such that, for i = 1, 2,..., K dim (Mi−1/Mi) 6< α. It is possible that there is no ordinal α such that K dim M = α, in which case we say M has no Krull dimension.

The Krull dimension of a ring R, denoted by K dim R, is defined to be the Krull dimension of the right R-module R.

The Krull dimension is a measure of how close a module or ring is to being ar- tinian. Notherian modules always have Krull dimension, but there are non-noetherian rings and modules that have Krull dimension, which have many noetherian-like con- sequences. The modules of Krull dimension 0 are precisely nonzero artinian modules, and K dim Z = 1. We note here that when F is a field with characteristic zero, then K dim An(F ) = n (see, for example, [17]), which differs from the other Krull dimensions mentioned above.

From [18, Proposition 7.9], if R is a ring with Krull dimension then cl K dim R

exists and

k dim R ≤ cl K dim R ≤ K dim R.

Gabriel Dimension

When studying the Krull dimension of modules, two special types of modules arise

(see [18]): critical modules and monoform modules. A nonzero module M is called

α-critical if K dim M = α and K dim M 0 < α for each proper homomorphic image

15 M 0 of M.A critical module is one which is α-critical for some ordinal α. Any nonzero module with Krull dimension has a critical module [18, Theorem 2.1].

If a nonzero module M has the property that, for each submodule N every homo- morphism ϕ : N → M is either zero or a monomorphism, then we call M a monoform module. Every critical module is monoform [18, Corollary 2.5]. Using this definition, one can describe, as in [18], monoform modules in terms of Serre subcategories. With this description, monoform modules show up in different places and in different forms: for instance, as simple objects in a quotient category or as supporting modules for kernel functions. Eventually, this leads to the definition of the Gabriel dimension given below (from [18]).

The Gabriel dimension is defined via a certain ascending chain, indexed by or- dinals, of localizing subcategories Mα of M = ModR, the category of R-modules.

Briefly, M0 = {0}, Mα+1 for α ≥ 0 is the smallest localizing subcategory containing all R-modules which have finite length considered as objects of the quotient category

M/Mα and, at limit ordinals, Mα is the smallest localizing subcategory containing S β<α Mβ.

If it happens that Mα = ModR for some ordinal α then the smallest such α is the

Gabriel dimension of R, which we denote G dim R. Similarly, if M ∈ Mα for some α, the least such α is the Gabriel dimension of M, denoted G dim M.

From [18, Page 16], every module with Krull dimension (K dim R) automatically has Gabriel dimension. Following [18], if a module contains no infinite direct sum of nonzero submodules, then there is a fixed bound on the number of summands. The module is said to have finite uniform dimension. As stated in [18] (proved in [19]), a module has Krull dimension if and only if it has Gabriel dimension and all its factors have finite uniform dimension. Furthermore, the dimensions differ by at most one.

We mention here that it is possible for a ring to have Gabriel dimension, but no

Krull dimension. We follow Hansen in [20]. Let K be the field of rational functions

16 over Q and let ’ denote the usual derivation on K. Also, let S = K [x;0 ] be the ring of differential polynomials where the coefficients are written on the right. For example, kx = xk + k0 for k ∈ K. It is known that S is a simple right and left ideal domain.

Let L be a proper, nonzero right ideal of S. The ideal L is a simple ring. Each nonzero ideal of Z + L = I contain L. Also, I has a Gabriel dimension of 3 and no Krull dimension.

We note here that Theorem 3.3.3 below shows that a semiprime ring with a nonzero nil ideal has no Gabriel dimension.

1.4 Definitions and Conventions

In this section we collect some standard facts from the literature and in the process

fix our notation for the rest of the paper. Nothing in this section is new.

∗ As mentioned in 1.1, the dual of the Steenrod algebra Ap is a commutative algebra.

Let R = (r1, r2,...) be a sequence of nonnegative integers which are almost all are zero; we will call such sequences Milnor sequences. Also, let E = (ε0, ε1,...) be a sequence of ones and zeros which are almost all zero; we call such sequences binary

ε0 ε1 r1 r2 sequences. Define τ(E) = τ0 τ1 ··· and ξ(R) = ξ1 ξ2 ··· . The elements {τ(E)ξ(R)}

∗ form an additive basis for Ap and are dual to some basis of Ap. Milnor [33, Theorem

4a] showed that this basis for Ap consists of elements of the form

Sq{R} for p = 2 and

ε0 ε1 Q0 Q1 ···P{R} for p > 2,

where R is a Milnor sequence and almost all of the εi ∈ {0, 1} are zero. (We use the notation P{R} in place of to Milnor’s notation PR in the interest of greater

ε0 ε1 legibility.) If we define Q(E) = Q0 Q1 ··· for a binary sequence E = (ε0, ε1,...),

17 then basis elements can be written in the form Q(E)P{R}, and we have

  1 if E = E0 and R = R0 hτ(E)ξ(R),Q(E0)P{R0}i =  0 otherwise.

For any Milnor sequence R, we can think of R as a finite sequence and write R =

(r1, . . . , rk) if ri = 0 for all i > k. Also, for R = (r1, . . . , rk), we’ll write P{r1, . . . , rk}

r1 r2 for P{R}. Note that P{r1, r2,...} 6= P P ··· in general, but we do have P{r} =

r P . The notation Pk(b) will be used to denote the element P{0,..., 0, b}, where b

th r is in the k position. For example, P1(r) = P . The same conventions are used for

Sq{R}. The elements Sq{R} = Sq{r1, . . . , rk} have degree

2
k
r1 (2 − 1) + r2 2 − 1 + ··· + rk 2 − 1 ,

k the elements Qk have degree 2p − 1, and the elements P{R} = P{r1, . . . , rk} have degree

2
k
r1 (2p − 2) + r2 2p − 2 + ··· + rk 2p − 2 .

From Milnor [33, Corollary 2], the elements Qk ∈ Ap can be defined inductively by the rule

h pk i Q0 = δ, Qk+1 = P ,Qk (1.3)

6 where [a, b] denotes the commutator ab − ba. Here, the elements Qk generate an exterior algebra (also known as a Grassmann algebra), which we denote by Q; that is, they satisfy

QjQk + QkQj = 0.

6Milnor [33] defines [a, b] = ab − (−1)|a||b|ba, where |a| denotes the dimension of a, but we define the commutator this way since |P{R}| ≡ 0 mod 2 for all Milnor sequences R.

18 d For brevity, we let M k = M − (0,..., 0, d, 0,...), where

M = (m1, . . . , mn, 0,...)

th and d appears in the k position. When we write (r1, r2,...) − (s1, s2,...), we mean

(r1 − s1, r2 − s2,...). It is understood that P{R} = 0 if rk < 0 for any k ≥ 1. In

 d d particular, P M k = 0 if mk < d. We let M 0 denote M for any value of d, and we

d,e d
e e d write M k,` in place of M k ` = (M `) k. According to Milnor [33, Theorem 4a], we have the permuting rule

n pk o n pk o P{R} Qk = QkP{R} + Qk+1P R1 + Qk+2P R2 + ···

X n pk o = Qk+iP Ri . (1.4) i≥0

We multiply the elements P{R} and P{S} as follows: Consider all infinite matrices

  ∗ x01 x02 ···     x10 x11 ····   X =   x ·····  20    ······

of non-negative integers, almost all zero, with leading entry x00 omitted. Alterna- tively, because of the constrictions we place on the infinite matrices, we can consider n × m matrices   ∗ x01 x02 . . . x0m     x10 x11 x12 . . . x1m X =   ,  . . . . .   ......      xn0 xn1 xn2 . . . xnm where n is the coordinate of the last nonzero entry in R and m is the coordinate of the

19 last nonzero entry in S, and there will be only finitely many such matrices. For each

P j th such X, define R(X) = (r1, r2,...), where ri = j p xij is the weighted i row sum; P th S(X) = (s1, s2,...), where sj = i xij is the j column sum; and T (X) = (t1, t2,...), P th where tn = i+j=n xij is the n diagonal sum. Define the coefficient

Q tn! b(X) = Q . (1.5) xi,j!

The multiplication in Ap is then as follows [33, Theorem 4b]:

X P{R}P{S} = b(X)P{T (X)} X

where the sum extends over all matrices X satisfying the conditions R(X) = R and

S(X) = S and the coefficients b(X) are in Zp. The elements Sq{R} multiply according to the same formula.

Given two sequences R and S, we write both sequences with the same length,

padding with zeros where necessary. We write R

largest index where R and S differ. We write R ≺ S if ri ≤ si for all i, but R 6= S.

Note that ≺ is a partial-order, whereas

We use the notation R ≡ 0 mod n if r` ≡ 0 mod n for all ` ≥ 1, where R =

(r1, r2,...), and we write R 6≡ 0 mod n if r` 6≡ 0 mod n for some `. For a matrix X,

we use a similar notation: we write X ≡ 0 mod n if xi,j ≡ 0 mod n for every entry

xi,j in X.

7 1 pn Letting Ap(n) denote the subalgebra generated by Q0, P ,..., P , Milnor [33,

Proposition 2] showed that the algebra Ap(n) is finite dimensional, having as basis the collection of all elements

ε0 εn+1 Q0 ··· Qn+1 P{r1, . . . , rn+1} (1.6)

7 1 pn−1 Note that Milnor defines Ap(n) in [33] as the subalgebra generated by Q0, P ,..., P .

20 which satisfy

n+1 n r1 < p , r2 < p , . . . , rn+1 < p.

When p = 2, there is a basis similar to (1.6) for A2(n), but no Qi’s are present. It

follows that Ap is the union of the finite dimensional subalgebras Ap(n). We define the multinomial coefficient as

 t  t! = x1, x2, . . . , xn x1!x2! ··· xn!

where t = x1 + x2 + ··· + xn. It is easy to show that we have the following identities

 t   t t − x  t − P`−1 x  = σ(1) ··· i=1 σ(i) , (1.7) x1, x2, . . . , xn xσ(1) xσ(2) xσ(`)

where σ is in the symmetric group Σn on n elements and

n n! = k (n − k)!k! is the regular binomial coefficient. We can represent b(X), given in Equation (1.5), as a product of multinomials as

  Y tn b(X) = . x , x , . . . , x n≥1 0,n 1,n−1 n,0

21 Chapter 2

The Steenrod algebra is a Prime

Ring

In this chapter, we extend the result of Kashkarev that the mod 2 Steenrod algebra

is a prime ring (see [25]) to all odd primes. We use his proof, replacing 2 with p and

Sq with P, and make use of Milnor’s permuting rule given in Equation (1.4) to handle

the Qk’s that are present in the Milnor basis when p is an odd prime.

2.1 Preliminary Lemmas

Before tackling our main theorem, Theorem 2.2.1, we present a few results that we will use later. In order to make our arguments in the proof of Theorem 2.2.1 more transparent we will need another basis for the Steenrod algebra in addition to

Milnor’s basis.

∗ In §1.4, we discussed an obvious basis, call it C, for Ap (see Page 17), and Milnor [33, Theorem 4a] showed that, remarkably, his basis consists of precisely those ele- ments of the Steenrod algebra that can be written as products of the form Q(E)P{R}.

Lemma 2.1.2, below, presents a basis of the Steenrod algebra that consists of all the products of the form P{R} Q(E). While clearly inspired by and similar in form to

22 Milnor’s basis, this basis isn’t dual to C or to any other obvious basis of the dual algebra. Consequently, the proof that it is a basis will require a little work.

We start with Lemma 2.1.1 which will be used in the proof of Lemma 2.1.2.

Lemma 2.1.1. Let F range over all binary sequences and S = (s1, . . . , sn) be a fixed Milnor sequence. Define

VS = span {Q(F )P{R} : R ≺ S} (2.1) and

WS = span {P{R} Q(F ): R ≺ S} . (2.2)

Then VS = WS, and this is a bimodule over the exterior algebra Q.

Proof. We first show that VS is a bimodule over Q. For Qk ∈ Q and Q(F )P{R} ∈ VS, it is clear that QkQ(F )P{R} ∈ VS. Using Property (1.4), we calculate

X n pk o Q(F )P{R} Qk = Q(F )Qk+jP Rj . j≥0

pk Since Rj  R ≺ S for all values of j, Q(F )P{R} Qk ∈ VS. So, VS is indeed a Q-bimodule.

Next, we show that WS is a bimodule over Q. For Qk ∈ Q and P{R} Q(F ) ∈ WS, it is clear that P{R} Q(F )Qk ∈ WS. To show QkP{R} Q(F ) ∈ WS, we use induction on R using the ordering ≺. The statement is clear for the zero sequence. Suppose the statement is true for all X ≺ R; that is, QkP{X} Q(F ) ∈ WS for all X ≺ R. Using Equation (1.4), we calculate

X n pk o QkP{R} Q(F ) = P{R} QkQ(F ) − Qk+jP Rj Q(F ). j≥1

23 n pk o By the induction hypothesis, we have Qk+jP Rj Q(F ) ∈ WS for all j ≥ 1, and since

P{R} QkQ(F ) ∈ WS, we have QkP{R} Q(F ) ∈ WS. So, WS is also a Q-bimodule.

To prove VS = WS, we first observe that P{R} ∈ VS ∩ WS, for all R ≺ S, since F can be the zero sequence. Since WS is a bimodule over Q, we have Q(F )P{R} ∈ WS so that VS ⊆ WS. Similarly, WS ⊆ VS. It follows that VS = WS.

Lemma 2.1.2. The set B = {P{S} Q(E)} is an additive basis for Ap. Here, S runs over all Milnor sequences, and E runs over all binary sequences.

Proof. Let S = (s1, . . . , sn) be a fixed Milnor sequence and define VS = WS as in

(2.1) and (2.2). Now let Gn(S) be the space of homogeneous elements of degree n in

VS. Observe that

Gn(S) = span {Q(E)P{R} ∈ VS : |Q(E)P{R}| = n} (2.3)

= span {P{R} Q(E) ∈ WS : |Q(E)P{R}| = n} . (2.4)

The spanning set in (2.3) is a subset of Milnor’s basis from [33] and so is linearly independent.

Of course Gn(S) is finite dimensional since Ap is of finite type; so, let N be the dimension of Gn(S). The spanning set in (2.4) must have at least N linearly independent elements. On the other hand there is a (well-defined) surjective map

Q(E)P{R} 7→ P{R} Q(E), so the spanning set in (2.4) can have no more than N distinct elements. It follows that this spanning set in (2.4) has exactly N distinct elements, and they are linearly independent. Thus, the spanning set in (2.4) is a basis for Gn(S).

Since VS is graded by the subspaces Gn(S), the set {P{R} Q(E): R ≺ S} is a basis for VS, where E runs aver all binary sequences. Furthermore, since the choice of S was arbitrary, we have that the set {P{R} Q(E)}, where E runs over all binary sequences and R runs over all Milnor sequences, is a basis for Ap.

24 Now that we have this new basis for Ap, we present a new basis for each of the

subalgebras Ap(n) that Milnor showed were finite dimensional in [33].

Corollary 2.1.3. Let S = (pn+1 − 1, pn − 1, . . . , p2 − 1, p − 1), Λ = {R : R  S}, and Γ denote the set of all binary sequences E with εi = 0 for all i > n + 1. The set

B = {P{R} Q(E): R ∈ Λ and E ∈ Γ} is an additive basis for Ap(n).

Proof. First note that C = {Q(E)P{R} : R ∈ Λ and E ∈ Γ} is exactly Milnor’s basis

given in (1.6) for Ap(n); that is, Ap(n) = span(C). From Lemma 2.1.2, the set B is linearly independent. Also, B and C have the same number of elements. Since

Q(E) ∈ span(C) and P{R} ∈ span(C), for R ∈ Λ and E ∈ Γ, we have P{R} Q(E) ∈

span(C) since Ap(n) is an algebra. Thus, B ⊆ span(C) = Ap(n), and B is a basis for

the finite dimensional algebra Ap(n).

The following lemma will help show uniqueness in the proof of Theorem 2.2.1.

Lemma 2.1.4. If Q(E)P{R} and Q(F )P{R} are two elements of the Steenrod al-

gebra having the same degree, then they are equal.

Proof. First note that if |Q(E)P{R}| = |Q(F )P{R}|, then |Q(E)| = |Q(F )|. Let

Q(E) = Qi1 Qi2 ··· Qih and Q(F ) = Qj1 Qj2 ··· Qjg , where i1 < i2 < ··· < ih and

j1 < j2 < ··· < jg are increasing sequences of nonnegative integers. Suppose that Q(E) 6= Q(F ). We will show |Q(E)|= 6 |Q(F )|. We calculate

h X |Q(E)| = 2pi1 − 1 + ··· + 2pih − 1 = 2 pik − h k=1 g X |Q(F )| = 2pj1 − 1 + ··· + 2pjg − 1 = 2 pj` − g. `=1

Without loss of generality, we may assume ih < jg. Indeed, if ih = jg, then

Qi1 ··· Qih−1 and Qj1 ··· Qjg−1 would have the same degree, and it would suffice to show that these elements are equal. To show |Q(E)| < |Q(F )|, it suffices to show

25 that |Q(E)| < Qjg . This holds by the calculations

Ph ik Ph ik |Q(E)| = 2 k=1 p − h ≤ 2 k=1 p − 1 (since h ≥ 1)

Pjg−1 k ≤ 2 k=0 p − 1 (since ih < jg)

jg Pn k n+1
< 2p − 1 since k=0 p < p

= Qjg ≤ |Q(F )| .

2.2 Main Primality Theorem

We now present our main theorem. Recall that a ring B is prime if aBb 6= 0 for all nonzero a, b ∈ B. Kashkarev [25] proved that the mod 2 Steenrod algebra A2 is prime.

Theorem 2.2.1. The Steenrod algebra Ap for odd p is a prime ring.

Proof. We assume that a 6= 0 and b 6= 0 and show that aApb 6= 0.

Let I be the ideal generated by Q0 and Bp = Ap/I, which is isomorphic to the algebra generated by P0, P1, P2,.... Kashkarev’s argument in [25] shows (if replace 2 with p and Sq with P) that Bp is a prime ring and I is a prime ideal (see Proposition 2.3.2 below). We use Kashkarev’s argument, but we slightly modify his his proof to handle the Qi’s that are present in Ap when p is an odd prime. Using Lemma 2.1.2, let

X X a = aijQ (Ei) P{Rj} and b = bk`P{Sk} Q (F`) , (2.5) (i,j)∈I (k,`)∈J where I and J are finite subsets of N × N, the aij’s and bk`’s are all nonzero, and

j j j
k k k
Rj = r1, r2, . . . , rm and Sk = s1, s2, . . . , sn . (2.6)

26 Here, the j’s and k’s are indexing superscripts and not exponents. For simplicity,

we write all the sequences Rj given in (2.6) with the same length for m = ` (a), where `(a) is defined below, adding zeros where necessary. Similarly, we write all the sequences Sk given in (2.6) with the same length n = ` (b), adding zeros where necessary.

We slightly modify the definitions of `(R), `(a), and w(a) from Kashkarev in [25] as follows. Given a sequence R = (r1, . . . , rγ) with rγ 6= 0, let `(R) = γ; that is, γ is the index of the last nonzero entry in R. If R is the zero sequence, we define `(R) = 0.

Call the number ` (a) = maxj {` (Rj)} the length of a, where a is as in (2.5). Also, call the width of a the integer w(a) determined by the inequalities

w(a)−1  j w(a) p ≤ max rα < p . j,α

For a binary sequence E = (τ0, τ1, . . . , τγ) with τγ = 1, we define σ(E) = γ; that is, the index of the last nonzero entry in E. If E is the zero sequence, we define

σ(E) = 0. For the element a given in (2.5), we define σ(a) = maxi {σ (Ei)}; that is, σ(a) is the largest index of the Q’s present in a. Similarly, for b defined in (2.5), we let σ(b) = max` {σ (F`)} . Since both a and b are nonzero, the sums given in (2.5) are nonempty. To begin showing that aApb 6= 0, we choose z, x, and c similar to Kashkarev in [25]. We set

Pn α j Pσ(b) i c = Pz(x), where x ≥ ( α=1 p sα) + i=0 p , for all j, and

z = max {σ(a), ` (a) , w (a) − 1} + 1.

Note that if every Rj and Sk in (2.5) consists entirely of zeros, then we let z = σ(a)+1

Pσ(b) i and x ≥ i=0 p and choose c accordingly. Below, Lemma 2.2.2 and Lemma 2.2.3 are from Kashkarev [25], where they appear as Lemma 1 and Lemma 2, respectively, with proofs. With our choice of x, z, and c,

27 the proofs of these lemmas are still valid when we replace 2 with p and Sq with P.

So, we state the main results here, and the reader may refer to [25] for details. From

(2.5), we write

    X X aApb =  aijQ (Ei) P{Rj} Ap  bk`P{Sk} Q (F`) , (i,j) (k,`) and with our choice of c, we write

    X X acb =  aijQ (Ei) P{Rj} Pz(x)  bk`P{Sk} Q (F`) (i,j) (k,`) X   = aijbk`Q (Ei) P{Rj}Pz(x) P{Sk} Q (F`) (i,j),(k,`) X    = aijbk`Q (Ei) P Rj P{Sk} Q (F`) , (2.7) (i,j),(k,`)

j j j
th where (i, j) ∈ I,(k, `) ∈ J, and Rj = r1, r2, . . . , rm, 0,..., 0, x with x in the z position. It is possible for x to be in the (m + 1)th position in T so that no zeros

j appear between rm and x. Note here that the products aijbk` are all nonzero since all aij and bk` are nonzero.

 Lemma 2.2.2. ([25, Lemma 1]) In the expansion of P Rj P{Sk} in terms of the
Milnor basis the right lexicographical maximal term M Rj,Sk corresponds to the matrix   ∗ 0 0 0 ... 0      rj 0 0 0 ... 0   1     j   r2 0 0 0 ... 0  X =   .  ......   ......       0 0 0 0 ... 0      P α k k k k k x − α p sα s1 s2 s3 . . . sn

28 ! X α Noticing that M(R,S) = r1, . . . , rm, 0,..., 0, x − p sα, s1, . . . , sn , we have α the following Lemma:

Lemma 2.2.3. ([25, Lemma 2]) Take R1 ≤r R2 and S1 ≤r S2. Then:

1. M(R1,S1) ≤r M(R2,S2)

2. M(R1,S1) = M(R2,S2) if and only if R1 = R2 and S1 = S2.

In the p = 2 case in [25], Kashkarev concluded from the two lemmas that if Ri

and Sj are the maximum multi-indices in the expression of ac and b, respectively, in
the right lexicographical sense, then M Rj,Sk will be maximal in the expansion of acb. The same applies for the case when p is an odd prime.  In the right lexicographical sense, choose P{R} maximum of all the P Rj and

P{S} maximum of all the P{Sk} that are present in (2.7). Note that the product P{R}P{S} does occur in (2.7).

At this point, we alter Kashkarev’s proof so that we can handle the Qi’s that are present. Choose Q(E) and Q(F ) such that Q(E)P{R}P{S} Q(F ) has a nonzero

coefficient. Let P{M} be the maximum summand, in the right lexicographical sense,

in the expansion of P{R}P{S}; that is, M = M(R,S). Recall from Lemma 2.2.2

that ! X α M = r1, r2, . . . , rm, 0,..., 0, x − p sα, s1, . . . , sn , (2.8) α

where S = (s1, . . . , sn). In the case every Rj and Sk in (2.5) consists entirely of zeros,

we simply have P{M} = c. For E = (ε0, ε1, . . . , εk) and F = (τ0, τ1, . . . , τ`), we write

ρ = Q (E) P{M} Q (F )

in terms of the Milnor basis.

29 Suppose that τi1 , . . . , τih are the only nonzero exponents in

τ0 τ1 τ`−1 τ` Q(F ) = Q0 Q1 ··· Q`−1 Q`

where ` = ih. So, we write

ρ = Q(E)P{M} Qi1 Qi2 ··· Qih−1 Qih

X n i1 o = Q(E)Q P M p Q Q ··· Q Q i1+j1 j1 i2 i3 ih−1 ih j1≥0 . .

X n i1 i2 ih o = Q(E)Q Q ··· Q P M p ,p ,...,p . i1+j1 i2+j2 ih+jh j1, j2, ...,jh j1,...,jh≥0

Consider the case j1 = ··· = jh = z. We have

n i i i o n i i i o n i i i o P M p 1 ,p 2 ,...,p h = P M p 1 ,p 2 ,...,p h = P M p 1 +p 2 +···+p h 6= 0 j1, j2, ...,jh z, z, ...,z z because of the choice of x. By the choice of z > σ(a), we see that

ε0 ε1 εk Q(E)Qi1+zQi2+z ··· Qih+z = Q0 Q1 ··· Qk Qi1+zQi2+z ··· Qih+z 6= 0 since all the subscripts are distinct. Thus,

n pi1 +pi2 +···+pih o µ = Q(E)Qi1+zQi2+z ··· Qih+zP M z

pi1 +pi2 +···+pih th is a nonzero basis element. Note that M and Mz differ only in the z position.

We now show that µ does not appear in the representation in terms of the Milnor basis of any other term in the sum (2.7). After reviewing Lemma 2.2.2, we note that  in the expansion of P Rj P{Sk}, an arbitrary summand is of the form uP{X}, with

30 u ∈ Zp and

! X α X = h1, h2, . . . , hz−1, x − p dα + y, hz+1, . . . , hz+n (2.9) α

P with 0 ≤ y = xij summing over all indices i and j with j 6= 0 and i + j = z. The terms are of this form because all possible matrices are of the form

  ∗ x01 x02 x03 . . . x0n 0      x x x x . . . x 0  10 11 12 13 1n     ......   ......       xm0 xm1 xm2 xm3 . . . xmn 0      0 0 0 0 ... 0 0 . (2.10)      ......   ......       0 0 0 0 ... 0 0     x − P pαdj dj dj dj . . . dj 0  α α 1 2 3 n    0 0 0 0 ... 0 0

Let Q(E0)P{R0}P{S0} Q(F 0) be a summand in (2.7), with

0 0 E = (ϕ0, ϕ1, . . . , ϕk0 ) and F = (ψ0, ψ1, . . . , ψ`0 ) .

Let P{X} be an arbitrary summand in the expansion of P{R0}P{S0} in terms of the

Milnor basis. Note that X is of the form given in (2.9). Using (1.4) and assuming

0 that ψe1 , . . . , ψeg are the only nonzero elements in F , we calculate

η = Q(E0)P{X} Q(F 0)

X n e1 e2 eg o = Q (E0) Q Q ··· Q P Xp ,p ,...,p . (2.11) e1+f1 e2+f2 eg+fg f1, f2, ...,fg f1,...,fg≥0

Recalling the form of M in (2.8), we have two cases.

31 0 0 Case 1: Suppose that R

Suppose that for some 1 ≤ i ≤ n, we have hz+i < si and hz+` = s` for i < ` ≤ n.

e e e Then for any values of f , . . . , f , the (z + i)th position in Xp 1 ,p 2 ,...,p g will still be 1 g f1, f2, ...,fg

th pi1 +pi2 +···+pih less than the (z + i) position in M z , which is si. So, no monomial in the expansion of η is equal to µ.

j Suppose that hz+` = s` for all 1 ≤ ` ≤ n. This implies that, in (2.10), dn = sn

th j (since (z + n) diagonal sum equals hz+n = dn), so that the column sum cn = P 0 sn + i xin. Since S ≤r S, we have xin = 0, for all i, and cn = sn. Consequently, j j sn−1 = hn−1+z = dn−1 + xz−1,n = dn−1, which again forces xi,n−1 = 0, for all i, and

cn−1 = sn−1. By induction, xi` = 0 for all i and `, and d` = s` for all `. It follows

P α P α that y = 0 and hz = x − α p dα = x − α p sα. Furthermore, this forces h` = 0

for m < ` < z, and there is some i between 1 and m with hi < ri and h` = r` for

e e e i < ` ≤ m. Yet again, for any values of f , . . . , f , the ith position in Xp 1 ,p 2 ,...,p g 1 g f1, f2, ...,fg

th pi1 +pi2 +···+pih will still be less than the i position in M z , which is ri. Therefore, no monomial in the expansion of η is equal to µ.

0 0 Case 2: Suppose that R = R and S = S. Since M is maximal, X ≤r M. The case where X

e e e Suppose f 6= z for some 1 ≤ i ≤ g. Then the fth position in M p 1 ,p 2 ,...,p g will be i i f1, f2, ...,fg

th pi1 +pi2 +···+pih strictly less than the fi position in M z . Thus, the particular monomial

e e e in η involving M p 1 ,p 2 ,...,p g with f 6= z is not equal to µ. f1, f2, ...,fg i

When f1 = ··· = fg = z in (2.11), we have the monomial

e e e ϕ0 ϕ1 ϕk0  p 1 +p 2 +···+p g π = Q0 Q1 ··· Qk0 Qe1+zQe2+z ··· Qeg+zP M z .

By way of contradiction, suppose π = µ. Thus,

pe1 +pe2 +···+peg pi1 +pi2 +···+pih M z = M z

32 and, in the zth position, we have

g h X α X eβ X α X iβ x − p sα − p = x − p sα − p α β=1 α β=1 so that

pe1 + pe2 + ··· + peg = pi1 + pi2 + ··· + pih .

0 It follows that g = h and eα = iα for 1 ≤ α ≤ g, which implies F = F . Furthermore, Lemma 2.1.4 quickly implies E = E0 since |π| = |µ|. This contradicts the fact that

the pair (E,F ) 6= (E0,F 0). Therefore, no monomial in the expansion of η is equal to

µ.

We have thus shown that µ is a nonzero element that does not cancel with any other monomial in the expansion of (2.7). Therefore, acb 6= 0, and aApb 6= 0. This concludes the proof of Theorem 2.2.1.

2.3 Miscellaneous

Similarly to Kashkarev, we can find an estimate for c. As previously stated in

Section 1.4, the Steenrod algebra Ap is the union of the subalgebras Ap(n), each

ε0 εn+1 of which has a linear basis formed by Q0 ··· Qn+1 P{R} for R = (r1, r2, . . . , rn+1)

n−k+2 with rk < p . By Corollary 2.1.3, Ap(n) also has a linear basis formed by

τ0 τn+1 n−k+2 P{R} Q0 ··· Qn+1 for R = (r1, r2, . . . , rn+1) with rk < p . Given a 6= 0 and

b 6= 0 in Ap(n), we exhibited some c, as in Theorem 2.2.1, such that acb 6= 0. Note that Proposition 2.3.1 below is an extension of Proposition 1 in [25] to odd primes.

Proposition 2.3.1. Given a, b ∈ Ap(n), there exists some c ∈ Ap(g) with acb 6= 0,   where g = logp (2n + 2) + 2n + 2 and dve denotes the smallest integer greater than or equal to v.

33 i n+1 Proof. If a ∈ Ap(n), then σ(a) ≤ n + 1 and rα < p , which implies w(a) ≤ n + 1 and `(a) ≤ n + 1. It follows that z ≤ n + 2. We also note that σ(b) ≤ n + 1 for

j n+2−α n+2 b ∈ Ap(n). Similarly, sα ≤ p . Choosing x = (2n + 2) p and c = Pz(x), we have

x = 2 (n + 1) pn+2

= (n + 1) pαpn+2−α + (n + 1) pn+2 ! n+1 ! X α j X i > p sα + p , α i=0

for all α. Furthermore,

x < (2n + 2) pn+2 · pn−z+2 ≤ pdlogp(2n+2)e+2n+2−z+2 = pg−z+2,

th which is the necessary condition on x in the z position for c ∈ Ap(g).

As an example, let p = 3, n = 1, and

a = b = Q0Q1Q2P{8, 2} = P{8, 2} Q0Q1Q2

so that z = 3, x = 108, g ≈ 5.261859507 < 6, and c = P{0, 0, 108} ∈ Ap(6). So,

    acb = Q0Q1Q2P{8, 2} P{0, 0, 108} P{8, 2} Q0Q1Q2

= Q0Q1Q2P{8, 2, 66, 8, 2} Q0Q1Q2

= Q0Q1Q2Q3Q4Q5P{8, 2, 53, 8, 2}= 6 0.

These calculations were confirmed using Sage [38].

Proposition 2.3.2. The ideal I generated by Q0 is a prime ideal of the Steenrod

0 1 algebra. Furthermore, Ap/I is isomorphic to the algebra Bp generated by P , P ,....

34 Proof. From (1.3), Qk ∈ I for all k ≥ 0. Therefore, Q(E)P{R} ∈ I precisely when E 6= (0, 0,...), and we have the one-to-one correspondence

P{R} ←→ P{R} + I,

∼ from which it follows that Bp = Ap/I via the natural homomorphism P{R} 7→ P{R} + I. As mentioned in the proof of Theorem 2.2.1, Kashkarev’s proof shows

that Bp is a prime ring.

Historical Note: At the time of writing this thesis, this chapter was accepted for publication under the title “The Steenrod Algebra is a Prime Ring” [46].

35 Chapter 3

The Krull Dimensions of the

Steenrod Algebra

In this chapter, we show that the Steenrod algebra Ap has infinite little Krull dimension; that is, k dim Ap ≥ ω, where ω is the first infinitely countable ordinal. In Section 3.2, we will show this result three different ways. First, in §3.2.1, we

1 p pj will define the ideals Dj of Bp generated by the elements P , P ,..., P . We give a characterization for N so that P{N} ∈ Dj for all j. Using this characterization, we

show that the ideals Dj form an infinite ascending chain of prime ideals. Then, in

§3.2.2, we define a surjective map ψ : Bp → Bp (given in Equation (3.20)) whose kernel is a nonzero prime ideal. The ideals ker ψj, for j ≥ 1, form an infinitely ascending

chain of prime ideals with strict containment. Finally, in §3.2.3, we will use the Hopf

algebra structure of the Steenrod algebra to construct our desired chain. As it turns

out, these three different approaches produce the same chain or prime ideals.

A consequence of the infinitely ascending chain of prime ideals Dj, which we discuss in §3.3, is that the Steenrod algebra has neither Krull dimension nor classical

Krull dimension. We also show that the Steenrod algebra has no Gabriel dimension.

36 3.1 Preliminary Results

We first discuss some number theoretic results. This will help us determine when a

multinomial coefficient is zero modulo a prime p, which in turn shows when b(X) ≡ 0

mod p.

Theorem 3.1.1. (Kummer’s Theorem1) If n, k are integers and p is a prime, then

n
the largest power of p dividing k is given by the number of borrows required when subtracting the base p representation of k from n.

An easily verifiable consequence of Kummer’s Theorem is the following:

k Corollary 3.1.2. Let x1, x2, . . . , xn be nonnegative integers. If xj 6≡ 0 mod p , for

k some 1 ≤ j ≤ n and k > 0, and t = x1 + ··· + xn ≡ 0 mod p , then

 t  ≡ 0 mod p. x1, . . . , xn

k Pr ` Proof. Since xj 6≡ 0 mod p , we write xj = `=0 c`p , where ci 6= 0 for the smallest

such i ≤ r and i < k. Thus, subtracting the base p representation of xj from t requires at least k − i borrows. By Kummer’s Theorem,

 t  ≡ 0 mod p, xj which, using Equation (1.7), implies

 t  ≡ 0 mod p. x1, . . . , xn

Lastly, we present Lucas’s Theorem.

1See [29, page 116] and also [31, Theorem 3]

37 Theorem 3.1.3. (Lucas’s Theorem, [14, Theorem 1]) Let p be a prime, and let

2 k M = M0 + M1p + M2p + ··· + Mkp (0 ≤ Mr < p) ,

2 k N = N0 + N1p + N2p + ··· + Nkp (0 ≤ Nr < p) .

Then M M M M  M  ≡ 0 1 2 ··· k mod p. N N0 N1 N2 Nk

3.2 The Little Krull Dimension

To show that k dim Ap ≥ ω, it will suffice to exhibit an infinitely ascending chain

C of prime ideals of Ap, where ω = λ(C) ≤ k dim Ap.

Recall Proposition 2.3.2, which states that the ideal generated by Q0 is a prime ideal and Bp is a prime ring. The author has not searched for any other prime ideals involving Qi’s. For the rest of Section 3.2, unless otherwise stated, we work with the algebra Bp.

3.2.1 The Ideals Dj

Below we define the ideals Dj, for all j ≥ 0. We find a characterization for N when P{N} ∈ Dj for all j ≥ 0 (see Theorem 3.2.2). With this characterization, we show that the ideals Dj for all j are prime ideals, and these ideals form an infinite ascending chain of prime ideals.

Definition 3.2.1. For all j ≥ 0, let Dj denote the ideal generated by the elements

1 p pj P , P ,..., P . We define D−1 to be the zero ideal.

Theorem 3.2.2. For j ≥ 0,

j+1 P{N} ∈ Dj ⇐⇒ N 6≡ 0 mod p .

38 Before proving Theorem 3.2.2, we present Lemma 3.2.3, which we will use in the proof of Theorem 3.2.2. From Milnor [33, Corollary 5], we have

  i−1  Pi(1) = P p , Pi−1(1) (3.1)

for all i ≥ 2. Lemma 3.2.3 is an extension of Equation (3.1), modulo Dj. Indeed, when j = −1, Equation (3.2) reduces to Equation (3.1).

Lemma 3.2.3. Let j ≥ 0 and suppose that P{N} ∈ Dj if and only if N 6≡ 0 mod pj+1. For i ≥ 2, we have

  i+j j+1
 j+1
P p + Dj, Pi−1 p + Dj = Pi p + Dj. (3.2)

j+1 Furthermore, Pi(p ) ∈ Dj+1 for all i ≥ 1.

Proof. We have two cases: i = 2 and i > 2.

2+j j+1 First let i = 2. Via Milnor [33, Corollary 3], P{p }P1(p ) + Dj equals

pj+1 X p2+j − px + pj+1 − x Pp2+j − px + pj+1 − x, x + D , (3.3) p2+j − px, pj+1 − x j x=0

j+1 2+j and P1(p ) P{p } + Dj equals

pj X pj+1 − px + p2+j − x Ppj+1 − px + p2+j − x, x + D . (3.4) pj+1 − px, p2+j − x j x=0

We obtain (3.3) and (3.4) since all matrices are of the form

    ∗ pj+1 − x 0 ... ∗ p2+j − x 0 ...         p2+j − px x 0 ... pj+1 − px x 0 ...       and   ,  0 0 0 ...  0 0 0 ...      . . . .   . . . .  ......

39 respectively. When x = 0,

p2+j + pj+1 Pp2+j + pj+1 + D p2+j, pj+1 j is common in both (3.3) and (3.4), which implies that this term cancels out in

  2+j j+1
 P p + Dj, P1 p + Dj .

j+1 2+j j+1 When 0 < x < p , we have P{p − px + p − x, x} ∈ Dj by assumption. When

j+1 j+1 x = p in equation (3.3), we have P{0, p } + Dj as a summand. Thus,

  2+j j+1
 j+1
P p + Dj, P1 p + Dj = P2 p + Dj.

i+j j+1 Suppose now that i > 2. We have P{p }Pi−1(p ) + Dj equals

pj+1 X  i+j i−1 j+1 P p − p x, 0,..., 0, p − x, x + Dj (3.5) x=0

j+1 i+j and Pi−1(p ) P{p } + Dj equals

pj X  i+j j+1 P p − x, 0,..., 0, p − px, x + Dj, (3.6) x=0

40 where the pj+1 − x and pj+1 − px are in the (i − 1)th position and the x is in the ith position. The summations in (3.5) and (3.6) are obtained via the matrices

  ∗ pi+j − x 0 ...        0 0 0 ... j+1   ∗ 0 ... 0 p − x 0 ...  . . . .     . . . ..     pi+j − pi−1x 0 ... 0 x 0 ...         and  0 0 0 ... ,  0 0 ... 0 0 0 ...         pj+1 − px x 0 ... ......   ......      0 0 0 ...  . . . .  . . . ..

respectively. In the first matrix x is in the 1st,(i − 1)th position, and in the second matrix x is in the (i − 1)th, 1st position. We see that when x = 0, the summand

 i+j j+1 P p , 0,..., 0, p + Dj

appears in both (3.5) and (3.6) so that this term cancels out in

  i+j j+1
 P p + Dj, Pi−1 p + Dj .

When 0 < x < pj+1, we see that

 i+j i−1 j+1  i+j j+1 P p − p x, 0,..., 0, p − x, x , P p − x, 0,..., 0, p − px, x ∈ Dj

by assumption. When x = pj+1, we have

 i+j i−1 j+1 j+1
P p − p x, 0,..., 0, p − x, x + Dj = Pi p + Dj

as a summand of (3.5).

41 Therefore, we have

  i+j j+1
 j+1
P p + Dj, Pi−1 p + Dj = Pi p + Dj (3.7)

j+1 for all i ≥ 2, which implies that Pi(p ) ∈ Dj+1 for all i ≥ 1.

We now prove Theorem 3.2.2, and the proof will proceed by induction on j. proof of Theorem 3.2.2. Let j = 0. We will show that

P{N} ∈ D0 ⇐⇒ N 6≡ 0 mod p.

(⇐) Suppose that N is a sequence where ni 6≡ 0 mod p for some i ≥ 1. Using

Equation (3.1), we have Pi(1) ∈ D0 for all i ≥ 1. Thus, we have

1  1 P{N} = Pi(1) P N i ∈ D0. ni

(⇒) Conversely, we prove the contrapositive; suppose N ≡ 0 mod p. Arguing by

way of contradiction, suppose P{N} ∈ D0. Thus, for Milnor sequences R` and S` (with repeats allowed), we can write

X P{N} = a`P{R`}P1(1) P{S`} ` X `
 −1 = a` 1 + s1 P{R`}P S`1 ` X `
= a` 1 + s1 b (Tα) P{Tα} , `,α

` where s1 (` being a superscript and not an exponent) is the first coordinate of S` and

b (Tα) is the coefficient that comes from the associated coefficient matrix that yielded

Tα. Since P{X}, for varying Milnor sequences X, forms a basis, the only Tα that

42 survive are the ones where Tα = N; that is,

Q X nk! P{N} = a 1 + s`
P{N} , (3.8) ` 1 Q xα `,α i,j

α where the xi,j come from the associated coefficient matrix. For brevity, let gk = P α P α≥1 p xk,α and hk = α≥1 xα,k. The associated coefficient matrices are of the form

  ∗ 1 + s1 − h1 s2 − h2 ...     r1 − g1 x1,1 x1,2 ...   X =   . r − g x x ...  2 2 2,1 2,2   . . . .  . . . ..

We note here that, for instance,

x1,0 = r1 − g1 and x0,1 = 1 + s1 − h1

x2,0 = r2 − g2 and x0,2 = s2 − h2,

and so on. By assumption, ni ≡ 0 mod p for all i ≥ 1. For b(X) 6≡ 0 mod p, we require  n  i 6≡ 0 mod p x0,i, x1,i−1, . . . , xi,0 for all i ≥ 1. Corollary 3.1.2 forces xi,j ≡ 0 mod p for all i, j ≥ 0; that is, xi,j = pyi,j. In particular, we have

X X py0,1 = x0,1 = 1 + s1 − h1 = 1 + s1 − xj,1 = 1 + s1 − p yj,1, (3.9) j j

from which it follows that 1 + s1 ≡ 0 mod p. From (3.8), we have

Q X nk! P{N} = a 1 + s`
P{N} , ` 1 Q xα `,α i,j

43 ` and Equation (3.9) implies 1 + s1 ≡ 0 mod p for all `, which implies P{N} = 0 — an impossibility. Therefore, P{N} 6∈ D0. We have now established the base case, and we will induce j. Let j ≥ 0 and

suppose that

j+1 P{N} ∈ Dj ⇐⇒ N 6≡ 0 mod p . (3.10)

With assumption (3.10), we will show

j+2 P{N} ∈ Dj+1 ⇐⇒ N 6≡ 0 mod p .

(⇒) We prove the contrapositive; suppose N ≡ 0 mod pj+2. From (3.10), P{N} 6∈

Dj. Arguing by way of contradiction, suppose that P{N} ∈ Dj+1; that is, P{N} is

j+1 an element of the principle ideal in Bp/Dj generated by P{p } + Dj. For Milnor

sequences Rα and Sα (with repeats allowed), we can write

X  j+1 P{N} + Dj = aαP{Rα}P p P{Sα} + Dj (3.11) α X = aαtγP{Tγ} + Dj α,γ X = aαtγP{N} + Dj, (3.12) α,γ

where the only Tγ that survive are the Tγ = N and tγ is the resulting coefficient after

expanding the multiplication. Our strategy is to show that tγ ≡ 0 mod p, which will

yield our contradiction of P{N} ∈ Dj.

j+1 For a given sequence S, the possible matrices for P{p }P{S} + Dj are

  ∗ s1 − k1 . . . sj − kj sj+1 ... K =    j+1 Pj α  p − α=1 p kα k1 . . . kj 0 ...

44 and   ∗ s . . . s s − 1 s ...  1 j j+1 j+2    . 0 0 ... 0 1 0 ...

Let j j+1 X α K1,0 = p − p kα. α=1

j+1 So, P{p }P{S} + Dj equals

   1, −1 X K1,0 + s1 − k1 (1 + sj+2) P Sj+1,j+2 + bK P{MK } + Dj, (3.13) K1,0, s1 − k1 K where the sum is over all possible matrices K,

  j   kj + sj+1 Y kα−1 + sα − kα b = , K k , s k , s − k j j+1 α=2 α−1 α α and MK = (mK,1, mK,2,...) with

  K1,0 + s1 − k1 if ` = 1,    k`−1 + s` − k` if 2 ≤ ` ≤ j, mK,` = (3.14)  k + s if ` = j + 1,  j j+1    s` if ` ≥ j + 2.

 1, −1 Given a sequence R, the only possible matrices for P{R}P Sj+1,j+2 + Dj are

  ∗ s − f . . . s − f s − 1 − f s + 1 − f ...  1 1 j j j+1 j+1 j+2 j+2    r − e y . . . y y y ...  1 1 1,1 1,j 1,j+1 1,j+2      , r2 − e2 y2,1 . . . y2,j y2,j+1 y2,j+2 ...     r3 − e3 y3,1 . . . y3,j y3,j+1 y3,j+2 ...    ......  ......

45 P β P where e` = β≥1 p y`,β and f` = β≥1 xβ,`. To obtain a nonzero coefficient when working in Dj, Corollary 3.1.2 requires yα,β ≡ 0 mod p for all α and all β, since

j+2 N ≡ 0 mod p by assumption; that is, yα,β = pzα,β. In particular, we have

X y0,j+2 = pz0,j+2 = 1 + sj+2 − fj+2 = 1 + sj+2 − p zβ,j+2 β

so that 1 + sj+2 ≡ 0 mod p, which in turn implies

 1, −1 (1 + sj+2) P Sj+1,j+2 + Dj = 0.

The possible matrices for P{R}P{MK } + Dj are

  ∗ m − h m − h ...  K,1 1 K,2 2    r − g x x ...  1 1 1,1 1,2    X =   , (3.15) r2 − g2 x2,1 x2,2 ...     r3 − g2 x3,1 x3,2 ...    . . . .  . . . ..

P β P where g` = β≥1 p x`,β and h` = β≥1 xβ,`. The first diagonal yields the multinomial coefficient r − g + m − h  1 1 K,1 1 , r1 − g1, mK,1 − h1

where

j+2 n1 = r1 − g1 + mK,1 − h1 ≡ 0 mod p ,

whilst the `th diagonal yields the multinomial coefficient

  r` − g` + x`−1,1 + ··· + x1,`−1 + mK,` − h` d`(X) = , r` − g`, x`−1,1, . . . , x1,`−1, mK,` − h`

46 where

j+2 n` = r` − g` + x`−1,1 + ··· + x1,`−1 + mK,` − h` ≡ 0 mod p .

Recalling the multinomial coefficient from (3.13), let

   r1 − g2 + mK,1 − h1 K1,0 + s1 − k1 cK,X = , r1 − g2, mK,1 − h1 K1,0, s1 − k1

which is a factor of some tγ given in Equation (3.12). Our strategy will be to show that for any values of r1 or s1, and hence any Rα and Sα in (3.11), we obtain cK,X ≡ 0 mod p.

j+2 Suppose that cK,X 6≡ 0 mod p. Since n` ≡ 0 mod p for all ` ≥ 1, Corollary

j+2 3.1.2 implies xα,β ≡ 0 mod p for every entry xα,β of the matrix X in (3.15). This

j+2 j+2 implies h1 ≡ 0 mod p , and since x0,1 = mK,1 − h1 ≡ 0 mod p , we conclude

j+2 mK,1 = K1,0 + s1 − k1 ≡ 0 mod p .

For cK,X 6≡ 0 mod p, we must have

j+2 j+2 K1,0 ≡ 0 mod p and s1 − k1 ≡ 0 mod p by Corollary 3.1.2, but

j j+1 X α j+2 K1,0 = p − p kα 6≡ 0 mod p , α=1 a contradiction. Therefore, we must have cK,X ≡ 0 mod p. Ergo, tγ = 0 for all γ so that P{N} + Dj = 0 and P{N} ∈ Dj, but this is an impossibility according to our

j+2 induction hypothesis (3.10) since N ≡ 0 mod p . Thus, P{N} 6∈ Dj+1. (⇐) We now show the converse by using a more direct approach. Suppose that

j+1 j+2 P{N} + Dj is such that N ≡ 0 mod p and N 6≡ 0 mod p . So, 0 6= nt 6≡ 0

47 j+2 mod p for some t ≥ 1. We note that P{N} + Dj 6= 0.

j+1 Our goal is to show that P{N}+Dj ∈ Span {P`(p ) + Dj : ` ≥ 1}, which implies,

by Lemma 3.2.3, that P{N} ∈ Dj+1. To begin, we calculate

n pj+1 o j+1
P N t Pt p + Dj.

The possible coefficient matrices are of the form

  j+1 P ∗ 0 ... 0 p − α xα,t 0    t   n1 − p x1,t 0 ... 0 x1,t 0    ......   ......      X =  t  . (3.16)  nt−1 − p xt−1,t 0 ... 0 xt−1,t 0    j+1 t  nt − p − p xt,t 0 ... 0 xt,t 0      t   nt+1 − p xt+1,t 0 ... 0 xt+1,t 0  ......  ......

n pj+1 o j+1 P We have P N t Pt(p ) + Dj = X b(X)P{MX } + Dj, where the sum is over all possible matrices X,

 t P   t  nt − p xt,t − xα,t Y nt+i − p xt+i,t + xi,t b(X) = α , n − pj+1 − ptx , pj+1 − P x n − ptx , x t t,t α α,t i≥1 t+i t+i,t i,t

and MX = (m1, m2,...) is a sequence where

  n − ptx if 1 ≤ k ≤ t − 1,  k k,t  m = t P k nt − p xt,t − α xα,t if k = t,   t  nk − p xk,t + xk−t,t if k > t.

t Note here that mt+i = nt+i − p xt+i,t + xi,t for i ≥ 1.

48 j+1 Suppose X is a matrix with xk,t = 0 for all k > `. If 0 < x`,t < p , then

t j+1 mt+` = nt+` − p xt+`,t + x`,t = nt+` + x`,t 6≡ 0 mod p

j+1 and P{MX } ∈ Dj. Thus, we only consider the cases x`,t = 0 or x`,t = p . In the

j+1 case x`,t = p , we have xα,t = 0 for all α 6= `. We conclude that either x`,t = 0 for

j+1 all ` or only one x`,t = p and xα,t = 0 for all α 6= `. With this, we can now write

n pj+1 o j+1 P N t Pt(p ) + Dj as

   j+1 nt X nt+` + p n pj+1,−pj+1 o j+1 j+1 P{N} + j+1 P N t, t+` + Dj. nt − p , p nt+`, p `≥1

j+1 j+2 Since nt ≡ 0 mod p and 0 6= nt 6≡ 0 mod p , we can write

j+1 j+2 j+h nt = aj+1p + aj+2p + ··· + aj+hp

where 0 ≤ ai < p and aj+1 6= 0. Thus, when every x`,t = 0 for ` > 0, Lucas’s Theorem shows

    j+1 nt nt nt − p j+1 j+1 = j+1 j+1 nt − p , p p nt − p a a  a  ≡ j+1 j+2 ··· j+h 1 0 0 j+1 = aj+1 mod p , (3.17)

where aj+1 is nonzero in Zp. Here we note that

 j+1 n pj+1 o j+1
X nt+` + p n pj+1,−pj+1 o P N t Pt+1 p + Dj = j+1 P N t, t+` + Dj nt+`, p `≥1

49 since all possible matrices are of the form

  j+1 P ∗ 0 ... 0 p − α xα,t+1 0 ...      n1 − x1,t+1 0 ... 0 x1,t+1 0 ...    ......   ......        , (3.18)  nt−1 − xt−1,t+1 0 ... 0 xt−1,t+1 0 ...    j+1  nt − p − xt,t+1 0 ... 0 xt,t+1 0 ...        nt+1 − xt+1,t+1 0 ... 0 xt+1,t+1 0 ...  ......  ......

j+1 where either x`,t+1 = 0 for all ` or only one x`,t+1 = p and xα,t+1 = 0 for all α 6= `. Notice that the matrices in (3.18) are similar to the matrices given in (3.16), but with the column of xk,`’s shifted to the right once. Thus,

n pj+1 o j+1
n pj+1 o j+1
P N t Pt p + Dj = aj+1P{N} + P N t Pt+1 p + Dj

where aj+1 is as in (3.17). Solving for P{N}, we have

1  n pj+1 o j+1
n pj+1 o j+1
 P{N} + Dj = P N t Pt p − P N t Pt+1 p + Dj, aj+1

which implies P{N} ∈ Dj+1 by Lemma 3.2.3.

Using the results of Theorem 3.2.2, we can prove the following.

Theorem 3.2.4. The ideals Dj for all j ≥ 0 are prime ideals.

Proof. We assume a and b are not elements of Dj and show that aBpb 6⊆ Dj; that is, P that there exists a c ∈ Bp such that acb 6∈ Dj. So, for elements a = i aiP{Ri} and P j+1 b = k bkP{Sk} in Bp which are not in Dj, we have Ri,Sk ≡ 0 mod p for all i

and k. We note that each ai and bj is nonzero so that the product aibj, for all i, j,

is nonzero in Zp. Recall the definitions of `(a) and w(a) given on page 27. Letting

50 P α k z = max {` (a) , w (a) − 1} + 1, we choose c = Pz(x) in Cj, with x ≥ α p sα, for

j+1 all k, and x ≡ 0 mod p . Note here that c 6∈ Dj. We have acb 6∈ Dj since the maximum summand, with respect to the ordering

n X k o P{M} = P r1, . . . , rn, 0 . . . , x − p sk, s1, . . . , sm ,

j+1 where M ≡ 0 mod p ; ergo, P{M} 6∈ Dj. Thus, Dj is a prime ideal.

With this, we conclude:

Corollary 3.2.5. For an odd prime p, k dim Ap ≥ ω.

Proof. By Theorem 3.2.4, we have the infinitely ascending chain of prime ideals

D0 ⊂ D1 ⊂ D2 ⊂ · · · , (3.19)

from which the result follows.

3.2.2 The map ψ

Before defining the map ψ given in Equation (3.20) below, we present here three

lemmas that we will use to show that ψ is a homomorphism. The ideals ker ψj, for

j ≥ 1, will be used to show that the little Krull dimension of the Steenrod algebra is

infinite. Recall the definition of T (X) on Page 20.

Lemma 3.2.6. Suppose R,S ≡ 0 mod p. If P{T (X)} is a nonzero summand of

P{R}P{S}, then T (X) ≡ 0 mod p if and only if X ≡ 0 mod p.

P Proof. If X ≡ 0 mod p, then xi,j ≡ 0 mod p for all i, j ≥ 0 so that tn = i+j=n xi,j ≡

0 mod p for all n ≥ 1 and T (X) ≡ 0 mod p. Conversely, if xi,j 6≡ 0 mod p for some entry in X, it follows from 3.1.2 that b(X) = 0, which contradicts the assumption

that P{T (X)} is nonzero. Thus, we must have, X ≡ 0 mod p.

51 Lemma 3.2.7. Let R and S be Milnor sequences such that R,S ≡ 0 mod p.A matrix X ≡ 0 mod p is a coefficient matrix of the product P{R}P{S} if and only if

1 n 1 o n 1 o p X is a coefficient matrix of the product P p R P p S .

n 1 o Proof. If P{T } is a nonzero summand of P{R}P{S}, then P p T comes from the 1 n 1 o n 1 o matrix p X, which is a coefficient matrix for the product P p R P p S . Recall  1  1  1  1 here that R(X) = R and S(X) = S so that R p X = p R and S p X = p S makes n 1 o n 1 o since. Multiplication by p takes a coefficient matrix of the product P p R P p S to a coefficient matrix of the product P{R}P{S}. Thus, there is a one-to-one corre-

n 1 o n 1 o spondence between coefficient matrices of the product P p R P p S and coefficient matrices of the product P{R}P{S} that are congruent to 0 mod p.

 1  Lemma 3.2.8. If X ≡ 0 mod p, then b (X) ≡ b p X .

Proof. This follows from Lucas’s Theorem (Theorem 3.1.3).

Define the map ψ : Bp → Bp on the Milnor basis elements by

 n o  P 1 R if R ≡ 0 mod p ψ (P{R}) = p (3.20)  0 otherwise,

and extend linearly. Note that this is a surjective map onto the prime ring Bp, so its kernel will be a prime ideal provided ψ is a ring homomorphism, and this is the content of the next theorem.

Theorem 3.2.9. The map ψ is a ring homomorphism.

Proof. Let R,S ≡ 0 mod p. Then

1  1  X P R P S = b(X)P{T } , p p X

52 1 1 where the sum is over matrices X with R(X) = p R and S(X) = p S. On the other hand, X P{R}P{S} = b(X0)P{T 0} , X0 where the sum is over matrices X0 with R(X0) = R and S(X0) = S.

Using Lemma 3.2.6, we write

! X ψ (P{R}P{S}) = ψ b(X0)P{T 0} X0 X 1  = b (X0) P T 0 p X0≡0 mod p X 1  1  = b X0 P T 0 by Lemma 3.2.8 p p X0≡0 mod p X = b (X) P{T } by Lemma 3.2.7 X n 1 o n 1 o = P p R P p S = ψ (P{R}) ψ (P{S}) .

Suppose now that R or S 6≡ 0 mod p. It is clear that ψ (P{R}) ψ (P{S}) = 0 in this case. Suppose that ψ (P{R}P{S}) 6= 0. Thus, b(X) 6≡ 0 mod p for some coefficient matrices X. By Lemma 3.1.2, X ≡ 0 mod p for all such matrices X, from

P α P which it follows that ri = α≥0 p xα,i ≡ 0 mod p for all i and sj = α≥0 xi,α ≡ 0 mod p for all j. This contradict the assumption that R or S 6≡ 0 mod p. Thus, we

must have ψ (P{R}P{S}) = 0.

The following three corollaries are consequences of Theorem 3.2.9.

Corollary 3.2.10. The ideals ker ψj for j ≥ 1 form an infinite strictly ascending

chain of prime ideals.

53 Proof. The ideals ker ψj for j ≥ 1 are certainly prime ideals and form a strictly ascending chain. In fact,

ker ψj = P{R} | R 6≡ 0 mod pj , (3.21)

the vector space span of the appropriate P{R}.

j+1 Corollary 3.2.11. For all j ≥ 0, Dj = ker ψ .

The next corollary is now immediate.

Corollary 3.2.12. For an odd prime p, k dim Ap ≥ ω.

3.2.3 The Hopf Algebra Approach

For Bp, defined in Proposition 2.3.2, the elements {ξ(R)} (defined on Page 17) in

∗ ∗ Bp are dual to the elements {P{R}} in Bp, and Bp is a polynomial ring Zp [ξ1, ξ2, ξ3,...]. With this, we present another proof that the little Krull dimension of the Steenrod

algebra is infinite.

Proposition 3.2.13. For an odd prime p, k dim Ap ≥ ω.

∗ ∗ p Proof. The Frobenius φ : Bp → Bp given by x 7→ x is a Hopf endomorphism (see Lemma B.0.1 in Appendix B). In fact, when we look at the image of φ, we have the

isomorphism of Hopf algebras (which does not preserve the grading)

∗
p p ∼ ∗ φ Bp = φ (Fp [ξ1, ξ2,...]) = Fp [ξ1 , ξ2 ,...] = Fp [ξ1, ξ2,...] = Bp,

∗
∗ ∗ j+1 ∗
where φ Bp is a Hopf subalgebra of Bp. Defining Cj = φ Bp for all j ≥ −1, we have the chain

∗ ∗ ∗ ∗ Bp = C−1 ⊃ C0 ⊃ C1 ⊃ · · ·

∗ ∼ ∗ of Hopf subalgebras with proper containment where Cj = Bp for all j.

54 ∼ Passing to the dual, each Cj is a quotient of Bp. Furthermore, each Cj = Bp and a prime ring. Let Ej denote the kernel of the map Bp → Cj, which is a prime ideal. We have the infinite ascending chain of prime ideals

0 = E−1 ⊂ E0 ⊂ E1 ⊂ · · · , from which the result follows.

Corollary 3.2.14. For all j ≥ 0, Dj = Ej.

Proof. By the construction of Cj, it is apparent that

j+1 Ej = P{R} | R 6≡ 0 mod p = Dj.

Historical Note: The author submitted Section 3.2.1, as well as Section 3.3, for

publication [47]. The referee thought that the argument using the classification of

the ideals Dj was overly complicated to only show that the little Krull dimension of the Steenrod algebra is infinite. The proof of Proposition 3.2.13 was suggested by

the referee in an attempt to simplify the arguments. This led to the content that is

presented in Section 3.2.2. At the time of writing this thesis, the topics covered in

Sections 3.2.2, 3.3, and 3.5 were accepted for publication in [47].

3.3 Other Dimensions

We briefly show here that the more general Krull dimensions do not exists.

Theorem 3.3.1. The Steenrod algebra Ap has no classical Krull Dimension.

Proof. Krause showed in [27] that having classical Krull dimension is equivalent to

having the ascending chain condition on prime ideals. The infinitely ascending chain

55 of prime ideals in (3.19) implies that the Steenrod algebra Ap cannot have a classical Krull dimension.

As mentioned on page 15, if the Krull dimension of a ring exists, then the ring

also has a classical Krull dimension. Theorem 3.3.1 implies the following.

Corollary 3.3.2. The Steenrod algebra Ap has no Krull Dimension.

On the subject of the Gabriel dimension of the Steenrod algebra, we recall the definition of a monoform module given on page 16, and use the results of [18] to obtain

Theorem 3.3.3. If a nonzero module M has the property that, for each submodule N every homomorphism ϕ : N → M is either zero or a monomorphism, then we call M a monoform module.

Theorem 3.3.3. Let R be a semiprime ring with a nonzero nil ideal J. Then R has no Gabriel dimension.

Proof. Suppose that R has Gabriel dimension. Then every nonzero R-module con- tains a monoform submodule (by [18, Corollary 2.10]). More specifically, J contains a monoform right ideal C. Since R is a semiprime ring, there is an element c ∈ C such that r(c) ∩ C = 0 (by [18, Lemma 3.2]), where r(c) denotes the right annihilator of c in R. Since J is a nil ideal, there exists a minimal n ∈ N such that cn = 0, which implies 0 6= cn−1 ∈ r(c). Since C is a right ideal, cn−1 ∈ C, which implies that

r(c) ∩ C 6= 0, a contradiction. Thus, R does not have Gabriel Dimension.

Corollary 3.3.4. The Steenrod algebra Ap has no Gabriel Dimension.

Proof. A consequence of Theorem 2.2.1 is that Ap is a semiprime ring. Following

Milnor [33, Corollary 7], the Jacobson radical of Ap is a nonzero nil ideal. The result follows by Theorem 3.3.3.

56 3.4 Future Work

We introduce a result below that raises the question about the order type of

the little Krull dimension of the Steenrod algebra. Does k dim Ap = ω1, the first uncountable ordinal? The following theorem shows that for a countable polynomial ring, the little Krull dimension is ω1.

Proposition 3.4.1. If F is a countable field and S = F [x1, x2,...] is a polynomial ring over F with an infinitely countable number of commuting indeterminates, then k dim R = ω1.

Proof. Let α be an arbitrary countable ordinal and consider the polynomial ring

Rα = F [xβ : β ∈ α], which is isomorphic to the ring S since α is countable and has

the same cardinality as the natural numbers. Define Γγ = {β : γ ≤ β < α} and let Iγ

denote the prime ideal of Rα generated by the elements of Γγ. If β < γ, then Γγ ⊂ Γβ

and Iγ ⊂ Iβ. We have the infinitely descending chain Cα of prime ideals

I0 ⊃ I1 ⊃ · · · ⊃ Iβ ⊃ Iβ+1 ⊃ · · · .

When we order Spec Rα with reverse inclusion, then we have the well-ordered chain

0 Cα

I0 ≺ I1 ≺ · · · ≺ Iβ ≺ Iβ+1 ≺ · · · .

0 If α is a limit ordinal, then λ (Cα) = α. If α is not a limit ordinal, then we consider

0 0
the chain Cα+1 in Rα+1 so that λ Cα+1 = α + 1 − 1 = α. Thus, for any countable ordinal α, we can order the indeterminates of S so that we have a chain of length

α. It follows that S has little Krull dimension of at least ω1, the first uncountable ordinal. Since F is countable and we have a countable number of indeterminates,

S is countable and cannot have a chain of prime ideals with uncountable length. It

57 follows that S cannot have a little Krull dimension larger then ω1. Thus, we have

k dim S = ω1.

Similar to S in the theorem above, the Steenrod algebra is countable so that k dim Ap 6> ω1. Some future work would be to try and find the exact order type of the little Krull dimension of the Steenrod algebra. One might use methods similar to

∗ those used in Section 3.2.3. Any Hopf subalgebra of Ap leads to a Hopf quotient of

Ap, which, by finding the kernels, is an easy way of finding ideals. One would only need to check if these ideals are prime.

∗ Consider the algebra Bp. Following Proposition 3.4.1, k dim Bp = ω1, which im-

∗ plies that k dim Ap = ω1. Since k dim Ap > ω0, there are two possibilities: either the little Krull dimension of the Steenrod algebra and its dual differ or there are a lot

more prime ideals waiting to be discovered.

3.5 The case p = 2

For the mod 2 Steenrod algebra A2, we don’t have (or need) the initial step of

j passing to B2; we can define our maps ψ (from Corollary 3.2.10) directly on A2.

Then, we can use [25], which states that A2 is a prime ring, in place of Theorem

2.2.1 to show that the little Krull dimension is infinite. Furthermore, if we let Dj,

j for j ≥ 0, denote the ideal generated by the elements Sq1, Sq2,..., Sq2 , then we still

j+1 have Dj = ker ψ . In short, all the results for this chapter hold when p = 2.

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63 Appendix A

Construction of the Steenrod

Squares and Powers

The Kan-Thurston theorem [24] states that there is no loss of generality to only

consider group cohomology instead of the more general construction of cohomology

of topological spaces [7]. With that in mind, we present here the construction of the

Steenrod squares and powers from the viewpoint of group cohomology. We follow

both Benson [7] and Evens [13]. We will simply construct the squares and powers,

and the reader may refer to both Benson and Evens for proofs. Our construction will

work with any group, finite or infinite.

For a topological construction, see [44] or [21]. Steenrod [44] first gives the con-

struction of the squares and powers for a finite complex; the existence of the powers

for more general spaces follows from standard arguments using the naturality of the

powers.

Let R be a commutative ring. Let G be a group and H a subgroup of G with finite index |G : H| = n. Let Σn denote the symmetric group on n elements. The wreath

1 product Σn o H consists of elements (σ; h1, . . . , hn) with σ ∈ Σn and h1, . . . , hn ∈ H,

1See [6]

64 and with multiplication given by

0 0 0 0 0 0
(σ ; h1, . . . , hn)(σ; h1, . . . , hn) = σ σ; hσ(1)h1, . . . , hσ(n)hn .

Let M be an RH-module. We can make a tensor product M ⊗n of n copies of M into a (Σn o H)-module by

(σ; h1, . . . , hn)(m1 ⊗ · · · ⊗ mn) = hσ−1(1)mσ−1(1) ⊗ · · · ⊗ hσ−1(n)mσ−1(n).

Let T = {g1, . . . , gn} be a set of coset representatives of H in G. We have an injective group homomorphism i : G → Σn o H as follows (see [6]): given g ∈ G, we write g · gj = gσ(j)hj for uniquely defined elements σ ∈ Σn and h1, . . . , hn ∈ H depending on g. We set i(g) = (σ; h1, . . . , hn) and use the notation G o H to indicate this imbedding. We note here that a different choice of coset representatives gives rise to a conjugate embedding of G into Σn o H. So, we fix our choice of T as above.

With this imbedding i : G,→ Σn o H (which depends on our choice of T ), we now have an action of G on M ⊗n. The tensor induced module, denoted M h⊗Gi, is defined to be

M h⊗Gi = i∗ M ⊗n
, and it is a G o H-module.2

Similarly, if η : U → k is an RH-projective resolution, then η induces an R (Hn)- projective resolution η⊗n : U ⊗n → k⊗k = k. (See [13].) Here, Hn = H × · · · × H, n

⊗n times. The action of Σn o H on η is given by

ν (σ; h1, . . . , hn)(x1 ⊗ · · · ⊗ xn) = (−1) hσ−1(1)xσ−1(1) ⊗ · · · ⊗ hσ−1(n)xσ−1(n)

2Our notation for the tensor induced module differs from the notation used by both Benson and Evens.

65 where X ν = |xj| |x`| . j<` σ(j)>σ(`)

Here |x| denotes the degree of x. If  : W → k is an RG-projective resolution, then

 ⊗ η⊗n : W ⊗ U ⊗n → k is an R (G o H)-projective resolution. Thus, if M is an H-module, then

H∗ G o H,M h⊗Gi
may be calculated as the cohomology of the complex (see [13])

⊗n
⊗n
∼ ⊗n
HomR(GoH) W ⊗ U ,M = HomRG W, HomRH (U, M) .

Suppose now that R = k is a field. Nakaoka’s Theorem (from [37], see also [7] and [13]) states that

H∗ (G o H, k) ∼= H∗ G, H∗ (H, k)⊗n
.

Following [13], and under the action of G, the ring H∗ (G o H, k) contains the subring G H0 G, H∗ (H, k)⊗n
= H∗ (H, k)⊗n
. If α ∈ Hr (H, k), where r is even, then

α⊗n ∈ H∗ (H, k)⊗n is an invariant under the action of G, and we will denote it 1 o α.

We can define these invariants at the level of cocycles (again see [13]). For a

r kH-resolution U → k, let f : Ur → k be a cocycle representing α ∈ H (H, k). Let  : W → k be the augmentation map. The cocycle

 o f =  ⊗ f ⊗n : W ⊗ U ⊗n → k ⊗ k⊗n = k

66 represents the class 1 o α ∈ Hrn (G o H, k). We note here that 1 o α is well defined and

does not depend on the choice of cocycle f (see [7] and [13] for details).

Evens [13] points out that this construction for 1 o α can also be done when r is

˜ ⊗n odd. One would only need to replace k with k = k , where G, embedded in Σn, permutes the factors and also multiples by the sign of the permutation.

We now introduce the Even’s norm map:

∗ ∗ ∗ normG,H : H (H, k) → H (G, k) defined by normG,H (α) = i (1 o α) ,

where i is the imbedding i : G,→ Σn o H from above. If α is homogeneous of degree

nr r, then normG,H (α) ∈ H (G, k). Though Evens defines this in [13] when r is even, Benson [7] also discusses the case when r is odd.

Now, let k = Fp be the field of p elements. Let G be a group and denote the group

Zp × G by Gp. (In this section, Zp denotes a group.) Following Benson [7], we define

r pr−j r the operations Dj : H (G, Fp) → H (G, Fp) as follows. If x ∈ H (G, Fp), set

X pr M j pr−j normG,Gp (x) = aj ⊗ Dj(x) ∈ H (Gp, Fp) = H (Zp, Fp) ⊗ H (G, Fp) . j j

The direct sum above is the result of the K¨unnethformulas. For odd r, the exact sign of Dj(x) depends on the order of the cosets of G in Gp, and the reader should refer to [7] for details about the ordering.

We are now able to define the Steenrod squares and Steenrod powers (see [7]). If

r p = 2 and x ∈ H (G, Fp), we define

i r+i Sq (x) = Dr−i(x) ∈ H (G, Fp) .

67 r If p is odd and x ∈ H (G, Fp), we define

i c −r r+2(p−1)i P = (−1) (m!) D(p−1)(r−2i)(x) ∈ H (G, Zp)

mr(r+1) (p−1) 2 where c = 1 + 2 and m = / . The choice here of the constant multiplier just ensures that P0 = 1. Benson [7] continues to show that the axioms and properties on Pages 6 and 7 hold for the maps Sqi and Pi. Both Steenrod [44] and Hatcher [21] show that these axioms and properties hold.

68 Appendix B

Hopf Algebras

We briefly define what a Hopf algebra is, along with prerequisite material, and

an interested reader may refer to [1], [35], and [48] for more detail. Many of the

definitions presented below are from [35].

Let k be a field and all tensor products are assumed to be over k.A k-algebra

(with unit) is a k-vector space A together with two k-linear maps, multiplication

m : A⊗A → A and unit u : k → A, such that the following diagrams are commutative:

a) associativity b) unit

m⊗id u⊗id id ⊗u A ⊗ A ⊗ A / A ⊗ A k ⊗ A / A ⊗ A o A ⊗ k K KK ∼ ∼ ss m KK m ss id ⊗m KK ss KK ss  m  K%  yss A ⊗ A / A A

Diagram b) gives the usual identity element in A by setting 1A = u (1k), and the maps labeled ∼ are given by scalar multiplication. We define the twist map

τ : V ⊗ W → W ⊗ V by τ (v ⊗ w) = w ⊗ v. With this map, A is commutative if and only if m ◦ τ = m on A ⊗ A. The algebra is sometimes denoted (A, m, u) to specify the multiplication and unit maps.

Let A and B be algebras, with multiplication mA and mB, and units uA and uB, respectively. A map f : A → B is an algebra morphism if f ◦ mA = mB ◦ (f ⊗ f) and

69 f ◦ uA = uB. By dualizing the diagrams above, we define the following. A k-coalgebra (with

counit) is a k-vector space C together with two k-linear maps, comultiplication ∆ :

C → C ⊗C and counit ε : C → k, such that the following diagrams are commutative:

a) coassociativity b) counit

∆ C / C ⊗ C C K ss KK 1⊗ ss KK ⊗1 ∆ ∆⊗id ss ∆ KK ss KK  id ⊗∆  yssε⊗id  id ⊗ε K% C ⊗ C / C ⊗ C ⊗ C k ⊗ C o C ⊗ C / C ⊗ k

We say that C is cocommutative if τ ◦ ∆ = ∆. The maps labeled with 1⊗ and ⊗1 are given by c 7→ 1 ⊗ c and c 7→ c ⊗ 1, respectively, for all c ∈ C. The coalgebra is sometimes denoted (C, ∆, ε) to specify the comultiplication and counit maps.

Let C and D be coalgebras, with comultiplications ∆C and ∆D, and counits εC and

εD, respectively. A map f : C → D is a coalgebra morphism if ∆D ◦ f = (f ⊗ f) ∆C and if εC = εD ◦ f. A subspace I is a coideal if ∆I ⊆ I ⊗ C + C ⊗ I and if ε(I) = 0. We define here sigma notation, or Sweedler notation. Let C be a coalgebra with comultiplication ∆ : C → C ⊗ C. The sigma notation for ∆ is given as follows: for any c ∈ C, we write X ∆c = c(1) ⊗ c(2).

P In general, we can simply write ∆n−1c = c(1) ⊗ · · · ⊗ c(n). This notation is very useful when dealing with the coassociative property, as can be seen by the example below:

X X ∆2c = (1 ⊗ ∆) ∆c = c(1) ⊗ c(2)(1) ⊗ c(2)(2) = c(1)(1) ⊗ c(1)(2) ⊗ c(2) = (∆ ⊗ 1) ∆c.

The sub-subscripts tell the reader which of the elements, c(1) or c(2), were expanded, but the resulting tensors will be the same because of the coassociativity property.

70 ∗ For a k-vector space V , let V = Homk (V, k) denote the linear dual of V . Let W be another k-vector space. The k-spaces V and V ∗ determine a non-degenerate

bilinear form h , i : V ∗ ⊗ V → k via hf, vi = f(v). If φ : V → W is k-linear, then the

transpose of φ is φ∗ : W ∗ → V ∗, given by

φ∗ (f)(v) = f (φ (v)) ,

for all f ∈ W ∗, v ∈ V .

Let A be a k-algebra. The finite dual of A is

A◦ = {f ∈ A∗ : f (I) = 0, for some ideal I of A such that dim A/I < ∞} .

If A is finite dimensional, then A◦ = A∗. With these definition, we presents the

following well-known relationships between algebras and coalgebras (see [35] for proofs

of the first two facts).

1. If C is a coalgebra, then C∗ is an algebra, with multiplication m = ∆∗ and unit

u = ε∗. If C is cocommutative, then C∗ is commutative.

2. If A is an algebra, then A◦ is a coalgebra, with comultiplication ∆ = m∗ and

counit ε = u∗. If A is commutative, then A◦ is cocommutative.

3. If A is of finite type, then A◦ = A∗ is a coalgebra (see [34]), where (A∗)n = (An)∗.

Here, An denotes the subspace consisting of the n degree elements of A.

A k-space H is a bialgebra if (H, m, u) is an algebra, (H, ∆, ε) is a coalgebra, and either of the following (equivalent) conditions holds:

1. ∆ and ε are algebra morphisms.

2. m and u are coalgebra morphisms.

71 Let H be a bialgebra. We define the convolution product as

(f ? g)(c) = m ◦ (f ⊗ g) (∆c)

for all f, g ∈ Homk (H,H). We call H a Hopf algebra if there exist an element

S ∈ Homk (H,H) which is an inverse to idH under convolution ?; that is

idH ?S = S? idH = u ◦ ε.

The map S is called the antipode of H, and it is unique. Furthermore, the antipode

is an antihomomorphism. If H is a connected (k ∼= H0), graded bialgebra, then H has an antipode and is therefore a Hopf algebra (see, for instance, [22, Proposition

3.8.8]).

A map f : H → K of two Hopf algebras is a Hopf morphism if it is a bialgebra mor-

phism (both an algebra morphism and coalgebra morphism) and f (SH h) = SK f(h). A subspace I of H is a Hopf ideal if it is a biideal (both an ideal and coideal) and

if SI ⊂ I; in this situation, the Hopf quotient H/I is a Hopf algebra with structure

induced from H.

Let H be a finite dimensional (or graded of finite type) Hopf algebra. If K is a

Hopf subalgebra of H, then K∗ is a Hopf quotient of H∗, and if J is a Hopf quotient

of H, then J ∗ is a Hopf subalgebra of H∗.

As an example of a Hopf morphism, we will briefly show a well-known result,

Lemma B.0.1, that the map which sends x to xp is a Hopf endomorphism on certain

Hopf algebras.

Lemma B.0.1. Let (H, m, u, ∆, ε, S) be a graded commutative Hopf algebra over the

field k with positive characteristic p and define the Frobenius φ : H → H given by

x 7→ xp. The map φ is a Hopf endomorphism.

72 Proof. Since H is commutative and char k = p > 0, the map φ is an algebra map.

Indeed, we have (a + b)p = ap + bp and (ab)p = apbp for all a, b, ∈ H. For any graded

Hopf algebra,   α if |h| = 0 ε (h) =  0 if |h| > 0 for all h ∈ H, from which it follows that the equation ε = ε ◦ φ is satisfied. Using that both ∆ and φ are algebra maps, we have, for all h ∈ H,

(∆ ◦ φ)(h) = ∆ (hp) = (∆ (h))p k !p k X X p p = h(1) ⊗ h(2) = h(1) ⊗ h(2) i=0 i=0 k ! X = (φ ⊗ φ) h(1) ⊗ h(2) i=0 = (φ ⊗ φ) ∆ (h) .

Thus, φ is a coalgebra morphism and thus also a bialgebra morphism. Since the antipode S is an antihomomorphism, the equation φ (Sh) = (Sh)p = S (hp) = Sφ (h)

is satisfied so that φ is a Hopf endomorphism.

73 Index of notation

d

Q(E), 17 δ, 5 Q(E)P{R}, 18 ∆, 70 Qi, 17 ε, 70 Pi, 6 λ (W ), 13 Pk(b), 18 σ(a), 27 P{R}, 17 Σn, 64 R ≡ 0 mod n, 20

[a, b], 18 R 6≡ 0 mod n, 20

i An(R), 13 Sq , 6

Ap, 8 Spec R, 12

Ap(n), 20 Sq{R}, 17

∗ Bp, 34 V , 71 cl K dim R, 14 w(a), 27

Dj, 38 X ≡ 0 mod n, 20 G o H, 65

H∗ (X; R), 3 H∗ (X; R), 3

K dim M, 15

k dim R, 13

`(a), 27

M(R,S), 28

74 Index

algebra, 69 homotopy-equivalent, 5 antipode, 72 Hopf algebra, 9, 72

Hopf morphism, 72 bialgebra, 71 binary sequences, 17 Krull dimension, 15

Bockstein homomorphism, 5 classical, 14

little, 13 chain complex, 3 classical Krull dimension, 14 length coalgebra, 70 of a Steenrod algebra element, 27 cochain complex, 3 of a well-ordered set, 13 cocommutative, 70 little Krull dimension, 13 cohomology, 3 Milnor sequences, 17 cohomology operation, 4 monoform module, 16, 56 comultiplication, 70 multinomial coefficient, 21 convolution product, 72 counit, 70 naturality, 4 critical module, 16 prime ideal, 11 cup product, 4 prime ring, 11

finite type, 9 sigma notation, 70

Gabriel dimension, 16 stable, 4 Steenrod algebra, 8 homology, 3 suspension, 4 homotopic, 5

75 twist map, 69

Weyl algebra, 13 width, 27 wreath product, 64

76