Table Of ContentLOW DIMENSIONAL HOMOLOGY OF LINEAR GROUPS
OVER HENSEL LOCAL RINGS
8
9
9 KEVIN P. KNUDSON
1
n
Abstract. WeprovethatifRisaHensellocalringwithinfiniteresidue
a
J fieldk,thenaturalmapHi(GLn(R),Z/p)→Hi(GLn(k),Z/p)isaniso-
1 morphism for i ≤ 3, p 6= char k. This implies rigidity for Hi(GLn),
i≤3, which in turn implies the Friedlander–Milnor conjecture in posi-
2
tive characteristic in degrees ≤3.
]
T
K
A fundamental question in the homology of linear groups is that of rigid-
.
h ity: given a smooth affine curve X over an algebraically closed field k and
t closedpointsx,y onX,dothecorrespondingspecialization homomorphisms
a
m
s ,s :H (G(k[X]),Z/p) −→ H (G(k),Z/p)
[ x y ∗ ∗
1 coincide? Here, G is a reductive algebraic group and p is a prime not equal
v to the characteristic of k. The answer is yes when X is the affine line
8
and G = SL ,GL ,PGL since the inclusion G(k) → G(k[t]) induces an
9 n n n
0 isomorphism
1 H (G(k),Z) −→ H (G(k[t]),Z)
0 ∗ ∗
8 (see [6]) and the map is split by evaluation at any x∈ A1.
9
/ Rigidity in algebraic K-theory has spectacular consequences, including
h
thecalculation oftheK-theoryof algebraically closed fieldsandthesolution
t
a of the Friedlander–Milnor conjecture for the stable general linear group GL.
m
Similarly,aproofofrigidityforG(k[X])wouldimplytheFriedlander–Milnor
:
v conjecture for G (see Section 5).
i Rigidity would follow if one could prove the following stronger result. Let
X
X beasmoothcurveover analgebraically closedfieldk andletxbeaclosed
r
a point on X. Denote by Oh the henselization of O .
x x
Conjecture. The inclusion G(k) → G(Oh) induces an isomorphism
x
H (G(k),Z/p) −→ H (G(Oh),Z/p)
∗ ∗ x
for p 6= char k.
Date: December4, 1997.
1991 Mathematics Subject Classification. 20G10.
Key words and phrases. rigidity, Hensel local ring, Friedlander–Milnor conjecture,
Bloch group.
Supported byan NSFPostdoctoral Fellowship, grant no. DMS-9627503.
1
2 KEVIN P. KNUDSON
InSection5wesketchaproofofrigidityassumingthisconjecture(thisar-
gument is well-known). Our results will allow us to deduce the Friedlander–
Milnor conjecture in positive characteristic for H (GL ), i ≤ 3. The conjec-
i n
ture is known to be true for H (SL (C)) by the work of Sah [9].
3 2
In this note we study H (G(R),Z/p) for G = SL and G = GL , where
∗ n n
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 (k) → GL (R) induces an isomorphism
n n
H (GL (k),Z/p) −→ H (GL (R),Z/p)
i n i n
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 (GL ). The above map is in fact an iso-
3 2
morphism 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 (PGL (k),Z/p) −→ H (PGL (R),Z/p)
2 2 2 2
is an isomorphism and use this to deduce the result for H (GL ). More-
2 2
over, 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 (GL ).
3 2
The corresponding statement for G = SL and i= 0,1 is almost obvious.
n
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 ,t ,...,t ) of
1 2 m
F[[t ,t ,...,t ]]. Consider the short exact sequence
1 2 m
1−→ C −→ SL (F[[t ,t ,...,t ]]) −→ SL (F) −→ 1. (1)
n,m n 1 2 m n
Here, the group C consists of those matrices which are congruent to the
n,m
identity modulo m. We have the following result.
Proposition. If n ≥ 3, then
m
H (C ,Z) = sl (F).
1 n,m n
i=1
M
When n = 2, assume further that char F 6= 2 and F 6= F . Then
3
m
H (C ,Z) = sl (F).
1 2,m 2
i=1
M
When p is a prime distinct from the characteristic of k, the proposition
implies that H (C ,Z/p) = 0 unless n = 2,char k = 2 or F = F . In
1 n,m 3
this case, however, one can still conclude that H (C ,Z/p) = 0 by other
1 2,m
LINEAR GROUPS OVER HENSEL RINGS 3
methods. One expects that H (C ,Z/p) vanishes in all positive degrees.
• n,m
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 is the kernel of the natural map
n,m
modm
SL (F[[t ,t ,...,t ]]) −→ SL (F)
n 1 2 m n
wherem is the ideal (t ,t ,...,t ). Define a sequence of subgroups Ci by
1 2 m n,m
Ci = {X ∈ SL (F[[t ,t ,...,t ]]) :X ≡ I mod mi}.
n,m n 1 2 m
If no confusion can result, we usually drop the subscripts from the notation.
Note that each Ci is a normal subgroup as it is the kernel of the map
SL (F[[t ,t ,...,t ]]) −→ SL (F[[t ,t ,...,t ]]/mi).
n 1 2 m n 1 2 m
For each i, define a homomorphism
ρ :Ci −→ sl (F)
i n,m n
λ
M
by
ρ (I + tk1tk2···tkmX +···) = (X )
i 1 2 m λ λ λ
λ
X
where λ = (k ,k ,...,k ) is an m-partition of i (i.e., λ is an m-tuple of
1 2 m
nonnegative integers whose sum is i) and (X ) denotes the element of
λ λ
sl (F) given by the various X . Note that ρ is well-defined since the
n λ i
equation
L
1= detX ≡ 1+ tk1···tkmtrX modmi+1
1 m λ
λ
X
implies that trX = 0 for each partition λ. One checks easily that ρ is a
λ i
surjective group homomorphism with kernel Ci+1 (see [7] for an analogous
n,m
result for congruence subgroups of SL (Z)). Hence, for each i we have an
n
isomorphism
Ci /Ci+1 ∼= sl (F).
n,m n,m n
λ
M
Remark. The above discussion remains valid for any simple algebraic group
G. The quotients Ci/Ci+1 are direct sums of copies of the lie algebra g. We
have chosen to work with SL just to fix ideas.
n
Lemma 1.1. For each i,j, [Ci,Cj]⊆ Ci+j.
Proof. This follows easily by writing out XYX−1Y−1 for X ∈ Ci, Y ∈ Cj
and noting that the inverse of I + Z +··· is I − Z +···.
λ λ
For any group G, denote by Γ• tPhe lower central sePries of G.
Corollary 1.2. For each i, Γi ⊆ Ci .
n,m
4 KEVIN P. KNUDSON
Proof. The series C• is a descending central series and as such contains
n,m
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 (A) are the congruence subgroups. If n= 2, the same is true as long
n
as the residue characteristic of A is not 2 or the residue field is not F .
3
Corollary 1.4. The filtration C• is the lower central series.
n,m
Proof. Note that the groups Γi are characteristic subgroups of C and
n,m
hence are normal in SL (F[[t ,...,t ]]). Since there are clearly elements
n 1 m
in Γi−Ci+1, it follows that Γi = Ci for all i.
n,m n,m
Corollary 1.5. For all n ≥ 3, H (C ,Z) = m sl (F). If char F 6= 2
1 n,m i=1 n
or F 6= F , then H (C ,Z)= m sl (F).
3 1 2,m i=1 2 L
Proof. Since C2 = Γ2 , we haLve
n,m n,m
m
H (C ,Z) = C /Γ2 = C /C2 = sl (F).
1 n,m n,m n,m n,m n
i=1
M
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 (C ,Z/p) = 0.
1 n,m
Proof. This follows from the previous corollary except in the case n = 2,
char F = 2 or F = F . However, one could also prove this corollary directly
3
by noting that each element of C has a pth root in C and hence that
n,m n,m
H (C ,Z/p) = H (C ,Z)⊗Z/p = 0.
1 n,m 1 n,m
Corollary 1.7. For each n,
H (SL (F[[t ,...,t ]]),Z/p) ∼= H (SL (F),Z/p).
1 n 1 m 1 n
Proof. This follows easily by considering the Hochschild–Serre spectral se-
quence associated to the extension (1).
2. Conjectures about H (C ,Z/p)
∗ n,m
The isomorphism
H•(SLn(k[[t1,t2,...,tm]]),Z/p) ∼= H•(SLn(k),Z/p)
would follow easily if one could prove the following.
Conjecture 2.1. For all l > 0, H (C (k[[t ,t ,...,t ]]),Z/p) = 0.
l n,m 1 2 m
We have just proved the case l = 1. We make the following observations.
LINEAR GROUPS OVER HENSEL RINGS 5
2.1. Continuous homology. For each i≥ 1, denote by Li the kernel of
n,m
the homomorphism
SL (k[t ,...,t ]) −→ SL (k[t ,...,t ]/(t ,...t )i).
n 1 m n 1 m 1 m
The group C is the inverse limit of the nilpotent groups L /Li .
n,m n,m n,m
One checks that these groups have no homology with Z/p coefficients by
noting that the successive graded quotients Lj /Lj+1 of L /Li are k-
n,m n,m n,m n,m
vector spaces (see for example [11]); an iterated use of the Hochschild–Serre
spectral sequence then shows that L /Li is Z/p–acyclic. It follows that
n,m n,m
the “continuous homology”
limH (C /Ci ,Z/p) = limH (L /Li ,Z/p)
←− • n,m n,m ←− • n,m n,m
vanishes and that
H•(SLn(k[t1,t2,...,tm]/mi),Z/p) ∼= H•(SLn(k),Z/p)
for all i ≥ 1.
2.2. Large acyclic subgroups. When m =1, it is easy to see that
H (L ,Z/p) = 0
• n,1
(see [6]) and the group L is a subgroup of C .
n,1 n,1
2.3. Product spaces. The group C is the subgroup of L /Li
n,m i n,m n,m
consisting of coherent sequences. A theorem of P. Goerss [4] implies that
Q
this product is Z/p–acyclic.
2.4. Cosimplicial replacement. We have seen that C is the inverse
n,m
limit of the nilpotent groups L /Li . Let X be a K(L /Li ,1)
n,m n,m i n,m n,m
space and consider the tower of fibrations
X ← X ← X ← ···
2 3 4
The homotopy inverse limit of this tower is a K(C ,1)-space; denote it
n,m
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 Xn is a certain
product of the spaces X of the tower (see Bousfield–Kan [1, XI, Sec. 5]).
i
Given a cosimplicial space X•, there is an associated homology spectral
sequence [2]
E2 = πmH (X•;A)
m,t t
forany abelian group A. Here, theterm on theright is themth cohomotopy
of the cosimplicial abelian group H (X•;A). The main problem is that of
t
convergence ofthisspectralsequence. Undercertain conditions,thespectral
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.
6 KEVIN P. KNUDSON
InthecaseofC ,weseethatthecosimplicialreplacementisparticularly
n,m
nice. Its spaces Xn are certain products of the X and by Goerss’ theorem
i
eachoftheseisZ/p–acyclic. Itfollowsthattheassociatedhomologyspectral
sequence has E2–term
Z/p (m,t) = (0,0)
E2 =
m,t
(0 (m,t) 6= (0,0).
The question is what, if anything, is this spectral sequence converging to?
Unfortunately,allknownconvergencetheoremsdonotapplytothesituation
at hand. Most of these require some hypotheses on the spaces Xn such as
nilpotency (which we do not have in this case—we would need the groups
L /Li to have bounded nilpotence degree). A proof that the spectral
n,m n,m
sequence converges to the homology of the total space of X• in this case
would immediately show that C is Z/p–acyclic.
n,m
Givenallofthis,itisdifficulttoimaginethatC hasanyZ/phomology.
n,m
Speaking heuristically, the group C is built up by successively attaching
n,m
copies of sl (k) and since this group is Z/p–acyclic, one would expect that
n
C is as well.
n,m
3. Spectral Sequences and the Second Homology Groups
Hereafter, R denotesaHensellocalk-algebra, andmdenotesthemaximal
ideal of R. We have R/m = k.
We begin by considering the group PGL (R). We will show that the in-
2
clusion PGL (k) → PGL (R) induces an isomorphism on second homology
2 2
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 ∈ A2 is unimodular if
the ideal generated by its entries is A itself. Denote by v the vector in F2
obtained by reducing the entries of v modulo m. We say that a collection
v ,v ,...,v of unimodular vectors is in general position if the collection
1 2 k
v ,v ,...,v is in general position (i.e., the matrix determined by any pair
1 2 k
of them is invertible).
Construct a simplicial set S as follows. The nondegenerate p–simplices
•
are collections v ,v ,...,v of projective equivalence classes of unimodular
0 1 p
vectors which are in general position (this amounts to saying that the v
i
are distinct closed points in P1). 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) →ǫ Z → 0 is acyclic.
•
Note that PGL (A) acts transitively on C (A) for i ≤ 2 (i.e., PGL (A)
2 i 2
1
acts 3-transitively on points in P1(A)). Denote by 0 the vector ;
0
(cid:18) (cid:19)
LINEAR GROUPS OVER HENSEL RINGS 7
0 1
by ∞ the vector ; and by 1 the vector . Then each orbit of
1 1
(cid:18) (cid:19) (cid:18) (cid:19)
the action of PGL (A) on C (A) has a unique representative of the form
2 p
(0,∞,1,v ,...,v ). Sinceeachv isingeneralpositionwith0,∞and1,we
1 p−2 i
see that both entries of v must be units in A. Since we are using projective
i
equivalenceclassesofvectors, wemay thenassumethatthevector v hasthe
i
1
form whereα is a unitin A. Moreover, the α satisfy theadditional
α i i
i
(cid:18) (cid:19)
conditions that 1 − α ∈ A× (since v is in general position with 1) and
i i
α −α ∈ A× for i 6= j (since v is in general position with v ).
i j i j
Denote by E• (A) the spectral sequence associated to the action of the
•,•
group PGL (A) on C (A). This spectral sequence converges to the homol-
2 •
ogy of PGL (A) and has E1–term
2
E1 = H (G )
p,q q σ
σM∈Σp
where Σ is a set of representatives of the orbits of the action of PGL (A)
p 2
onC (A)andG isthestabilizer of σ inPGL (A). Note thatthestabilizers
p σ 2
of the PGL (A) action are
2
G = B(A)/D(A), G = T(A)/D(A), G = D(A)/D(A)
0 (0,∞) (0,∞,1)
where B(A) is the upper triangular subgroup, T(A) is the diagonal sub-
group, and D(A) is the subgroupof scalar matrices. Thegroup D(A)/D(A)
is also the stabilizer of each orbit (0,∞,1,v ,...,v ). Moreover, we have
1 p−2
H•(B(A)/D(A),Z) ∼= H•(T(A)/D(A),Z) (see [8]). It follows that our spec-
tral sequence E• (A) has E1–term (with Z/p coefficients)
•,•
H (A×,Z/p) H (A×,Z/p) H ({1},Z/p) H ({1},Z/p) ···
• • • •
αM∈A×
1−α∈A×
and converges to H (PGL (A),Z/p).
• 2
Now consider the cases A = k and A = R where k is an infinite field.
The inclusion PGL (k) → PGL (R) induces an injective map of spectral
2 2
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
• (G⊗Z/p)⊗Γ ( G)−∼=→ H (G,Z/p),
Z/p • p •
^
8 KEVIN P. KNUDSON
where G is the subgroupof G killed by p andΓ is a divided power algebra.
p •
The lemma will follow if we can show that
k× ⊗Z/p ∼= R×⊗Z/p and k× ∼= R×.
p p
We have a natural homomorphism
π :R× −→ k×/(k×)p
givenbycomposingreductionmodulomwiththeprojectionk× → 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. Sup-
pose x ∈ R× maps to 0 under π, and consider the equation tp −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 yp −x = 0.
That is, x∈ (R×)p.
Note that k× is the subgroup of pth roots of unity in k. A similar
p
application of Hensel’s Lemma shows that this coincides with R×. This
p
completes the proof of the lemma.
Corollary 3.3. For each n, the natural map
H (GL (k),Z/p) −→ H (GL (R),Z/p)
1 n 1 n
is an isomorphism.
Corollary 3.4. The map of spectral sequences
E1 (k) −→ E1 (R)
p,q p,q
is an isomorphism for p ≤ 2.
Corollary 3.5. The natural map
H (PGL (k),Z/p) −→ H (PGL (R),Z/p)
2 2 2 2
is an isomorphism.
Proof. The E2–terms of the spectral sequences look as follows:
E2 E2 0
E2 : 0,2 1,2
0 0 0
Z/p 0 0 E2
3,0
By considering the commutative diagram
E3 (k) →d3 E2 (k) → H (PGL (k),Z/p) → 0
3,0 0,2 2 2
↓ || ↓
E3 (R) →d3 E2 (R) → H (PGL (R),Z/p) → 0
3,0 0,2 2 2
we see that the map
H (PGL (k),Z/p) −→ H (PGL (R),Z/p)
2 2 2 2
is an isomorphism.
LINEAR GROUPS OVER HENSEL RINGS 9
Corollary 3.6. The natural map
H (GL (k),Z/p) → H (GL (R),Z/p)
2 2 2 2
is an isomorphism.
Proof. TheHochschild–Serrespectralsequencesassociatedtotheextensions
1 → k× → GL (k) → PGL (k) → 1
2 2
↓ ↓ ↓
1 → R× → GL (R) → PGL (R) → 1
2 2
are isomorphic at E2 for 0 ≤ p ≤ 2, 0 ≤ q ≤ 2. This yields a commutative
p,q
diagram
H (PGL (k),Z/p) →d3 E3 (k) → H (GL (k),Z/p) → ···
3 2 0,2 2 2
↓ || ↓
H (PGL (R),Z/p) →d3 E3 (R) → H (GL (R),Z/p) → ···
3 2 0,2 2 2
··· → H (PGL (k),Z/p) → 0
2 2
||
··· → H (PGL (R),Z/p) → 0
2 2
The result follows from the Five Lemma.
Corollary 3.7. The natural map
H (GL (k),Z/p) → H (GL (R),Z/p)
2 n 2 n
is an isomorphism for all n.
Proof. Consider the commutative diagram
∼
H (GL (k),Z/p) −=→ H (GL (R),Z/p)
2 2 2 2
↓ ↓
H (GL (k),Z/p) −→ H (GL (R),Z/p)
2 n 2 n
The vertical arrows are isomorphisms by Suslin’s stability theorem [8].
Remark. The last two corollaries actually follow from the corresponding
statement for H (GL) [11]. Our approach provides an alternate proof of
2
this fact and also has the advantage of proving the correspondingstatement
for PGL .
2
Remark. Since the map E1 (k) → E1 (R) is an isomorphism for p ≤ 2, to
p,q p,q
show that H•(PGL2(k),Z/p) ∼=H•(PGL2(R),Z/p) it would suffice to show
that the chain complexes
D (k) = E1 (k) ← E1 (k) ← ···
• 3,0 4,0
and
D (R)= E1 (R)← E1 (R) ← ···
• 3,0 4,0
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 ,...,t ]/ml for each l ≥ 2
• 1 m
10 KEVIN P. KNUDSON
(see 2.1). One would hope to be able to write down a contracting homotopy
whichisindependentofl(ormoregenerally, whichisindependentofthefact
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 → Q .
0 1
4. Bloch Groups and Third Homology
We now turn our attention to the computation of H (GL (R),Z/p). The
3 2
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 (GM (k),Z) −→ H (GL (k),Z) −→ B(k) −→ 0
3 2 3 2
and
π0(BGM(k)+) −→ K (k) −→ B(k)−→ 0.
3 3
Here, B(k) denotes the Bloch group of k, GM(k) denotes the group of
monomial matrices over k, and π0(BGM(k)+) denotes the kernel of the
3
canonical map
π (BGM(k)+)−→ π (BΣ+)
3 3
induced by the projection GM(k) → Σ where Σ is the infinite symmetric
group (Σ = Σ ).
n≥2 n
The amazing fact is that the above theorem holds for any local k-algebra,
S
as we now describe.
For any local ring A with infinite residue field k, denote by D(A) the free
abeliangroupwithbasis[x],wherex,1−x∈ A× anddefineahomomorphism
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×) the quotient of A× ⊗A× by the
σ
action of σ. Define a group p(A) by
p(A) = D(A)/h[x]−[y]+[y/x]−[(1−x−1)/(1−y−1)]+[(1−x)/(1−y)]i.
One checks easily (see Lemma 1.1 of [12]) that φ induces a homomorphism
φ: p(A) → (A×⊗A×) . By definition, the Bloch group, B(A), is the kernel
σ
of φ.
ConsidertheactionofGL (A)onthechaincomplexC (A)oftheprevious
2 •
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 (GM (k),Z) −→ H (GL (k),Z) −→ B(k) −→ 0
3 2 3 2
(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.
