THE STEENROD PROBLEM OF REALIZING POLYNOMIAL COHOMOLOGY RINGS

KASPER K. S. ANDERSEN AND JESPER GRODAL

Abstract. In this paper we completely classify which graded polynomial R–algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring R satisfying mild conditions. In the fundamental case R = ℤ, our result states that the only polynomial cohomology rings over ℤ which can occur, are tensor products of copies of H^{∗}(ℂP^{∞}; ℤ) ≅ ℤ[x_{2}], H^{∗}(B SU(n); ℤ) ≅ ℤ[x_{4}, x_{6}, …, x_{2n}], and H^{∗}(B Sp(n); ℤ) ≅ ℤ[x_{4}, x_{8}, …, x_{4n}], confirming an old conjecture. Our classification extends Notbohm’s solution for R = 𝔽_{p}, p odd. Odd degree generators, excluded above, only occur if R is an 𝔽_{2}–algebra and in that case the recent classification of 2–compact groups by the authors can be used instead of the present paper. Our proofs are short and rely on the general theory of p–compact groups, but not on classification results for these.

1. Introduction

In 1960, N. E. Steenrod [35] asked which graded polynomial rings occur as the cohomology ring of a space. In this paper we answer this question.

Theorem 1.1. If H^{∗}(X; ℤ) is a finitely generated polynomial algebra over ℤ, for some space X, then H^{∗}(X; ℤ) is isomorphic, as a graded algebra, to a tensor product of copies of H^{∗}(ℂP^{∞}; ℤ) ≅ ℤ[x_{2}], H^{∗}(B SU(n); ℤ) ≅ ℤ[x_{4}, x_{6}, …, x_{2n}], and H^{∗}(B Sp(n); ℤ) ≅ ℤ[x_{4}, x_{8}, …, x_{4n}].

This has been a standard conjecture since the 1970s. In fact we solve the question over any ground ring R, satisfying mild assumptions. As usual, ℂP^{∞} denotes infinite complex projective space, and B SU(n) and B Sp(n) are the classifying spaces of the special unitary group and the symplectic group respectively. We first describe the background of the problem and previously known results.

Steenrod proved in his 1960 paper [35] that if H^{∗}(X; ℤ) ≅ ℤ[x] then |x| = 2 or 4, as a consequence of his newly introduced Steenrod operations, settling the one-variable case over ℤ. In contrast, when taking coefficients in a field R of characteristic zero every graded polynomial ring on even degree generators can occur, since H^{∗}(K(ℤ, 2n); R) ≅ R[x_{2n}], as proved by Serre in his thesis [33, Ch. VI, §3, Prop. 4]. Note that by anti-commutativity, all generators have to be in even degrees, whenever 2 ≠ 0 in R.

Restrictions on realizable polynomial rings over 𝔽_{p} were studied in the 1960s and 1970s, mainly by ingenious use of Steenrod operations (see e.g., [40, 41, 38, 36]). Key progress was made with the discovery of a connection with finite p–adic reflection groups by Sullivan [39], Clark–Ewing [16], and others in the early 1970s. Further significant progress was made with the connection to the category of unstable modules over the Steenrod algebra starting with Wilkerson [45] and Adams-Wilkerson [1] in the late 1970s and early 1980s. Aguadé in his 1981 paper [2] used Adams-Wilkerson’s work [1] to obtain a partial version of Theorem 1.1, under the strong additional hypothesis that all generators are in different degrees (referring to the general result in Theorem 1.1 as “the standing conjecture”; see e.g., [3] for a historical account and references as of 1981). Following additional partial results, Notbohm [29] in 1999 used the more powerful techniques of p-compact groups, as developed by Dwyer-Wilkerson [21, 19] and others since the early 1990s, to obtain a full answer in the case where the ground ring is 𝔽_{p} for p odd. Note however that 2-primary information is needed for Theorem 1.1 as e.g., H^{∗}(B Spin(2n); ℤ[\frac{1}{2}]) ≅ ℤ[\frac{1}{2}][x_{4}, x_{8}, …, x_{4(n−1)}, x_{2n}].

2000 Mathematics Subject Classification. Primary: 55N10; Secondary: 55R35, 55R40.

The second named author was partially supported by NSF grant DMS-0354633, the Alfred P. Sloan Foundation, and the Danish Natural Science Research Council.

In the present paper we obtain a solution for any commutative Noetherian ring R of finite Krull dimension, under the assumption that all generators are in even degrees. The even degrees assumption is automatic unless R is an 𝔽_{2}–algebra; in that case a solution however follows as a consequence of our recent classification of 2–compact groups; see [6, Thm. 1.4]. We use standard results about p-compact groups in the present paper, but remark that classification results for these, as well as Notbohm’s results in the case R = 𝔽_{p}, p odd, are not used. We also note that Theorem 1.1 can alternatively be deduced from our aforementioned result [6, Thm. 1.4], but the argument we present here is significantly more direct, as explained below. By the type of a graded polynomial ring we mean the multiset (i.e., unordered tuple) of degrees of its generators. The general result which we prove here is as follows.

Theorem 1.2. Let R be a commutative Noetherian ring of finite Krull dimension, P the set of prime numbers p which are not units in R, and A a graded polynomial R–algebra in finitely many variables, all in positive even degrees. Then, there exists a space Y such that A ≅ H^{∗}(Y; R) as graded R–algebras if and only if for each prime p ∈ P the type of A is a union of multisets of degrees in Tables 1–3 which occur at the prime p.

Lie group  Degrees  Occur for
S^{1}  2  all p
SU(n)  4, 6, …, 2n  all p
Sp(n)  4, 8, …, 4n  all p
Spin(2n)  4, 8, …, 4(n − 1), 2n  p ≥ 3
G_{2}  4, 12  p ≥ 3
F_{4}  4, 12, 16, 24  p ≥ 5
E_{6}  4, 10, 12, 16, 18, 24  p ≥ 5
E_{7}  4, 12, 16, 20, 24, 28, 36  p ≥ 5
E_{8}  4, 16, 24, 28, 36, 40, 48, 60  p ≥ 7

Table 1. Lie group cases

W  Degrees  Conditions  Occur for ; \begin{matrix}G(m, r, n) \\ D_{2m} \\ C_{m} \end{matrix} \begin{matrix}2m, 4m, …, 2(n − 1)m, 2mn/r \\ 4, 2m \\ 2m \end{matrix} \begin{matrix}n ≥ 2, m ≥ 3, r |m \\ m ≥ 5, m ≠ 6 \\ m ≥ 3 \end{matrix} p ≡ 1 (mod m) p ≡ ±1 (mod m) p ≡ 1 (mod m)

Table 2. Exotic cases, first part (family 2a, 2b, and 3)

W Degrees  Occur for ; \begin{matrix}G_{8} \\ G_{9} \\ G_{12} \\ G_{14} \\ G_{16} \\ G_{17} \\ G_{20} \\ G_{21} \\ G_{22} \end{matrix} \begin{matrix}16, 24 \\ 16, 48 \\ 12, 16 \\ 12, 48 \\ 40, 60 \\ 40, 120 \\ 24, 60 \\ 24, 120 \\ 24, 40 \end{matrix} p ≡ 1  (mod 4) p ≡ 1  (mod 8) p ≡ 1, 3 (mod 8) p ≡ 1, 19 (mod 24) p ≡ 1  (mod 5) p ≡ 1  (mod 20) p ≡ 1, 4 (mod 15) p ≡ 1, 49 (mod 60) p ≡ 1, 9 (mod 20) W  Degrees  Occur for ; \begin{matrix}G_{23} \\ G_{24} \\ G_{29} \\ G_{30} \\ G_{31} \\ G_{32} \\ G_{33} \\ G_{34} \end{matrix} \begin{matrix}4, 12, 20 \\ 8, 12, 28 \\ 8, 16, 24, 40 \\ 4, 24, 40, 60 \\ 16, 24, 40, 48 \\ 24, 36, 48, 60 \\ 8, 12, 20, 24, 36 \\ 12, 24, 36, 48, 60, 84 \end{matrix} p ≡ 1, 4 (mod 5) p ≡ 1, 2, 4 (mod 7), p ≠ 2 p ≡ 1  (mod 4) p ≡ 1, 4 (mod 5) p ≡ 1  (mod 4) p ≡ 1  (mod 3) p ≡ 1  (mod 3) p ≡ 1  (mod 3)

Table 3. Exotic cases, continued

The space Y is rarely unique; see Remark 3.3. We briefly explain the origin of Tables 1–3: Table 1 lists the simple simply connected compact Lie groups and S^{1}, and the primes for which the homology of the group is p–torsion free, together with the degrees of the polynomial generators of the 𝔽_{p}–cohomology of their classifying spaces, but with Spin(2n + 1), p odd, left out for simplicity, since its 𝔽_{p}–cohomology agrees with that of Sp(n). This information goes back to Borel [12] (see Proposition 4.3).

To explain Tables 2 and 3 we recall some facts about reflection groups. The Shephard– Todd–Chevalley theorem [11, Thm. 7.2.1] says that, for a field K of characteristic zero and a K–vector space V, a finite group W ≤ GL(V) is a reflection group if and only if the ring K[V]^{W} of W –invariant polynomial functions on V is a graded polynomial ring, where we grade K[V] by giving the elements of V^{∗} degree 2. The degrees of W ≤ GL(V) is defined to be the type of this invariant ring. (We warn the reader that degrees are sometimes defined as half of what is the convention in this paper.) Shephard–Todd [34] classified the finite irreducible complex reflection groups as falling into 3 infinite families (labeled 1–3) and 34 sporadic cases (labeled G_{i}, 4 ≤ i ≤ 37). Clark–Ewing [16] (with additional clarification by Dwyer–Miller–Wilkerson [19, Prop. 5.5, Pf. of Thm. 1.5]) used this to give a classification of finite ℚ_{p}–reflection groups. This classification says that for a given prime p, a finite ℂ–reflection group gives rise to a unique ℚ_{p}–reflection group if and only if its character field embeds in ℚ_{p}, and all finite ℚ_{p}–reflection groups arise this way. Tables 2 and 3 essentially list the finite irreducible ℚ_{p}–reflection groups which are exotic, i.e., those whose character field is not ℚ, except the ℚ_{2}–reflection group G_{24}; but for simplicity we have also removed the groups G_{i} for i = 4, 5, 6, 7, 10, 11, 13, 15, 18, 19, 25, 26, and 27, whose multiset of degrees is readily obtainable as a union of the multiset of degrees of the remaining groups, at any prime for which they exist as ℚ_{p}–reflection groups.

To illustrate Theorem 1.2, if P = ∅ (i.e., if R is a ℚ–algebra) there are no restrictions, recovering Serre’s result discussed earlier. If 2 ∈ P, the possible types of A are unions of the multisets {2}, {4, 6, …, 2n}, and {4, 8, …, 4n}, in particular recovering Theorem 1.1. If P = {3} the type of A is a union of {2}, {4, 6, …, 2n}, {4, 8, …, 4n}, {4, 8, …, 4(n − 1), 2n}, {4, 12}, and {12, 16}, and similar lists can easily be compiled for P = {p} for any individual prime p. For arbitrary P, it is a simple combinatorial problem to check if any given multiset of degrees is realizable, as is described in the following corollary.

Corollary 1.3. For any multiset of degrees {2d_{1}, …, 2d_{r}}, there exist integers a_{1}, …, a_{m} and N such that the following conditions are equivalent for any commutative Noetherian ring R of finite Krull dimension:

(1) The graded polynomial R–algebra R[x_{1}, …, x_{r}], with |x_{i}| = 2d_{i}, is isomorphic to H^{∗}(Y; R) for some space Y.

(2) Every prime number p which is not a unit in R satisfies p ≡ a_{i} (mod N) for some i.

We give an algorithm for finding N and a_{1}, …, a_{m} in the course of the proof. Finding explicit generators for the commutative monoid of realizable degree lists for a given collection of primes P is a harder combinatorial problem in general.

The proof of the main theorem spans three short sections. The first two sections reduce the problem to p–compact groups: In Section 2 we show that if H^{∗}(Y; R) is a polynomial algebra then the same holds for H^{∗}(Y; 𝔽_{p}) for all prime numbers p which are not units in R, and hence that the 𝔽_{p}–completion Y _{ p̂} is the classifying space of a p–compact group—the proof has a small ring theoretic twist due to finite-type problems coming from the fact that we are working with cohomology. Conversely Section 3 constructs a space Y with polynomial R– cohomology from a collection of p–compact groups with compatible cohomology rings, for p any non-unit in R. Finally, in Section 4, we determine the graded polynomial 𝔽_{p}–algebras on even degree generators which occur as the 𝔽_{p}–cohomology of the classifying space of a p–compact group. The key Proposition 4.1 reduces this question to separate questions for classifying spaces of compact Lie groups and classifying spaces of exotic p–compact groups (without using classification results on p–compact groups). These are then easily solved, the first one essentially by Borel, and the second just using earlier existence results for exotic p–compact groups. Combining the earlier steps now easily establishes the main result, Theorem 1.2, and its corollaries. We remark that Notbohm’s approach to the case R = 𝔽_{p}, p odd, in [29] differs from the one given here in Section 4. Notbohm instead uses an argument relying on various earlier fairly elaborate case-by-case calculations of invariant rings, which in particular does not extend to the case of 𝔽_{2}.

Notation and recollections: We stress that by a graded polynomial algebra we mean a graded algebra generated by a set of homogeneous elements, which are algebraically independent (i.e., we do not allow odd degree exterior generators). For the theorems of the paper a space can be taken to mean any topological space, though for the purposes of the proofs we may without loss of generality restrict ourselves to CW–complexes (or even simplicial sets), as we will often tacitly do. We constantly use the theory of p–compact groups, and here recall some pertinent facts, referring to e.g., [21, 18, 7, 6] for more information. A p–compact group consists of a triple (X, BX, e: X ⟶^{≃} ΩBX) such that BX is an 𝔽_{p}–complete space and X has finite 𝔽_{p}–cohomology. Proposition 2.2 recalls that spaces with polynomial 𝔽_{p}–cohomology ring always come from p–compact groups. A standard result, central to this paper, states that for a connected p–compact group X, H^{∗}(BX; 𝔽_{p}) is a polynomial algebra concentrated in even degrees if and only if H^{∗}(X; ℤ_{p}) is torsion free if and only if H^{∗}(BX; ℤ_{p}) ⟶^{≅} H^{∗}(BT; ℤ_{p})^{W_{X}}, where T and W_{X} are respectively the maximal torus and the Weyl group of X. (Parts of this result are due to Borel and parts to Dwyer–Miller–Wilkerson; see [7, Thm. 12.1]).) In this case the degrees of the generators of H^{∗}(BX; 𝔽_{p}) equal the degrees of W_{X}, which act as a reflection group on π_{2}(BT)⊗ℚ with invariant ring ℚ_{p}[π_{2}(BT)⊗ℚ]^{W_{X}} ≅ H^{∗}(BX; ℤ_{p})⊗ℚ [21, Thm. 9.7]—this provides the link between polynomial cohomology rings concentrated in even degrees, torsion free p–compact groups, and finite ℤ_{p}– and ℚ_{p}–reflection groups. A couple of times in the proofs we use the word root datum for which we refer to [6, Section 8]; although we do not use anything which is not already in [22] in a slightly different language.

Acknowledgments: We thank Bill Dwyer for helpful conversations and Hans-Bjørn Foxby for reminding us of what he taught us as undergraduates about finitistic dimensions of commutative Noetherian rings.

2. From R to 𝔽_{p}

In this section we show how a polynomial cohomology ring over R produces polynomial cohomology rings over 𝔽_{p} for the prime numbers p which are not units in R.

Proposition 2.1. Suppose that R is a commutative Noetherian ring of finite Krull dimension, and Y is a space such that H^{∗}(Y; R) is a polynomial R–algebra in finitely many variables, each in positive degree. If a prime p is not a unit in R then H^{∗}(Y; 𝔽_{p}) is a polynomial 𝔽_{p}–algebra with generators in the same degrees and Y _{ p̂} is the classifying space of a p–compact group.

The proof will occupy the rest of this section. The proof is straightforward if e.g. R ⊆ ℚ (using the exact sequence 0 → R ⟶^{p} R → 𝔽_{p} → 0), but in general it is a bit more subtle, with complications arising even for R = ℤ/4. First we explain the well known fact that spaces with polynomial 𝔽_{p}–cohomology ring give rise to p–compact groups.

Proposition 2.2. Let Y be a space. If H^{∗}(Y; 𝔽_{p}) is a finitely generated polynomial algebra over 𝔽_{p}, then Y _{ p̂} ≃ BX for a p–compact group X. If furthermore H^{1}(Y; 𝔽_{p}) = 0, then X is connected.

Proof. By considering the fiber of the canonical map Y → K(H_{1}(Y; 𝔽_{p}), 1) one easily reduces to the case where H_{1}(Y; 𝔽_{p}) = 0 (cf. [6, Pf. of Thm. 1.4]), and this is in fact also the only case relevant for our main theorem. By [13, Prop. VII.3.2] Y is 𝔽_{p}–good and Y _{ p̂} is 𝔽_{p}– complete and simply connected. In particular the Eilenberg–Moore spectral sequence of the path-loop fibration of Y _{ p̂} converges [17] and shows that H^{∗}(ΩY _{ p̂}; 𝔽_{p}) is finite dimensional over 𝔽_{p}. Hence Y _{ p̂} is the classifying space of a p–compact group, which is connected since Y _{ p̂} is simply connected. □

Next we deal with the finiteness restrictions usually associated with universal coefficient theorems in cohomology, which is where the ring theoretic properties of R enter in a slightly subtle way. Recall that the finitistic flat dimension of a commutative ring R is defined to be the supremum over the flat dimensions of all R–modules with finite flat dimension.

Lemma 2.3. Let R be a commutative ring, and C_{∗} a chain complex of R–modules concentrated in non-positive degrees such that C_{n} and H_{n}(C_{∗}) are flat for all n. If the finitistic flat dimension of R is finite, then H_{n}(C_{∗}) ⊗_{R} M ⟶^{≅} H_{n}(C_{∗} ⊗_{R} M) for any R–module M.

Proof. Let fd(−) denote the flat dimension of an R–module. By the Künneth formula [42, Thm. 3.6.1] it suffices to show that im(C_{n} ⟶^{d_{n}} C_{n−1}) is flat for all n. To see this, first note that by the short exact sequence

0 → im(d_{n+1}) → ker(d_{n}) → H_{n}(C_{∗}) → 0,

we have fd(im(d_{n+1})) = fd(ker(d_{n})), since H_{n}(C_{∗}) is flat, cf. [42, Ex. 4.1.2(3)]. Now, since C_{n} is flat, the short exact sequence

0 → ker(d_{n}) → C_{n} → im(d_{n}) → 0

shows that either im(d_{n}) and im(d_{n+1}) are flat or fd(im(d_{n+1})) = fd(ker(d_{n})) = fd(im(d_{n}))−1. Since im(d_{n}) = 0 is flat for n positive, it follows that either im(d_{n}) is flat for all n or there exists an n_{0} such that im(d_{n}) is flat for n ≥ n_{0} and fd(im(d_{n})) = n_{0} − n for n ≤ n_{0}. Since R has finite finitistic flat dimension the second possibility cannot occur, and we are done. □

Remark 2.4. A way to view the assumptions in the lemma is that they ensure convergence of the Künneth spectral sequence, derived from the spectral sequence of a double complex (see [42, §5.6]). An alternative assumption ensuring convergence would be that M has finite flat dimension over R. In complete generality the Künneth spectral sequence need not converge, the standard counterexample being R = ℤ/4, C_{∗} = ⋯ ⟶^{2} ℤ/4 ⟶^{2} ℤ/4 ⟶^{2} ⋯ and M = ℤ/2.

The preceding lemma, together with a result in commutative ring theory, gives us the following result about the cohomology of spaces.

Proposition 2.5. Let R be a commutative Noetherian ring of finite Krull dimension and p a prime number. If Y is a space such that H^{n}(Y; R) is finite free over R for each n then

H^{∗}(Y; R) ⊗_{R} R/p ⟶^{≅} H^{∗}(Y; R/p) ⟵^{≅} H^{∗}(Y; 𝔽_{p}) ⊗_{𝔽_{p}} R/p

as R/p–algebras.

Proof. Since R is a commutative Noetherian ring, the singular cochain complex C^{∗}(Y; R) is a complex of flat modules, since over a commutative Noetherian ring arbitrary products of flat modules are again flat (i.e., commutative Noetherian rings are coherent; see [15, Thm. 2.1] or [14, Ex. VI.4, p. 122]). By a result of Auslander–Buchsbaum [8, Thm. 2.4], the finitistic flat dimension of R is bounded above by its Krull dimension (in fact they differ by at most 1 by a result of Bass [10, Cor. 5.3]), and in particular the assumptions imply that it is finite. Hence, since obviously C^{∗}(Y; R) ⊗_{R} R/p ≅ C^{∗}(Y; R/p), Lemma 2.3 implies that H^{∗}(Y; R) ⊗_{R} R/p ⟶^{≅} H^{∗}(Y; R/p). Now

H^{∗}(Y; R/p) = H(C^{∗}(Y; R/p)) = H(Hom_{𝔽_{p}}(C_{∗}(Y; 𝔽_{p}), R/p)) ⟶^{≅} Hom_{𝔽_{p}}(H_{∗}(Y; 𝔽_{p}), R/p)

as R/p–modules, and in particular H_{∗}(Y; 𝔽_{p}) is finite in each degree. Hence H^{∗}(Y; 𝔽_{p}) ⊗_{𝔽_{p}}

R/p ⟶^{≅} Hom_{𝔽_{p}}(H_{∗}(Y; 𝔽_{p}), R/p), which combined with the previous isomorphisms gives the result. □

Lemma 2.6. Let A be a graded connected 𝔽_{p}–algebra which is finite dimensional in each degree, and B an 𝔽_{p}–algebra (viewed as a graded algebra concentrated in degree 0). If A ⊗_{𝔽_{p}} B is a graded polynomial B–algebra, then A is a graded polynomial 𝔽_{p}–algebra with generators in the same degrees.

Proof. By reducing B modulo a maximal ideal, we can without restriction assume that B is a field. Since B is a flat 𝔽_{p}–module, Q(A) ⊗_{𝔽_{p}} B ⟶^{≅} Q(A ⊗_{𝔽_{p}} B), where Q(·) denotes the module of indecomposable elements. Picking a basis for the 𝔽_{p}–vector space Q(A) and lifting these to A produces a set of generators for A as an 𝔽_{p}–algebra, which maps to a set of generators for A⊗_{𝔽_{p}}B as a B–algebra. By construction they form a B–basis for Q(A⊗_{𝔽_{p}}B). Hence they have to be algebraically independent, since A⊗_{𝔽_{p}}B is assumed to be a graded polynomial B–algebra (to see this note that mapping polynomial generators of A ⊗_{𝔽_{p}} B to the constructed generators will produce an epimorphism A ⊗_{𝔽_{p}} B → A ⊗_{𝔽_{p}} B which also has to be a monomorphism since A ⊗_{𝔽_{p}} B is finite dimensional over B in each degree). Since B is a faithfully flat 𝔽_{p}–module, the generators are algebraically independent in A as well, and hence A is a graded polynomial 𝔽_{p}–algebra. □

Proof of Proposition 2.1. By Proposition 2.5 and Lemma 2.6, H^{∗}(X; 𝔽_{p}) is a polynomial algebra with generators in the same degrees as H^{∗}(X; R). It hence follows from Proposition 2.2 that X is 𝔽_{p}–good and X_{p̂} is the classifying space of a p–compact group. □

3. From 𝔽_{p} to R

In this section we show how spaces with polynomial 𝔽_{p}–cohomology ring at different primes can be glued together, provided that the generators have the same degrees.

Proposition 3.1. Let P be a set of prime numbers, J the set of prime numbers not in P, and {2d_{1}, …, 2d_{r}} a fixed multiset. Suppose that for each p ∈ P there exists a space B_{p} such that H^{∗}(B_{p}; 𝔽_{p}) is a polynomial algebra with generators in degrees {2d_{1}, …, 2d_{r}}. Then there exists a simply connected ℤ[J^{−1}]–local space Y such that π_{n}(Y) is finite over ℤ[J^{−1}] for each n, and such that H^{∗}(Y; ℤ[J^{−1}]) is a polynomial algebra over ℤ[J^{−1}] with generators in degrees {2d_{1}, …, 2d_{r}}. More generally, if R is a commutative ring with R = R[J^{−1}], then H^{∗}(Y; R) is also a polynomial R–algebra with generators in the same degrees.

Before the proof we need the following lemma, which is similar to a lemma used by Baker– Richter [9, Prop. 2.4] in a different context.

Lemma 3.2. Let J be a set of prime numbers and set R = ℤ[J^{−1}]. Suppose that A is a graded connected R–algebra, finite free over R in each degree. If A⊗_{R} ℚ is a graded polynomial ℚ–algebra and A ⊗_{R} ℤ_{p} is a graded polynomial ℤ_{p}–algebra for all primes p ∉ J, then A is a graded polynomial R–algebra as well (in the same degrees as both A⊗_{R} ℤ_{p}, p ∉ J, and A⊗_{R} ℚ).

Proof. The proof is similar to the proof of Lemma 2.6. Since ℤ_{p} is flat over R for p ∉ J, we have Q(A) ⊗_{R} ℤ_{p} ⟶^{≅} Q(A ⊗_{R} ℤ_{p}), which is free over ℤ_{p} in each degree by assumption. Hence Q(A) does not have p–torsion when p ∉ J, and since Q(A) is finite over R in each degree, we conclude by the structure theorem for finite ℤ[J^{−1}]–modules that Q(A) is finite free over R in each degree. Likewise, since ℚ is flat over R, Q(A) ⊗_{R} ℚ ⟶^{≅} Q(A ⊗_{R} ℚ). Pick homogeneous elements in A which project to an R–basis of Q(A). Clearly these elements generate A as an R–algebra. They are also algebraically independent, since they, via the isomorphism Q(A) ⊗_{R} ℚ ⟶^{≅} Q(A ⊗_{R} ℚ), give rise to generators of the graded polynomial ring A ⊗_{R} ℚ. Hence A is a graded polynomial R–algebra, with generators in the same degrees as both A ⊗_{R} ℚ and A ⊗_{R} ℤ_{p}, p ∉ J. □

Proof of Proposition 3.1. The proof is reminiscent of [5, Pf. of Lem. 1.2], to which we also refer. By Proposition 2.2, (B_{p})_{p̂} is 𝔽_{p}–complete with the same 𝔽_{p}–cohomology as B_{p}, so we may assume that B_{p} is 𝔽_{p}–complete. Set K = K(ℚ, 2d_{1}) × ⋯ × K(ℚ, 2d_{r}). We will define Y as a homotopy pull-back

(3.1)  \xymatrix{Y \ar[r] \ar[d] & ∏_{p∈P} B_{p} \ar[d] \\ K \ar[r]^{f} & (∏_{p∈P} B_{p})_{ℚ}}

where the map f has to be constructed.

Since B_{p} is the classifying space of a connected p–compact group by Proposition 2.2, π_{∗}(B_{p}) and H^{∗}(B_{p}; ℤ_{p}) are finite over ℤ_{p} in each degree. It follows that H^{∗}(B_{p}; ℤ_{p}) is also a polynomial algebra over ℤ_{p}, with generators in the same degrees. By an essentially classical result (see [5, Thm. 2.1])

π_{n}(B_{p}) ≅ π_{n−1} ((S^{2d_{1}−1} × ⋯ × S^{2d_{r}−1})_{p̂}) ,

for p > max{d_{1}, ⋯, d_{r}} and similarly for the ℚ–localization, so

π_{n}((∏_{p∈P} B_{p})_{ℚ}) ≅ π_{n−1} (S^{2d_{1}−1} × ⋯ × S^{2d_{r}−1}) ⊗ (∏_{p∈P} ℤ_{p}) ⊗ ℚ.

Since these homotopy groups are concentrated in even degrees, (∏_{p∈P} B_{p})_{ℚ} is a product of

Eilenberg–Mac Lane spaces, as follows from standard obstruction theory [43, Ch. IX].

Hence we can define f: K → (∏_{p∈P} B_{p})_{ℚ} to be the canonical map induced by the unique ring map ℚ → (∏_{p∈P} ℤ_{p}) ⊗ ℚ. By the Mayer–Vietoris sequence in homotopy applied to the pull-back square (3.1) we see that

π_{n}(Y) = ( ⨁_{i,2d_{i}=n} ℤ[J^{−1}]) ⊕ (⨁_{p∈P} Tor (ℚ/ℤ, π_{n}(B_{p}))) .

In particular π_{n}(Y) is finite over ℤ[J^{−1}], since for a fixed n, π_{n}(B_{p}) is p–torsion free, for p large [33, Ch. V, §5, Prop. 4]. Note also that Y is ℤ[J^{−1}]–local by [13, Prop. V.3.3].

For p ∈ P, the pull-back square above shows that the map Y → B_{p} induces an isomorphism on mod p homology so H^{∗}(Y; ℤ_{p}) ⟵^{≅} H^{∗}(B_{p}; ℤ_{p}). Hence H^{∗}(Y; ℤ_{p}) is a polynomial ring over ℤ_{p} with generators in the same degrees as H^{∗}(B_{p}; 𝔽_{p}). Since H_{∗}(Y; ℤ[J^{−1}]) is finite over ℤ[J^{−1}] in each degree, the universal coefficient theorem [42, 3.6.5] (applied to the ℤ[J^{−1}]–module ℤ_{p}) shows that H_{∗}(Y; ℤ[J^{−1}]) is in fact finite free over ℤ[J^{−1}] in each degree, and hence so is H^{∗}(Y; ℤ[J^{−1}]). In particular

H^{∗}(Y; ℤ[J^{−1}]) ⊗_{ℤ[J^{−1}]} ℤ_{p} ≅ H^{∗}(Y; ℤ_{p}) ≅ H^{∗}(B_{p}; ℤ_{p}).

Lemma 3.2 now shows that H^{∗}(Y; ℤ[J^{−1}]) is a polynomial ℤ[J^{−1}]–algebra as wanted. Finally, let R be a commutative ring with R = R[J^{−1}]. Since H_{∗}(Y; ℤ[J^{−1}]) is finite free over ℤ[J^{−1}] in each degree, the universal coefficient theorem again implies that H^{∗}(Y; R) ≅ H^{∗}(Y; ℤ[J^{−1}]) ⊗_{ℤ[J^{−1}]} R, proving the last claim. □

Remark 3.3. The space Y in Proposition 3.1 is rarely unique as a ℤ[J^{−1}]–local space, due to a multitude of factors: First, given a multiset of degrees and a prime p, there may be several choices of p–compact groups with the same degrees; this is a finite, essentially combinatorial, problem (see [6, Thm. 1.5] and [29, Cor. 1.8]).

Secondly, the classifying space BX of a connected p–compact group may lift to many ℤ_{(p)}– homotopy types of spaces of finite type over ℤ_{(p)}, with examples at least going back to Bousfield and Mislin [30, Appendix]. This uses a variant of Wilkerson’s double coset formula [44, Proof of Thm. 3.8] resulting from the pull-back K → (BX)_{ℚ} ← BX, for K a rational space with the same rational degrees as BX. In fact a more careful analysis along the same lines reveals that BX has a unique lifting as above if and only if X is a p–compact torus or X is a product of rank one p–compact groups and the rational degrees {2d_{1}, …, 2d_{r}} has the property that none of the degrees can be written as a non-negative integral linear combination of the others. Moreover, if the lift is non-unique there are uncountably many lifts.

Thirdly, given a set of ℤ_{(p)}–homotopy types Y_{p} as in the second step, one for each prime in P (i.e., p ∉ J), with the same rational degrees, Zabrodsky mixing associated to the pullback K → (∏_{p∈P} Y_{p})_{ℚ} ← ∏_{p∈P} Y_{p} in general produces infinitely many ℤ[J^{−1}]–homotopy types. Again a more careful analysis with the corresponding double coset formula reveals that if 2 ≤ |P| < ∞, then there are countably infinitely many lifts, unless (Y_{p})_{p̂} satisfies the conditions of the second step for each p ∈ P, in which case there is only one lift. If P is infinite then there are uncountably many lifts unless the rational degrees are {2, 2, …, 2}, i.e., the torus case. (See also e.g., [32, 44, 30, 27, 26] for related information.)

4. p–compact groups with polynomial 𝔽_{p}–cohomology concentrated in even degrees

In this section we provide the necessary results on p–compact groups, and use this, together with the results of the previous sections, to prove Theorem 1.2 and its corollaries. The key step is the following result, which in fact also holds true without the even degree assumption [6, Thm. 1.4], though we do not know a proof in that generality which does not go via the classification of 2–compact groups [6, Thm. 1.2].

Proposition 4.1. If X is a p–compact group such that H^{∗}(BX; 𝔽_{p}) is a finitely generated polynomial algebra over 𝔽_{p} concentrated in even degrees, then we can write

H^{∗}(BX; 𝔽_{p}) ≅ H^{∗}(BG; 𝔽_{p}) ⊗ H^{∗}(BY; 𝔽_{p})

as 𝔽_{p}–algebras (and in fact as algebras over the Steenrod algebra), for G a compact connected Lie group and Y a product of exotic p–compact groups.

We first need a lemma.

Lemma 4.2. Let X be a p–compact group such that H^{∗}(BX; 𝔽_{p}) is a finitely generated polynomial algebra over 𝔽_{p} concentrated in even degrees. Then C_{X}(ν) is connected for any elementary abelian p–subgroup ν: BE → BX of X.

Proof. Let ν: BE → BX be an elementary abelian p–subgroup of X. By [22, Pf. of Thm. 8.1] we have H^{∗}(BC_{X}(ν); 𝔽_{p}) ≅ T_{E,ν^{∗}}(H^{∗}(BX; 𝔽_{p})), where T_{E,ν^{∗}} denotes the component of Lannes’ T –functor corresponding to ν^{∗} [25, 2.5.2]. Since T_{E,ν^{∗}} preserves objects concentrated in even degrees [25, Prop. 2.1.3] as well as finitely generated graded polynomial algebras [24, Thm. 1.2], we conclude that H^{∗}(BC_{X}(ν); 𝔽_{p}) is also a finitely generated polynomial algebra over 𝔽_{p} concentrated in even degrees and in particular C_{X}(ν) is connected by Proposition 2.2. □

Proof of Proposition 4.1. By Proposition 2.2, X is connected. The classification of ℤ_{p}–root data [6, Thm. 8.1] says that the ℤ_{p}–root datum D_{X} of X can be written as a direct product D_{X} ≅ (D_{G} ⊗ ℤ_{p}) × D_{1} × … × D_{n}, where G is a compact connected Lie group and the D_{i} are exotic ℤ_{p}–root data. By the product decomposition theorem [23, Thm. 1.4] we have an associated decomposition BX ≃ BX′ × BX_{1} × … × BX_{n}, where X′ is a connected p–compact group with D_{X′} ≅ D_{G} ⊗ ℤ_{p} and the X_{i} are exotic p–compact groups with D_{X_{i}} ≅ D_{i}.

We finish the proof by showing that H^{∗}(BX′; 𝔽_{p}) ≅ H^{∗}(BG; 𝔽_{p}) as 𝔽_{p}–algebras. Lemma 4.2 shows that C_{X′}(ν) is connected for every elementary abelian p–subgroup ν: BE → BX′ of X′. If ν is toral, i.e., if ν factors through the maximal torus BT → BX′ of X′, a result of Dwyer–Wilkerson [22, Thm. 7.6] (see also [6, Prop. 8.4(3)]) shows that π_{0}(C_{X′}(ν)) can be computed from the root datum D_{X′}. Since D_{X′} ≅ D_{G_{p̂}} it follows that C_{G_{p̂}}(ν) is also connected for every toral elementary abelian p–subgroup ν of G_{p̂}. If E is an elementary abelian subgroup of G, and i: E → G is the inclusion we have C_{G}(E)_{p̂} ≅ C_{G_{p̂}}(Bi), so C_{G}(E) is connected for every elementary abelian p–subgroup E of G which is contained in a maximal torus. By results of Steinberg [37, Thms. 2.27 and 2.28] and Borel [12, Thm. B] it follows that G does not have p–torsion, or equivalently that H^{∗}(BG; 𝔽_{p}) is a polynomial algebra concentrated in even degrees. But, by [7, Thm. 12.1], this implies that we have isomorphisms

H^{∗}(BX′; ℤ_{p}) ⟶^{≅} H^{∗}(BT; ℤ_{p})^{W} ⟵^{≅} H^{∗}(BG; ℤ_{p})

since the Weyl group W and its action of T is determined by D_{X′} ≅ D_{G_{p̂}}. In particular H^{∗}(BX′; 𝔽_{p}) ≅ H^{∗}(BG; 𝔽_{p}) as algebras over the Steenrod algebra, as wanted. □

The following proposition determining the compact Lie groups whose classifying space has polynomial cohomology basically goes back to Borel [12, Thm. 2.5].

Proposition 4.3. A finitely generated graded polynomial 𝔽_{p}–algebra A concentrated in even degrees is isomorphic to H^{∗}(BG; 𝔽_{p}) for some compact connected Lie group G if and only if the type of A is union of multisets of degrees occurring in Table 1.

Proof. In [12, Thm. 2.5] Borel determined for all simple simply connected compact Lie groups G, the prime numbers p such that H^{∗}(BG; 𝔽_{p}) is a polynomial algebra concentrated in even degrees, and this is exactly the data indicated in Table 1. (G = Spin(2n + 1), p ≥ 3, is omitted since H^{∗}(B Spin(2n + 1); 𝔽_{p}) ≅ H^{∗}(B Sp(n); 𝔽_{p}) as 𝔽_{p}–algebras.)

Assume now that G is an arbitrary compact connected Lie group such that H^{∗}(BG; 𝔽_{p}) is a polynomial 𝔽_{p}–algebra concentrated in even degrees. There exists a fibration BG′ → BG → BS, where G′ denotes the commutator subgroup of G and S is a torus. Since π_{2}(BG) → π_{2}(BS) is surjective by the long exact sequence in homotopy, H^{2}(BS; 𝔽_{p}) → H^{2}(BG; 𝔽_{p}) is injective. Because H^{∗}(BS; 𝔽_{p}) is generated by elements in degree 2 and H^{∗}(BG; 𝔽_{p}) is polynomial, the homomorphism H^{∗}(BS; 𝔽_{p}) → H^{∗}(BG; 𝔽_{p}) is injective and H^{∗}(BG; 𝔽_{p}) is free over H^{∗}(BS; 𝔽_{p}). Hence the Eilenberg–Moore spectral sequence for the fibration collapses and we get an isomorphism H^{∗}(BG; 𝔽_{p}) ≅ H^{∗}(BS; 𝔽_{p}) ⊗ H^{∗}(BG′; 𝔽_{p}) of 𝔽_{p}–algebras. This shows that H^{∗}(BG′; 𝔽_{p}) is a polynomial algebra concentrated in even degrees as well, so we can assume that G has finite fundamental group. But, since H^{3}(BG; 𝔽_{p}) = 0, π_{1}(G) cannot contain p–torsion, so we can assume that G is simply connected. Hence G splits as a product of simple simply connected compact Lie groups, and in particular the type of H^{∗}(BG; 𝔽_{p}) is a union of multisets of degrees from Table 1. □

The next proposition covers the exotic case.

Proposition 4.4. A finitely generated graded polynomial 𝔽_{p}–algebra A concentrated in even degrees is isomorphic to H^{∗}(BY; 𝔽_{p}) for Y a product of exotic p–compact groups if and only if the type of A is union of multisets of degrees occurring in Tables 2 and 3.

Proof. The multiset of degrees of every exotic ℚ_{p}–reflection group, p odd, is realized as the type of a polynomial 𝔽_{p}–cohomology ring; this is the culmination of the work of a number of people: Sullivan [39, p. 166–167], Clark–Ewing [16], Quillen [31, §10], Zabrodsky [46, 4.3], Aguadé [4] and Notbohm–Oliver [28]. See also [29] and [7, §7] for a unified treatment. In particular the multisets of degrees in Tables 2 and 3 are all realized.

Conversely suppose that X is an exotic p–compact group, with H^{∗}(BX; 𝔽_{p}) a polynomial 𝔽_{p}–algebra concentrated in even degrees. As used before, this implies that H^{∗}(BX; ℤ_{p}) is also polynomial and H^{∗}(BX; ℤ_{p}) ≅ H^{∗}(BT; ℤ_{p})^{W}. In particular, the degrees of H^{∗}(BX; 𝔽_{p}) agree with the degrees of H^{∗}(BT; ℤ_{p})^{W} ⊗ ℚ, i.e., with the degrees of the ℚ_{p}–reflection group W, which by assumption is exotic. For p = 2 the classification of finite ℚ_{2}–reflection groups says that there is only one exotic ℚ_{2}–reflection group, namely G_{24}. By the classification of ℤ_{p}–root data [6, Thm. 8.1], there is a unique ℤ_{2}–root datum associated to G_{24}, namely D_{X} = D_{DI(4)}, the root datum of the exotic 2–compact group DI(4) constructed by Dwyer– Wilkerson [20]. Using the formula for the component group of a centralizer [22, Thm. 7.6] (see also [6, Prop. 8.4(3)]) one sees that C_{X}(_{2}T̆) is not connected, where _{2}T̆ denotes the maximal elementary abelian 2–subgroup of the discrete torus T̆. This contradicts Proposition 2.2, so the classifying space of a 2–compact group with this root datum cannot have polynomial 𝔽_{2}– cohomology ring concentrated in even degrees, and we must have p odd. (Alternatively, one can see directly that the polynomial algebra 𝔽_{2}[x_{8}, x_{12}, x_{28}] cannot be an unstable algebra over the mod 2 Steenrod algebra A_{2}: For degree reasons the ideal generated by x_{8} and x_{12} is closed under A_{2}, so the quotient 𝔽_{2}[x_{28}] would have to be an unstable algebra over A_{2}. However this is not possible since 28 is not a power of 2.)

It only remains to show that for p odd, the multiset of degrees of any exotic ℚ_{p}–reflection group which does not occur in Tables 2 and 3, can be written as a union of the multiset of degrees of those which do: The cases G_{4}, G_{5}, G_{6}, G_{7}, G_{10}, G_{11}, G_{13}, G_{15}, G_{18}, G_{19}, G_{25}, G_{26} and G_{27} are left out of Table 3 since these are covered by the groups G(6, 3, 2), G(6, 1, 2), C_{4} × C_{12}, C_{12} × C_{12}, G(12, 1, 2), C_{24} × C_{24}, G_{8}, G(12, 1, 2), G(30, 1, 2), C_{60} × C_{60}, G(6, 2, 3), G(6, 1, 3) and C_{6} × G_{20} respectively. □

Proof of Theorem 1.2. Suppose that we are in the setup of the theorem, with a fixed commutative Noetherian ring R of finite Krull dimension. If H^{∗}(Y; R) is a polynomial R–algebra, then Proposition 2.1 shows that for p ∈ P, H^{∗}(Y; 𝔽_{p}) is a polynomial 𝔽_{p}–algebra on generators in the same degrees and Y _{ p̂} is the classifying space of a p–compact group. Hence by Propositions 4.1, 4.3 and 4.4, these degrees satisfy the restrictions of Theorem 1.2, as wanted.

Conversely, suppose that we are given a multiset of degrees, which satisfy the degree restrictions for all p ∈ P. Then for each p ∈ P there exists by Propositions 4.3 and 4.4 a space B_{p} such that H^{∗}(B_{p}; 𝔽_{p}) is a polynomial 𝔽_{p}–algebra in those degrees. By Proposition 3.1 we can construct a space Y, such that H^{∗}(Y; R) is a polynomial R–algebra with generators in the same degrees. □

Proof of Corollary 1.3. By inspection a given multiset of degrees occurs at most finitely many times in Tables 1–3. It follows that a given finite multiset L can only be decomposed in finitely many ways as a union of multisets of degrees from Tables 1–3 (ignoring for now restrictions on primes p). We next have to determine the prime numbers p for which each of these decompositions can occur.

For short, say that a set of integers is listable if it can be written as a union of congruence classes modulo some number N. Finite intersections and finite unions of listable sets are again listable and there is an obvious algorithm for computing them as listable sets. Notice that each entry in the tables occur for a prime number p if and only if p belongs to some listable set. (This follows directly from the tables, together with the observation that for any integer k we have p ≥ k if and only if p belongs to a union of congruence classes modulo N = ∏_{l<k, l prime} l; e.g., p ≥ 5 iff p ≡ ±1 mod 6.) A given decomposition of L (as a union of multisets of degrees corresponding to entries in the tables), can be used at a prime p if and only if each of the entries occur at p, and by the above this is equivalent to p belonging to an explicit listable set depending only on the decomposition. Since there are only finitely many decompositions of L as above, we conclude that the set of primes p for which there exists some decomposition of L at p is a listable set. The result now follows from Theorem 1.2. □

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Department of Mathematical Sciences, University of Aarhus

   E-mail address: kksa@imf.au.dk

Department of Mathematical Sciences, University of Copenhagen E-mail address: jg@math.ku.dk