LOW DIMENSIONAL HOMOLOGY OF LINEAR GROUPS OVER HENSEL LOCAL RINGS

KEVIN P. KNUDSON

Abstract. We prove that if R is a Hensel local ring with infinite residue field k, the natural map H_{i}(GL_{n}(R), ℤ/p) → H_{i}(GL_{n}(k), ℤ/p) is an isomorphism for i ≤ 3, p ≠ char k. This implies rigidity for H_{i}(GL_{n}), i ≤ 3, which in turn implies the Friedlander–Milnor conjecture in positive characteristic in degrees ≤ 3.

A fundamental question in the homology of linear groups is that of rigidity: given a smooth affine curve X over an algebraically closed field k and closed points x, y on X, do the corresponding specialization homomorphisms

s_{x}, s_{y} : H_{∗}(G(k[X]), ℤ/p) ⟶ H_{∗}(G(k), ℤ/p)

coincide? Here, G is a reductive algebraic group and p is a prime not equal to the characteristic of k. The answer is yes when X is the affine line and G = SL_{n}, GL_{n}, P GL_{n} since the inclusion G(k) → G(k[t]) induces an isomorphism

H_{∗}(G(k), ℤ) ⟶ H_{∗}(G(k[t]), ℤ)

(see [6]) and the map is split by evaluation at any x ∈ 𝔸^{1}.

Rigidity in algebraic K-theory has spectacular consequences, including the calculation of the K-theory of algebraically closed fields and the solution of the Friedlander–Milnor conjecture for the stable general linear group GL. Similarly, a proof of rigidity for G(k[X]) would imply the Friedlander–Milnor conjecture for G (see Section 5).

Rigidity would follow if one could prove the following stronger result. Let X be a smooth curve over an algebraically closed field k and let x be a closed point on X. Denote by O_{x}^{h} the henselization of O_{x}.

Conjecture. The inclusion G(k) → G(O_{x}^{h}) induces an isomorphism

H_{∗}(G(k), ℤ/p) ⟶ H_{∗}(G(O_{x}^{h}), ℤ/p)

for p ≠ char k.

Date: December 4, 1997.

1991 Mathematics Subject Classification. 20G10.

Key words and phrases. rigidity, Hensel local ring, Friedlander–Milnor conjecture, Bloch group.

Supported by an NSF Postdoctoral Fellowship, grant no. DMS-9627503.

In Section 5 we sketch a proof of rigidity assuming this conjecture (this argument is well-known). Our results will allow us to deduce the Friedlander– Milnor conjecture in positive characteristic for H_{i}(GL_{n}), i ≤ 3. The conjecture is known to be true for H_{3}(SL_{2}(ℂ)) by the work of Sah [9].

In this note we study H_{∗}(G(R), ℤ/p) for G = SL_{n} and G = GL_{n}, where R is a Hensel local k-algebra and k is an infinite field. Our results are far from complete. We prove the following result.

Theorem. The inclusion GL_{n}(k) → GL_{n}(R) induces an isomorphism

H_{i}(GL_{n}(k), ℤ/p) ⟶ H_{i}(GL_{n}(R), ℤ/p)

for i ≤ 3.

Note that this map is split injective for all i; the only issue is surjectivity. The expert reader by now will have noticed that this theorem contains

only one new case, namely H_{3}(GL_{2}). The above map is in fact an isomorphism for i ≤ n by a combination of Suslin’s stability theorem [8] and rigidity in K-theory. However, we show a bit more than the theorem states. We first prove that the map

H_{2}(P GL_{2}(k), ℤ/p) ⟶ H_{2}(P GL_{2}(R), ℤ/p)

is an isomorphism and use this to deduce the result for H_{2}(GL_{2}). Moreover, our approach is different from that used by Suslin [11] to compute H_{•}(GL(R), ℤ/p). We then study Bloch groups to obtain the result for H_{3}(GL_{2}).

The corresponding statement for G = SL_{n} and i = 0, 1 is almost obvious. However, we provide an alternate proof for i = 1 in a special case as follows. Let F be any field and denote by 𝔪 the maximal ideal (t_{1}, t_{2}, …, t_{m}) of F [[t_{1}, t_{2}, …, t_{m}]]. Consider the short exact sequence

1 ⟶ C_{n,m} ⟶ SL_{n}(F [[t_{1}, t_{2}, …, t_{m}]]) ⟶ SL_{n}(F) ⟶ 1.  (1)

Here, the group C_{n,m} consists of those matrices which are congruent to the identity modulo 𝔪. We have the following result.

Proposition. If n ≥ 3, then

H_{1}(C_{n,m}, ℤ) = ⨁_{i=1}^{m} 𝔰𝔩_{n}(F).

When n = 2, assume further that char F ≠ 2 and F ≠ 𝔽_{3}. Then

H_{1}(C_{2,m}, ℤ) = ⨁_{i=1}^{m} 𝔰𝔩_{2}(F).

When p is a prime distinct from the characteristic of k, the proposition implies that H_{1}(C_{n,m}, ℤ/p) = 0 unless n = 2, char k = 2 or F = 𝔽_{3}. In this case, however, one can still conclude that H_{1}(C_{2,m}, ℤ/p) = 0 by other methods. One expects that H_{•}(C_{n,m}, ℤ/p) vanishes in all positive degrees. We provide evidence for this in Section 2.

Conventions. Throughout, k denotes a field and p is a prime distinct from the characteristic of k.

1. The Congruence Subgroup

Recall that the group C_{n,m} is the kernel of the natural map

SL_{n}(F [[t_{1}, t_{2}, …, t_{m}]]) ⟶^{mod 𝔪} SL_{n}(F)

where 𝔪 is the ideal (t_{1}, t_{2}, …, t_{m}). Define a sequence of subgroups C_{n,m}^{i} by

C_{n,m}^{i} = {X ∈ SL_{n}(F [[t_{1}, t_{2}, …, t_{m}]]) : X ≡ I mod 𝔪^{i}}.

If no confusion can result, we usually drop the subscripts from the notation. Note that each C^{i} is a normal subgroup as it is the kernel of the map

SL_{n}(F [[t_{1}, t_{2}, …, t_{m}]]) ⟶ SL_{n}(F [[t_{1}, t_{2}, …, t_{m}]]/𝔪^{i}).

For each i, define a homomorphism

ρ_{i} : C_{n,m}^{i} ⟶ ⨁_{λ} 𝔰𝔩_{n}(F)

by

ρ_{i}(I + ∑_{λ} t_{1}^{k_{1}}t_{2}^{k_{2}} ⋯ t_{m}^{k_{m}}X_{λ} + ⋯) = (X_{λ})_{λ}

where λ = (k_{1}, k_{2}, …, k_{m}) is an m-partition of i (i.e., λ is an m-tuple of nonnegative integers whose sum is i) and (X_{λ})_{λ} denotes the element of ⨁ 𝔰𝔩_{n}(F) given by the various X_{λ}. Note that ρ_{i} is well-defined since the equation

1 = det X ≡ 1 + ∑_{λ} t_{1}^{k_{1}} ⋯ t_{m}^{k_{m}}trX_{λ} mod 𝔪^{i+1}

implies that tr X_{λ} = 0 for each partition λ. One checks easily that ρ_{i} is a surjective group homomorphism with kernel C_{n,m}^{i+1} (see [7] for an analogous result for congruence subgroups of SL_{n}(ℤ)). Hence, for each i we have an isomorphism

C_{n,m}^{i}/C_{n,m}^{i+1} ≅ ⨁_{λ} 𝔰𝔩_{n}(F).

Remark. The above discussion remains valid for any simple algebraic group G. The quotients C^{i}/C^{i+1} are direct sums of copies of the lie algebra 𝔤. We have chosen to work with SL_{n} just to fix ideas.

Lemma 1.1. For each i, j, [C^{i}, C^{j}] ⊆ C^{i+j}.

Proof. This follows easily by writing out XY X^{−1}Y^{−1} for X ∈ C^{i}, Y ∈ C^{j} and noting that the inverse of I + ∑ Z_{λ} + ⋯ is I − ∑ Z_{λ} + · · ·.

For any group G, denote by Γ^{•} the lower central series of G.

Corollary 1.2. For each i, Γ^{i} ⊆ C_{n,m}^{i}.

Proof. The series C_{n,m}^{•} is a descending central series and as such contains the lower central series.

We now recall the following theorem of Klingenberg [5].

Theorem 1.3. If A is a local ring then for n ≥ 3 the only normal subgroups of SL_{n}(A) are the congruence subgroups. If n = 2, the same is true as long as the residue characteristic of A is not 2 or the residue field is not 𝔽_{3}.

Corollary 1.4. The filtration C_{n,m}^{•} is the lower central series.

Proof. Note that the groups Γ^{i} are characteristic subgroups of C_{n,m} and hence are normal in SL_{n}(F [[t_{1}, …, t_{m}]]). Since there are clearly elements in Γ^{i} − C_{n,m}^{i+1}, it follows that Γ^{i} = C_{n,m}^{i} for all i.

Corollary 1.5. For all n ≥ 3, H_{1}(C_{n,m}, ℤ) = ⨁_{i=1}^{m} 𝔰𝔩_{n}(F). If char F ≠ 2 or F ≠ 𝔽_{3}, then H_{1}(C_{2,m}, ℤ) = ⨁_{i=1}^{m} 𝔰𝔩_{2}(F).

Proof. Since C_{n,m}^{2} = Γ_{n,m}^{2}, we have

H_{1}(C_{n,m}, ℤ) = C_{n,m}/Γ^{2} = C_{n,m}/C_{n,m}^{2} = ⨁_{i=1}^{m} 𝔰𝔩_{n}(F).

Remark. This corollary holds for power series rings over more general rings (for example, rings of the form ℤ[J^{−1}] for a set of primes J).

Corollary 1.6. If p is a prime different from the characteristic of F, then

H_{1}(C_{n,m}, ℤ/p) = 0.

Proof. This follows from the previous corollary except in the case n = 2, char F = 2 or F = 𝔽_{3}. However, one could also prove this corollary directly by noting that each element of C_{n,m} has a pth root in C_{n,m} and hence that H_{1}(C_{n,m}, ℤ/p) = H_{1}(C_{n,m}, ℤ) ⊗ ℤ/p = 0.

Corollary 1.7. For each n,

H_{1}(SL_{n}(F [[t_{1}, …, t_{m}]]), ℤ/p) ≅ H_{1}(SL_{n}(F), ℤ/p).

Proof. This follows easily by considering the Hochschild–Serre spectral sequence associated to the extension (1).

2. Conjectures about H_{∗}(C_{n,m}, ℤ/p)

The isomorphism

H_{•}(SL_{n}(k[[t_{1}, t_{2}, …, t_{m}]]), ℤ/p) ≅ H_{•}(SL_{n}(k), ℤ/p)

would follow easily if one could prove the following.

Conjecture 2.1. For all l > 0, H_{l}(C_{n,m}(k[[t_{1}, t_{2}, …, t_{m}]]), ℤ/p) = 0.

We have just proved the case l = 1. We make the following observations.

2.1. Continuous homology. For each i ≥ 1, denote by L_{n,m}^{i} the kernel of the homomorphism

SL_{n}(k[t_{1}, …, t_{m}]) ⟶ SL_{n}(k[t_{1}, …, t_{m}]/(t_{1}, … t_{m})^{i}).

The group C_{n,m} is the inverse limit of the nilpotent groups L_{n,m}/L_{n,m}^{i}. One checks that these groups have no homology with ℤ/p coefficients by noting that the successive graded quotients L_{n,m}^{j}/L_{n,m}^{j+1} of L_{n,m}/L_{n,m}^{i} are kvector spaces (see for example [11]); an iterated use of the Hochschild–Serre spectral sequence then shows that L_{n,m}/L_{n,m}^{i} is ℤ/p–acyclic. It follows that the “continuous homology”

\underleftarrow{lim} H_{•}(C_{n,m}/C_{n,m}^{i}, ℤ/p) = \underleftarrow{lim} H_{•}(L_{n,m}/L_{n,m}^{i}, ℤ/p)

vanishes and that

H_{•}(SL_{n}(k[t_{1}, t_{2}, …, t_{m}]/𝔪^{i}), ℤ/p) ≅ H_{•}(SL_{n}(k), ℤ/p)

for all i ≥ 1.

2.2. Large acyclic subgroups. When m = 1, it is easy to see that

H_{•}(L_{n,1}, ℤ/p) = 0

(see [6]) and the group L_{n,1} is a subgroup of C_{n,1}.

2.3. Product spaces. The group C_{n,m} is the subgroup of ∏_{i} L_{n,m}/L_{n,m}^{i} consisting of coherent sequences. A theorem of P. Goerss [4] implies that this product is ℤ/p–acyclic.

2.4. Cosimplicial replacement. We have seen that C_{n,m} is the inverse limit of the nilpotent groups L_{n,m}/L_{n,m}^{i}. Let X_{i} be a K(L_{n,m}/L_{n,m}^{i}, 1) space and consider the tower of fibrations

X_{2} ← X_{3} ← X_{4} ← ⋯

The homotopy inverse limit of this tower is a K(C_{n,m}, 1)-space; denote it by X. Computing the homology of homotopy inverse limits is an important problem in homotopy theory, and has been studied intensely by Bousfield [2], Goerss [4], Shipley [10], and others.

The standard technique is to consider the cosimplicial replacement of the tower. This is a cosimplicial space X^{•} whose nth space X^{n} is a certain product of the spaces X_{i} of the tower (see Bousfield–Kan [1, XI, Sec. 5]). Given a cosimplicial space X^{•}, there is an associated homology spectral sequence [2]

E_{m,t}^{2} = π^{m}H_{t}(X^{•}; A)

for any abelian group A. Here, the term on the right is the mth cohomotopy of the cosimplicial abelian group H_{t}(X^{•}; A). The main problem is that of convergence of this spectral sequence. Under certain conditions, the spectral sequence converges to the homology of the total space of X^{•}. In the case of the cosimplicial replacement of a tower of fibrations, the total space is the homotopy inverse limit of the tower.

In the case of C_{n,m}, we see that the cosimplicial replacement is particularly nice. Its spaces X^{n} are certain products of the X_{i} and by Goerss’ theorem each of these is ℤ/p–acyclic. It follows that the associated homology spectral sequence has E^{2}–term

E_{m,t}^{2} = \begin{cases}ℤ/p & (m, t) = (0, 0) \\ 0 & (m, t) ≠ (0, 0). \end{cases}

The question is what, if anything, is this spectral sequence converging to? Unfortunately, all known convergence theorems do not apply to the situation at hand. Most of these require some hypotheses on the spaces X^{n} such as nilpotency (which we do not have in this case—we would need the groups L_{n,m}/L_{n,m}^{i} to have bounded nilpotence degree). A proof that the spectral sequence converges to the homology of the total space of X^{•} in this case would immediately show that C_{n,m} is ℤ/p–acyclic.

Given all of this, it is difficult to imagine that C_{n,m} has any ℤ/p homology. Speaking heuristically, the group C_{n,m} is built up by successively attaching copies of 𝔰𝔩_{n}(k) and since this group is ℤ/p–acyclic, one would expect that C_{n,m} is as well.

3. Spectral Sequences and the Second Homology Groups

Hereafter, R denotes a Hensel local k-algebra, and 𝔪 denotes the maximal ideal of R. We have R/𝔪 = k.

We begin by considering the group P GL_{2}(R). We will show that the inclusion P GL_{2}(k) → P GL_{2}(R) induces an isomorphism on second homology with coefficients in ℤ/p.

Let A be a local ring with maximal ideal 𝔪 and residue field F, which we assume to be infinite. Recall that a column vector v ∈ A^{2} is unimodular if the ideal generated by its entries is A itself. Denote by \overline{v} the vector in F^{2} obtained by reducing the entries of v modulo 𝔪. We say that a collection v_{1}, v_{2}, …, v_{k} of unimodular vectors is in general position if the collection \overline{v}_{1}, \overline{v}_{2}, …, \overline{v}_{k} is in general position (i.e., the matrix determined by any pair of them is invertible).

Construct a simplicial set S_{•} as follows. The nondegenerate p–simplices are collections v_{0}, v_{1}, …, v_{p} of projective equivalence classes of unimodular vectors which are in general position (this amounts to saying that the v_{i} are distinct closed points in ℙ^{1}). Denote by C_{•}(A) the associated simplicial chain complex. A proof of the following may be found in Nesterenko–Suslin [8].

Lemma 3.1. The augmented complex C_{•}(A) →^{ϵ} ℤ → 0 is acyclic.

Note that P GL_{2}(A) acts transitively on C_{i}(A) for i ≤ 2 (i.e., P GL_{2}(A)

acts 3-transitively on points in ℙ^{1}(A)). Denote by 0 the vector ( \begin{matrix}1 \\ 0 \end{matrix} ); by ∞ the vector ( \begin{matrix}0 \\ 1 \end{matrix} ); and by 1 the vector ( \begin{matrix}1 \\ 1 \end{matrix} ). Then each orbit of

the action of P GL_{2}(A) on C_{p}(A) has a unique representative of the form (0, ∞, 1, v_{1}, …, v_{p−2}). Since each v_{i} is in general position with 0, ∞ and 1, we see that both entries of v_{i} must be units in A. Since we are using projective equivalence classes of vectors, we may then assume that the vector v_{i} has the

form ( \begin{matrix}1 \\ α_{i} \end{matrix} ) where α_{i} is a unit in A. Moreover, the α_{i} satisfy the additional

conditions that 1 − α_{i} ∈ A^{×} (since v_{i} is in general position with 1) and α_{i} − α_{j} ∈ A^{×} for i ≠ j (since v_{i} is in general position with v_{j}).

Denote by E_{•,•}^{•}(A) the spectral sequence associated to the action of the group P GL_{2}(A) on C_{•}(A). This spectral sequence converges to the homology of P GL_{2}(A) and has E^{1}–term

E_{p,q}^{1} = ⨁_{σ∈Σ_{p}} H_{q}(G_{σ})

where Σ_{p} is a set of representatives of the orbits of the action of P GL_{2}(A) on C_{p}(A) and G_{σ} is the stabilizer of σ in P GL_{2}(A). Note that the stabilizers of the P GL_{2}(A) action are

G_{0} = B(A)/D(A),  G_{(0,∞)} = T (A)/D(A),  G_{(0,∞,1)} = D(A)/D(A)

where B(A) is the upper triangular subgroup, T (A) is the diagonal subgroup, and D(A) is the subgroup of scalar matrices. The group D(A)/D(A) is also the stabilizer of each orbit (0, ∞, 1, v_{1}, …, v_{p−2}). Moreover, we have H_{•}(B(A)/D(A), ℤ) ≅ H_{•}(T (A)/D(A), ℤ) (see [8]). It follows that our spectral sequence E_{•,•}^{•}(A) has E^{1}–term (with ℤ/p coefficients)

H_{•}(A^{×}, ℤ/p) H_{•}(A^{×}, ℤ/p) H_{•}(\{1\}, ℤ/p) ⨁_{\begin{matrix}α∈A^{×} \\ 1−α∈A^{×} \end{matrix}} H_{•}(\{1\}, ℤ/p) ⋯

and converges to H_{•}(P GL_{2}(A), ℤ/p).

Now consider the cases A = k and A = R where k is an infinite field. The inclusion P GL_{2}(k) → P GL_{2}(R) induces an injective map of spectral sequences

E_{•,•}^{•}(k) ⟶ E_{•,•}^{•}(R).

Lemma 3.2. The inclusion k^{×} → R^{×} induces an isomorphism

H_{•}(k^{×}, ℤ/p) ⟶ H_{•}(R^{×}, ℤ/p).

Proof. If G is an abelian group, we have an isomorphism

⋀_{ℤ/p}^{•}(G ⊗ ℤ/p) ⊗ Γ_{•}(_{p}G) ⟶^{≅} H_{•}(G, ℤ/p),

where _{p}G is the subgroup of G killed by p and Γ_{•} is a divided power algebra. The lemma will follow if we can show that

k^{×} ⊗ ℤ/p ≅ R^{×} ⊗ ℤ/p  and  _{p}k^{×} ≅ _{p}R^{×} .

We have a natural homomorphism

π : R^{×} ⟶ k^{×}/(k^{×})^{p}

given by composing reduction modulo 𝔪 with the projection k^{×} → k^{×}/(k^{×})^{p}. This map is clearly surjective; we need only check that the kernel coincides with (R^{×})^{p}. This is an immediate consequence of Hensel’s Lemma. Suppose x ∈ R^{×} maps to 0 under π, and consider the equation t^{p} − x = 0 in R[t]. Since π(x) = 0, the reduced equation has a solution in k. Since R is Henselian and p is invertible in R, there exists a y ∈ R with y^{p} − x = 0. That is, x ∈ (R^{×})^{p}.

Note that _{p}k^{×} is the subgroup of pth roots of unity in k. A similar application of Hensel’s Lemma shows that this coincides with _{p}R^{×}. This completes the proof of the lemma.

Corollary 3.3. For each n, the natural map

H_{1}(GL_{n}(k), ℤ/p) ⟶ H_{1}(GL_{n}(R), ℤ/p)

is an isomorphism.

Corollary 3.4. The map of spectral sequences

E_{p,q}^{1}(k) ⟶ E_{p,q}^{1}(R)

is an isomorphism for p ≤ 2.

Corollary 3.5. The natural map

H_{2}(P GL_{2}(k), ℤ/p) ⟶ H_{2}(P GL_{2}(R), ℤ/p)

is an isomorphism.

Proof. The E^{2}–terms of the spectral sequences look as follows:

E^{2} : \begin{matrix}E_{0,2}^{2} & E_{1,2}^{2} & 0 \\ 0 & 0 & 0 \\ ℤ/p & 0 & 0 & E_{3,0}^{2} \end{matrix}

By considering the commutative diagram

E_{3,0}^{3}(k) →^{d^{3}} E_{0,2}^{2}(k) → H_{2}(P GL_{2}(k), ℤ/p) → 0
↓  ||  ↓
E_{3,0}^{3}(R) →^{d^{3}} E_{0,2}^{2}(R) → H_{2}(P GL_{2}(R), ℤ/p) → 0

we see that the map

H_{2}(P GL_{2}(k), ℤ/p) ⟶ H_{2}(P GL_{2}(R), ℤ/p)

is an isomorphism.

Corollary 3.6. The natural map

H_{2}(GL_{2}(k), ℤ/p) → H_{2}(GL_{2}(R), ℤ/p)

is an isomorphism.

Proof. The Hochschild–Serre spectral sequences associated to the extensions

1 → k^{×} → GL_{2}(k) → P GL_{2}(k) → 1
↓ ↓ ↓
1 → R^{×} → GL_{2}(R) → P GL_{2}(R) → 1

are isomorphic at E_{p,q}^{2} for 0 ≤ p ≤ 2, 0 ≤ q ≤ 2. This yields a commutative diagram

H_{3}(P GL_{2}(k), ℤ/p) →^{d^{3}} E_{0,2}^{3}(k) → H_{2}(GL_{2}(k), ℤ/p) → ⋯
↓  ||  ↓
H_{3}(P GL_{2}(R), ℤ/p) →^{d^{3}} E_{0,2}^{3}(R) → H_{2}(GL_{2}(R), ℤ/p) → ⋯
⋯ → H_{2}(P GL_{2}(k), ℤ/p) → 0
||
⋯ → H_{2}(P GL_{2}(R), ℤ/p) → 0

The result follows from the Five Lemma.

Corollary 3.7. The natural map

H_{2}(GL_{n}(k), ℤ/p) → H_{2}(GL_{n}(R), ℤ/p)

is an isomorphism for all n.

Proof. Consider the commutative diagram

H_{2}(GL_{2}(k), ℤ/p) ⟶^{≅} H_{2}(GL_{2}(R), ℤ/p)
↓  ↓
H_{2}(GL_{n}(k), ℤ/p) ⟶ H_{2}(GL_{n}(R), ℤ/p)

The vertical arrows are isomorphisms by Suslin’s stability theorem [8].

Remark. The last two corollaries actually follow from the corresponding statement for H_{2}(GL) [11]. Our approach provides an alternate proof of this fact and also has the advantage of proving the corresponding statement for P GL_{2}.

Remark. Since the map E_{p,q}^{1}(k) → E_{p,q}^{1}(R) is an isomorphism for p ≤ 2, to show that H_{•}(P GL_{2}(k), ℤ/p) ≅ H_{•}(P GL_{2}(R), ℤ/p) it would suffice to show that the chain complexes

D_{•}(k) = E_{3,0}^{1}(k) ← E_{4,0}^{1}(k) ← ⋯

and

D_{•}(R) = E_{3,0}^{1}(R) ← E_{4,0}^{1}(R) ← ⋯

are quasi-isomorphic. One way to do this is to write down a contracting homotopy for the quotient complex Q_{•} = D_{•}(R)/D_{•}(k). Note that the complex Q_{•} must be acyclic in the case R = k[t_{1}, …, t_{m}]/𝔪^{l} for each l ≥ 2 (see 2.1). One would hope to be able to write down a contracting homotopy which is independent of l (or more generally, which is independent of the fact that R is a truncated polynomial ring, but is really just a local k-algebra). However, this seems to be a very difficult question (even in the simplest case m = 1, l = 2). In fact, so far we have been unable to write down a contracting map in the first case Q_{0} → Q_{1}.

4. Bloch Groups and Third Homology

We now turn our attention to the computation of H_{3}(GL_{2}(R), ℤ/p). The case R = k was treated fully by Suslin in his beautiful paper [12]. We recall the following theorem.

Theorem 4.1. There are exact sequences

H_{3}(GM_{2}(k), ℤ) ⟶ H_{3}(GL_{2}(k), ℤ) ⟶ B(k) ⟶ 0

and

π_{3}^{0}(BGM(k)^{+}) ⟶ K_{3}(k) ⟶ B(k) ⟶ 0.

Here, B(k) denotes the Bloch group of k, GM(k) denotes the group of monomial matrices over k, and π_{3}^{0}(BGM(k)^{+}) denotes the kernel of the canonical map

π_{3}(BGM(k)^{+}) ⟶ π_{3}(BΣ^{+})

induced by the projection GM(k) → Σ where Σ is the infinite symmetric group (Σ = ⋃_{n≥2} Σ_{n}).

The amazing fact is that the above theorem holds for any local k-algebra, as we now describe.

For any local ring A with infinite residue field k, denote by D(A) the free abelian group with basis [x], where x, 1−x ∈ A^{×} and define a homomorphism

ϕ : D(A) ⟶ A^{×} ⊗ A^{×}

by ϕ([x]) = x ⊗ (1 − x). Denote by σ the involution of A^{×} ⊗ A^{×} given by σ(x ⊗ y) = −y ⊗ x, and by (A^{×} ⊗ A^{×})_{σ} the quotient of A^{×} ⊗ A^{×} by the action of σ. Define a group 𝔭(A) by

𝔭(A) = D(A)/⟨[x] − [y] + [y/x] − [(1 − x^{−1})/(1 − y^{−1})] + [(1 − x)/(1 − y)]⟩.

One checks easily (see Lemma 1.1 of [12]) that ϕ induces a homomorphism \overline{ϕ} : 𝔭(A) → (A^{×} ⊗ A^{×})_{σ}. By definition, the Bloch group, B(A), is the kernel of \overline{ϕ}.

Consider the action of GL_{2}(A) on the chain complex C_{•}(A) of the previous section. The resulting spectral sequence is studied by Suslin in Section 2 of [12]. The main result of that section is the existence of the exact sequence

H_{3}(GM_{2}(k), ℤ) ⟶ H_{3}(GL_{2}(k), ℤ) ⟶ B(k) ⟶ 0

(Theorem 2.1). The proof of Theorem 2.1 goes through word for word with k replaced by A. Hence we have the following result.

Proposition 4.2. There is an exact sequence

H_{3}(GM_{2}(A), ℤ) ⟶ H_{3}(GL_{2}(A), ℤ) ⟶ B(A) ⟶ 0.

To construct the second exact sequence, Suslin considers the action of GL_{3}(k) on the chain complex C_{•}^{2}(k), where each C_{p}^{2}(k) is the free abelian group on (p + 1)-tuples of points in general position in ℙ^{2}(k). The construction works equally well over A if we recall that closed points in ℙ^{2}(A) consist of projective equivalence classes of unimodular vectors in A^{3}. The resulting complex C_{•}^{2}(A) is acyclic [8] and the proofs of all the results in Section 3 of [12] go through with one minor modification. Lemma 3.4 of [12] asserts that the restriction of the homomorphism ∂ : H_{3}(GL_{3}(k)) → B(k) to H_{3}(GL_{2}(k)) coincides with the homomorphism above. The proof makes use of the projection ℙ^{2}(k) → ℙ^{1}(k) centered at (0, 0, 1), which is not welldefined for ℙ^{2}(A). However, a careful reading of the proof shows that one need only construct maps among the various (C_{•})_{GL_{n}(k)} under consideration. The projection ℙ^{2}(A) → ℙ^{1}(A) is well-defined on these coinvariants since each orbit has a unique representative for which each entry of each vector is a unit in A. This allows the proof of Lemma 3.4 to go through for A.

The results in Section 4 and those through Proposition 5.1 in Section 5 may be applied word for word to A. The end result is the following.

Proposition 4.3 (cf. Prop. 5.1, [12]). There is an exact sequence

π_{3}^{0}(BGM(A)^{+}) ⟶ K_{3}(A) ⟶ B(A) ⟶ 0.

Remark. The rest of the proofs in Section 5 of [12] do not go through for A as stated. Thus, we cannot immediately get the exact sequence

0 ⟶ Tor(μ(A), μ(A))^{∼} ⟶ K_{3}(A)_{ind} ⟶ B(A) ⟶ 0.

However, since μ(R) = μ(k) for a Hensel local k-algebra R (see Lemma 3.2) and K_{∗}^{M}(R) ⊗ ℤ/p = K_{∗}^{M}(k) ⊗ ℤ/p, we see that the exact sequence exists for R after tensoring with ℤ/p (see Proposition 4.4 below). Moreover, the above results hold even more generally (e.g., semilocal rings with infinite residue fields).

We now compare the exact sequences for k and R, where R is a Hensel local k-algebra. Let p be a prime distinct from the characteristic of k. Since tensor product is right exact, we have a commutative diagram

π_{3}^{0}(BGM(k)^{+}) ⊗ ℤ/p ⟶ K_{3}(k) ⊗ ℤ/p ⟶ B(k) ⊗ ℤ/p ⟶ 0
↓ ↓ ↓
π_{3}^{0}(BGM(R)^{+}) ⊗ ℤ/p ⟶ K_{3}(R) ⊗ ℤ/p ⟶ B(R) ⊗ ℤ/p ⟶ 0.

The map B(k)⊗ℤ/p → B(R)⊗ℤ/p is injective and the map K_{3}(k)⊗ℤ/p → K_{3}(R) ⊗ ℤ/p is an isomorphism [11]. A simple diagram chase gives the following result.

12 KEVIN P. KNUDSON

Proposition 4.4. The natural map

B(k) ⊗ ℤ/p ⟶ B(R) ⊗ ℤ/p

is an isomorphism.

We are now in a position to compute H_{3}(GL_{2}(R), ℤ/p). Proposition 4.5. The natural map

H_{3}(GL_{2}(k)) ⊗ ℤ/p ⟶ H_{3}(GL_{2}(R)) ⊗ ℤ/p

is an isomorphism. Proof. Consider the commutative diagram

H_{3}(GM_{2}(k)) ⊗ ℤ/p → H_{3}(GL_{2}(k)) ⊗ ℤ/p → B(k) ⊗ ℤ/p → 0
↓ ↓ ↓
H_{3}(GM_{2}(R)) ⊗ ℤ/p → H_{3}(GL_{2}(R)) ⊗ ℤ/p → B(R) ⊗ ℤ/p → 0.

The map H_{3}(GM_{2}(k))⊗ℤ/p → H_{3}(GM_{2}(R))⊗ℤ/p is an isomorphism since we have an isomorphism

H_{•}(GM_{2}(k), ℤ/p) ⟶ H_{•}(GM_{2}(R), ℤ/p).

The latter isomorphism is proved by considering the Hochschild–Serre spectral sequences associated to the extension

1 ⟶ T ⟶ GM_{2} ⟶ Σ_{2} ⟶ 1

and noting that H_{•}(T (k), ℤ/p) ≅ H_{•}(T (R), ℤ/p). By the Five Lemma, the map

H_{3}(GL_{2}(k)) ⊗ ℤ/p ⟶ H_{3}(GL_{2}(R)) ⊗ ℤ/p

is an isomorphism. Corollary 4.6. The natural map

H_{3}(GL_{2}(k), ℤ/p) ⟶ H_{3}(GL_{2}(R), ℤ/p)

is an isomorphism. Proof. Consider the commutative diagram of universal coefficient sequences

H_{3}(GL_{2}(k)) ⊗ ℤ/p → H_{3}(GL_{2}(k), ℤ/p) → _{p}H_{2}(GL_{2}(k))
↓ ↓ ↓
H_{3}(GL_{2}(R)) ⊗ ℤ/p → H_{3}(GL_{2}(R), ℤ/p) → _{p}H_{2}(GL_{2}(R)).

The last map is an isomorphism by stability for H_{2}(GL_{n}) [8] and the corresponding statement for H_{2}(GL). The result now follows from the Five Lemma.

5. Unstable Rigidity and the Friedlander–Milnor Conjecture

We now apply our results to the study of the Friedlander–Milnor conjecture in low degree. I thank Andrei Suslin for showing me the proofs of Propositions 5.1 and 5.4 below; these are well-known to the experts.

Proposition 5.1. Let X be a smooth affine curve over an algebraically closed field k and let x, y be closed points on X. Suppose that for all z ∈ X, the inclusion G(k) → G(O_{z}^{h}) induces an isomorphism on homology with ℤ/p coefficients. Then the specialization homomorphisms

s_{x}, s_{y} : H_{•}(G(k[X]), ℤ/p) ⟶ H_{•}(G(k), ℤ/p)

coincide.

Proof. (Sketch) Observe that

H_{•}(G(O_{x}^{h}), ℤ/p) = \underrightarrow{lim}_{Y/Xétale}H_{•}(G(k[Y]), ℤ/p).

Let α ∈ H_{•}(G(k[X]), ℤ/p). Denote by α(x) the image of α under the specialization map s_{x}. Note that since H_{•}(GL_{n}(k), ℤ/p) is a direct summand of H_{•}(G(k[X]), ℤ/p), the class α(x) also lies in the latter group. Consider α−α(x). This class has specialization 0 and maps to 0 in H_{•}(G(O_{x}^{h}), ℤ/p). It follows that there is a curve Y which is étale over X such that α−α(x) maps to 0 in H_{•}(G(k[Y]), ℤ/p). Denote by α_{Y} the image of α in H_{•}(G(k[Y]), ℤ/p). Then we have that α_{Y} = α(x), which is a constant class. It follows that the specialization of α_{Y} at all points of Y is the same and hence there is an affine open neighborhood of x ∈ X on which the specialization of α is constant. Now if y ∈ X, find an affine neighborhood of y on which the specialization is constant. Since these two neighborhoods have nontrivial intersection, we see that α(x) = α(y); i.e., the homomorphisms s_{x} and s_{y} agree.

Corollary 5.2. If X is a smooth affine curve and x, y are closed points on X, then the specialization homomorphisms

s_{x}, s_{y} : H_{i}(GL_{n}(k[X]), ℤ/p) ⟶ H_{i}(GL_{n}(k), ℤ/p)

coincide for i ≤ 3.

Proof. By Corollary 4.6, H_{i}(GL_{n}(O_{z}^{h}), ℤ/p) ≅ H_{i}(GL_{n}(k), ℤ/p) for i ≤ 3. The result follows.

Recall the Friedlander–Milnor conjecture [3]. Suppose that G is a reductive group scheme over an algebraically closed field k. Denote by BG_{k} the simplicial classifying scheme of G and by BG(k) the classifying space of the discrete group G(k) of k-rational points of G.

Conjecture 5.3. The comparison map

H_{ét}^{•}(BG_{k}, ℤ/p) ⟶ H^{•}(BG(k), ℤ/p)

is an isomorphism.

This is known to hold when k = \overline{𝔽}_{l} [3], and for the stable general linear group GL [11]. A few other special cases are known [9].

Rigidity implies the following result.

Proposition 5.4. Let k be an algebraically closed field and denote by K the algebraic closure of k(T). Then if rigidity holds, the natural homomorphism

i_{∗} : H_{•}(G(k), ℤ/p) ⟶ H_{•}(G(K), ℤ/p)

is an isomorphism.

Proof. (Sketch) Injectivity does not require rigidity. Indeed, K = \underrightarrow{lim} k[C], where C ranges over the smooth affine curves over k, and each map

H_{•}(G(k), ℤ/p) ⟶ H_{•}(G(k[C]), ℤ/p)

is split injective. It follows that i_{∗} is injective (but not necessarily split).

To prove surjectivity, note that each class in H_{•}(G(K), ℤ/p) comes from some α ∈ H_{•}(G(k[C]), ℤ/p), where C is a smooth affine curve over k. Denote by C_{K} the curve C ×_{Spec k} Spec K and by x the point of C_{K} given by the inclusions k[C] ↪ k(C) ↪ K. Consider the class α_{K} ∈ H_{•}(G(K[C]), ℤ/p) arising from the homomorphism

H_{•}(G(k[C]), ℤ/p) ⟶ H_{•}(G(K[C]), ℤ/p)

and denote by α_{K}(x) the image of α in H_{•}(G(K), ℤ/p). Choose a rational point y : Spec k → C. This yields another map y : Spec K → C. Rigidity implies that α_{K}(x) = α_{K}(y) and hence α_{K}(x) = α(y) (since y factors through Spec k) so that α arises from a class in H_{•}(G(k), ℤ/p). That is, the map i_{∗} is surjective.

Corollary 5.5. If rigidity holds and char k > 0, then the Friedlander– Milnor conjecture is true for G.

Proof. Note that if k is an algebraically closed field of characteristic l > 0, then k is a union of transcendental extensions of \overline{𝔽}_{l}. It follows that the natural map

H_{•}(G(\overline{𝔽}_{l}), ℤ/p) ⟶ H_{•}(G(k), ℤ/p)

is an isomorphism. The result follows by considering the commutative diagram

H_{ét}^{•}(BG_{k}, ℤ/p) ⟶ H^{•}(BG(k), ℤ/p)
↓  ↓
H_{ét}^{•}(BG_{\overline{𝔽}_{l}}), ℤ/p) ⟶ H^{•}(BG(\overline{𝔽}_{l}), ℤ/p)

where the left vertical arrow is an isomorphism by the proper base change theorem for étale cohomology and the bottom arrow is an isomorphism by [3].

Corollary 5.6. The Friedlander–Milnor conjecture holds in positive characteristic for H_{i}(GL_{n}), i ≤ 3.

Acknowledgements

I would like to thank Andrei Suslin for many fascinating discussions about rigidity and homology of congruence subgroups. Thanks also go to Paul Goerss and Brooke Shipley for patiently answering my questions about cosimplicial spaces and the associated homology spectral sequence. I also thank Philippe Elbaz-Vincent for his careful reading of the manuscript and for many useful comments. Finally, I thank Mark Walker for listening to my ramblings.

References

[1] A. Bousfield, D. Kan, Homotopy limits, completions, and localizations, Lecture Notes in Mathematics 304, Springer–Verlag, Berlin/Heidelberg/New York, 1972.

[2] A. Bousfield, On the homology spectral sequence of a cosimplicial space, Amer. J. Math. 109 (1987), 361–394.

[3] E. Friedlander, G. Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete groups, Comment. Math. Helv. 49 (1984), 347–361.

[4] P. Goerss, The homology of homotopy inverse limits, J. Pure Appl. Algebra 111 (1996), 83–122.

[5] W. Klingenberg, Lineare Gruppen über lokalen Ringen, Amer. J. Math. 83 (1961), 137–153.

[6] K. Knudson, The homology of special linear groups over polynomial rings, Ann. Sci. École Norm. Sup. (4) 30 (1997), 385–416.

[7] R. Lee, R. Szcarba, On the homology and cohomology of congruence subgroups, Invent. Math. 33 (1976), 15-53.

[8] Yu. Nesterenko, A. Suslin, Homology of the general linear group over a local ring and Milnor’s K–theory, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 121–146.

[9] C. Sah, Homology of classical Lie groups made discrete, III, J. Pure Appl. Algebra 56 (1989), 269–312.

[10] B. Shipley, Convergence of the homology spectral sequence of a cosimplicial space, Amer. J. Math. 118 (1996), 179–207.

[11] A. Suslin, On the K–theory of local fields, J. Pure Appl. Algebra 34 (1984), 301–318.

[12] A. Suslin, K_{3} of a field and the Bloch group, Trudy Mat. Inst. Steklov. 183 (1990), 180–199.

Department of Mathematics, Northwestern University, Evanston, IL 60208

E-mail address: knudson@math.nwu.edu