Using again that cd(k(xp)) ≤3−p for any xp ∈X(p) ([31, §4.2, Proposition 11]), we get HpZar(X,Hq(m))=0 for any q≥4. Since X is affine, this implies H2(X,H3(m))= H5et(X, μ⊗lnm)=0 by [21, Chapter VI, Theorem 7.2].
Now there is a commutative diagram ([9, Theorem 2.3])
x1∈X(1)
K2(k(x1))/ln
x2∈X(2)
K1(k(x2))/ln
x3∈X(3)
K0(k(x3))/ln
x1∈X(1)
H2(k(x1), μ⊗ln2)
x2∈X(2)
H1(k(x2), μln)
x3∈X(3)
H0(k(x3),Z/ln)
whose vertical maps are isomorphisms ([20]). This gives H2
X,K3/ln
=H2
X,H3(3)
=0.
The result follows.
7. The main theorem
In this section, we prove that any stably free module P of rank (d −1) over a d-dimensional normal affine algebra R over an algebraically closed field k, for which gcd((d−1)!,char(k))=1, is indeed free.
Recall first that a unimodular row of length n≥2 over R is a row a=(a1, . . . ,an) with ai ∈R such that there exist b1, . . . ,bn∈R with
aibi =1. We denote by Umn(R) the set of unimodular rows of length n and we consider it as a pointed set with base point e1:=(1,0, . . . ,0). If M∈GLn(R)and a∈Umn(R), then aM is also unimodular, and in this way GLn(R)acts on Umn(R). If a is any unimodular row, the exact sequence
0 P(a) Rn a R 0
yields a projective module P(a)which is free if and only if there exists M∈GLn(R)such that aM=e1. It is clear that any projective module P such that P⊕RRncomes from a unimodular row, and it suffices therefore to understand the (pointed) set Umn(R)/GLn(R) to understand stably free modules of rank n−1 satisfying P⊕RRn.
Of course, any subgroup of GLn(R)acts on Umn(R), and one is classically inter-ested in the (pointed) orbit set Umn(R)/En(R) which classifies unimodular rows up to elementary homotopies. An orbit in Umn(R)/En(R)is usually called an elementary orbit.
We now recall a lemma of L. N. Vaserstein ([39, Corollary 2]).
Lemma 7.1. — Let(a1,a2,a3)∈Um3(R), u∈R, and Ra1+Ru=R. Then the rows (a1,a2,a3)and(a1,ua2,ua3)are in the same elementary orbit.
Consequently, by Whitehead’s lemma the rows(a1,ua2,ua3)and(a1,a2,u2a3)can be seen to be in the same elementary orbit. In particular, if −1 is a square in R, then (−a1,a2,a3)and(a1,a2,a3)are in the same elementary orbit.
Given v=(v0, v1, v2), w=(w0, w1, w2)∈Um3(R), withv·wt=1, we shall de-note by V(v, w)the class in WE(R)of the 4×4 skew-symmetric matrix
⎛
⎜⎜
⎝
0 v0 v1 v2
−v0 0 w2 −w1
−v1 −w2 0 w0
−v2 w1 −w0 0
⎞
⎟⎟
⎠
of Pfaffian 1. For simplicity, we may just write V(v), as the class of V(v, w) does not depend on the choice of w such that v·wt =1. Associating V(v) to v yields a well defined map V:Um3(R)/E3(R)→WE(R)called the Vaserstein symbol ([40, §5]).
Theorem7.2. — Let R be a smooth affine algebra over a field k of cohomological dimension≤1 which is perfect if the characteristic is 2 or 3. Then the Vaserstein symbol
V:Um3(R)/E3(R)→WE(R) is a bijection.
Proof. — See [27, Corollary 3.5].
This bijection gives rise to an abelian group structure on the set Um3(R)/E3(R), when R is as above. We recall Vaserstein’s rule in WE(R)(see [40, Theorem 5.2 (iii)]). If (a,b,c)and(a,x,y)are in Um3(R)/E3(R), then
V(a,b,c)⊥V(a,x,y)=V(a, (b,c)
x y
−y x
)
for any x,ysuch that xx+yy=1 modulo(a).
Lemma7.3. — Let R be a ring. The following formulas hold in WE(R):
(i) V(a,b,c)−1=V(−a,b,c), where a is an inverse of a modulo(b,c).
(ii) V(ab2,c,d)=V(a,c,d)⊥V(b2,c,d).
Proof. — Since Vaserstein’s rule holds in WE(R), it suffices to follow the proof of
[38, Lemma 3.5(iii), Lemma 3.5(v)].
We now recall the Antipodal Lemma of Rao in [25, Lemma 1.3.1].
Lemma 7.4. — Let R be a ring and (a,b,c)∈Um3(R). Suppose that (a,b,c) and (−a,b,c)are in the same elementary orbit. Then for any n∈N, we have
nV(a,b,c)=V(an,b,c) in WE(R).
Proof. — The proof of [25, Lemma 1.3.1] applies mutatis mutandis using the
for-mulas of Lemma7.3.
We now state and prove our main theorem:
Theorem7.5. — Let R be a d-dimensional normal affine algebra over an algebraically closed field k such that gcd((d −1)!,char(k))=1. If d =3, suppose moreover that R is smooth. Then every stably free R-module P of rank d−1 is free.
Proof. — Let P be a stably free module of rank d−1. Since the result is clear when d ≤2, we assume that d ≥3. Using Suslin’s cancellation theorem [34, Theorem 1], we can suppose that there is an isomorphism P⊕RRd, and therefore that P is given by a unimodular row(a1, . . . ,ad). In view of [34, Theorem 2], to prove that P is free it suffices to show that there exists a unimodular row(b1, . . . ,bd)such that (a1, . . . ,ad)= (b(d1−1)!, . . . ,bd)in Umd(R)/Ed(R).
Suppose that d≥4. Let J be the ideal of the singular locus of R. Since R is normal, J has height at least 2 and dim(R/J)≤d −2. It follows from [6, Theorem 3.5] that Umd(R/J)=e1Ed(R/J)and we can therefore assume, performing elementary operations if necessary, that ad≡1 (mod J)and a1, . . . ,ad−1∈J. Using now Swan’s Bertini theorem [37, Theorem 1.5], we can perform elementary operations on(a1, . . . ,ad)such that B:=
R/(a4, . . . ,ad)is either empty, either a non-singular threefold outside the singular locus of R. In the first case, the row(a4, . . . ,ad)is unimodular, and therefore the row(a1, . . . ,ad) is completable in an elementary matrix. Thus we can restrict to the second case. In this situation, we see that B is actually smooth since ad≡1 (mod J).
Given a unimodular row(a,b,c)on B, we can choose lifts a,b,c∈R and consider the unimodular row(a,b,c,a4, . . . ,ad)on R. It is straightforward to check that this gives a well-defined map
Um3(B)/E3(B)→Umd(R)/Ed(R),
showing that(a1, . . . ,ad)comes from the unimodular row(a1,a2,a3)on B. We are thus reduced to the case where R is the affine algebra of a smooth threefold. By Theo-rem 7.2, the set Um3(R)/E3(R)is in bijection with WE(R) and is thus endowed with the structure of an abelian group. Since −1 is a square in k, Lemma 7.4 shows that n·(a1,a2,a3)=(a1n,a2,a3)in Um3(R)/E3(R)for any n∈N. Now Propositions 5.1and 6.1show that Um3(R)/E3(R)is a divisible group prime to the characteristic of k. Since gcd((d−1)!,char(k))=1, there exists a unimodular row(b1,b2,b3)∈Um3(R)such that
(a1,a2,a3)=(d−1)! ·(b1,b2,b3)=(b(d−1)1 !,b2,b3)
in Um3(R)/E3(R). The result follows.
Remark 7.6. — The proof of the theorem actually gives a bit more. Let R be an affine algebra of dimension d≥4 over an algebraically closed field k such that gcd((d− 1)!,char(k))=1. Let J be the ideal of the singular locus of R, and let(a1, . . . ,ad)be a unimodular row of rank d which is in the elementary orbit of e1 in Umd(R/J)/Ed(R/J).
Then the projective module associated to(a1, . . . ,ad)is free. To see this, observe that we can use [28, Theorem 2] to reduce to the case of a reduced ring and then apply the proof of the theorem.
To conclude, we derive from the above theorem some stability results for the group K1(R)where R is a smooth affine algebra over an algebraically closed field.
Corollary7.7. — Let R be a smooth algebra of dimension d≥3 over an algebraically closed field k. Assume that d!k=k. Then SLd(R)∩Ed+1(R)=Ed(R).
Proof. — Letσ ∈SLd(R)∩Ed+1(R). In view of T. Vorst’s theorem in [42], it suffices to show that there exists a matrix N(X)∈SLd(R[X])with the property that N(0)=Id and N(1)=σ.
Since 1⊥σ ∈Ed+1(R), there exists α(X)∈GLd+1(R[X]) with α(0)=Id and α(1)=1⊥σ. It follows that the unimodular row v(X):= e1α(X)∈Umd+1(R[X]) is congruent to e1 modulo f , where f :=X2−X. We now prove that there exists a ma-trix M(X)∈GLd+1(R[X])congruent to Id modulo f satisfying the equality e1M(X)= e1α(X). This will prove the corollary since these properties imply that
α(X)M(X)−1=
1 0 N(X)
with N(X)∈GLd(R[X])such that N(0)=Id and N(1)=σ.
To find M(X), we argue as in [27, Proposition 3.3]. Set B=R[X,T]/(T2−Tf) and write e1α(X)=e1+(X2−X)w(X), withw(X)∈R[X]d+1. Observe that u(T):=e1+ Tw(X)is a unimodular row of length d+1 over B with u(f)=v(X)and u(0)=e1. Since R is smooth, it follows that the ideal of the singular locus of B is precisely(T,f). Now
u(T)is congruent to e1modulo T and we can use Remark7.6to findε(T)∈Ed+1(B)such that u(T)ε(T)is of the form(bd1!, . . . ,bd+1). By [27, Proposition 3.3] or [26, Proposition 5.4], there is aδ(T)∈SLd+1(B)congruent to Id modulo f such that u(T)ε(T)=e1δ(T).
Finally, we set M(X):=β(0)β(f)−1 where β(T)=ε(T)δ(T)−1. It is straightforward to check that M(X)satisfies the conditions stated above, and therefore the result is proved.
Corollary 7.8. — Let R be a smooth algebra over the algebraic closure k of a finite field of dimension d≥3. Assume that d!k=k. Then the natural map SLd(R)/Ed(R)→SK1(R) is an isomorphism.
Proof. — Using [40, Corollary 17.3], we see that Umn(R)=e1En(R)for n≥d. It follows that SLd(R)/Ed(R)→SK1(R)is onto. We conclude using Corollary7.7.
Corollary7.9. — Let R be a smooth algebra of dimension d≤4 over an algebraically closed field k. Assume that d!k=k. Then the Vaserstein symbol
V:Um3(R)/E3(R)→WE(R) is an isomorphism.
Proof. — L.N. Vaserstein in [40, Theorem 5.2(c)] has shown that V is onto. Apply [40, Lemma 5.1] to conclude that V is injective as SL4(R)∩E(R)=E4(R) in view of
Corollary7.7.
Acknowledgements
The first author wishes to thank the Swiss National Science Foundation, grant PP00P2_129089/1, for support.
REFERENCES
1. J. KRISTINNARASONand A. PFISTER, Beweis des Krullschen Durchschnittsatzes für den Wittring, Invent. Math.,12 (1971), 173–176.
2. P. BALMER, An introduction to triangular Witt groups and a survey of applications, in Algebraic and Arithmetic Theory of Quadratic Forms, Contemp. Math., vol. 344, pp. 31–58, Amer. Math. Soc., Providence, 2004.
3. P. BALMERand C. WALTER, A Gersten-Witt spectral sequence for regular schemes, Ann. Sci. Éc. Norm. Super.,35(1) (2002), 127–152.
4. J. BARGEand J. LANNES, Suites de Sturm, indice de Maslov et périodicité de Bott, Progress in Mathematics, vol. 267, Birkhäuser, Basel, 2008.
5. H. BASS, K-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math.,22 (1964), 5–60.
6. H. BASS, Algebraic K-theory, Benjamin, New York, 1968.
7. H. BASS, Libération des modules projectifs sur certains anneaux de polynômes, in Séminaire Bourbaki, 26e année (1973/1974), Exp. No. 448, Lecture Notes in Math., vol. 431, pp. 228–354, Springer, Berlin, 1975.
8. S. M. BHATWADEKAR, A cancellation theorem for projective modules over affine algebras over C1-fields, J. Pure Appl.
Algebra,1–3 (2003), 17–26.
9. S. BLOCH, Torsion algebraic cycles, K2, and Brauer groups of function fields, in The Brauer Group (Sem., Les Plans-sur-Bex, 1980), Lecture Notes in Math., vol. 844, pp. 75–102, Springer, Berlin, 1981.
10. S. BLOCHand A. OGUS, Gersten’s conjecture and the homology of schemes, Ann. Sci. Éc. Norm. Super.,7 (1975), 181–
201, 1974.
11. J.-L. COLLIOT-THÉLÈNEand C. SCHEIDERER, Zero-cycles and cohomology on real algebraic varieties, Topology,35(2) (1996), 533–559.
12. J. FASELand V. SRINIVAS, A vanishing theorem for oriented intersection multiplicities, Math. Res. Lett.,15(3) (2008), 447–458.
13. J. FASELand V. SRINIVAS, Chow-Witt groups and Grothendieck-Witt groups of regular schemes, Adv. Math.,221 (2009), 302–329.
14. J. FASEL, Stably free modules over smooth affine threefolds, Duke Math. J.,156 (2011), 33–49.
15. D. R. GRAYSON, Universal exactness in algebraic K-theory, J. Pure Appl. Algebra,36 (1985), 139–141.
16. J. HORNBOSTEL, Constructions and dévissage in Hermitian K-theory, K-Theory,26 (2002), 139–170.
17. M. KAROUBI, Périodicité de la K-théorie hermitienne, in Algebraic K-theory, III: Hermitian K-Theory and Geometric Appli-cations (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., vol. 343, pp. 301–411, Springer, Berlin, 1973.
18. M. KAROUBI, Le théorème fondamental de la K-théorie hermitienne, Ann. Math.,112 (1980), 259–282.
19. N. MOHANKUMAR, Stably free modules, Am. J. Math.,107 (1985), 1439–1444.
20. A. S. MERKUR´EVand A. A. SUSLIN, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat.,46 (1982), 1011–1046, 1135–1136
21. J. S. MILNE, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, 1980.
22. J. MILNOR, Algebraic K-theory and quadratic forms, Invent. Math.,9 (1969/1970), 318–344.
23. J. MILNOR, Introduction to algebraic K-theory, Annals of Mathematics Studies, vol. 72. Princeton University Press, Prince-ton 1971
24. D. ORLOV, A. VISHIK, and V. VOEVODSKY, An exact sequence for KM∗/2 with applications to quadratic forms, Ann.
Math.,165 (2007), 1–13.
25. R. A. RAO, The Bass-Quillen conjecture in dimension three but characteristic=2,3 via a question of A. Suslin, Invent.
Math.,93 (1988), 609–618.
26. R. A. RAO, A stably elementary homotopy, Proc. Am. Math. Soc.,137 (2009), 3637–3645.
27. R. A. RAOand W. van der KALLEN, Improved stability for SK1and WMSdof a non-singular affine algebra, Astérisque, 226 (1994), 411–420. K-theory (Strasbourg, 1992)
28. M. ROITMAN, On unimodular rows, Proc. Am. Math. Soc.,95 (1985), 184–188.
29. M. SCHLICHTING, Hermitian K-theory of exact categories, K-Theory,5 (2010), 105–165.
30. M. SCHLICHTING, The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes, Invent. Math.,179 (2010), 349–433.
31. J.-P. SERRE, Cohomologie Galoisienne, Lecture Notes in Mathematics, vol. 5, 5th ed. Springer, Berlin, 1994.
32. A. A. SUSLIN, Torsion in K2of fields, K-Theory,1 (1987), 5–29.
33. A. A. SUSLIN, A cancellation theorem for projective modules over algebras, Dokl. Akad. Nauk SSSR,236 (1977), 808–811 (in Russian).
34. A. A. SUSLIN, On stably free modules, Math. U.S.S.R. Sbornik, (1977), 479–491. doi:10.1070/
SM1977v031n04ABEH003717
35. A. A. SUSLIN, Cancellation for affine varieties. (russian) modules and algebraic groups, Zap. Nauchn. Sem. Leningrad.
Otdel. Mat. Inst. Steklov. (L.O.M.I.),114 (1982), 187–195.
36. A. A. SUSLIN, Torsion in K2of fields, K-Theory,1 (1987), 5–29.
37. R. G. SWAN, A cancellation theorem for projective modules in the metastable range, Invent. Math.,27 (1974), 23–43.
38. W. van der KALLEN, A module structure on certain orbit sets of unimodular rows, J. Pure Appl. Algebra,57(3) (1989), 281–316.
39. L. N. VASERSTEIN, Operations on orbits of unimodular vectors, J. Algebra,100 (1986), 456–461.
40. L. N. VASERSTEINand A. A. SUSLIN, Serre’s problem on projective modules over polynomial rings, and algebraic k-theory, Izv. Akad. Nauk SSSR, Ser. Mat.,40 (1976), 993–1054.
41. V. VOEVODSKY, Motivic cohomology withZ/2-coefficients, Publ. Math. Inst. Hautes Études Sci.,98 (2003), 59–104.
42. T. VORST, The general linear group of polynomial rings over regular rings, Commun. Algebra,9 (1981), 499–509.
J. Fasel
Mathematisches Institut der Universität München, Theresienstrasse 39,
80333 München, Germany jean.fasel@gmail.com R.A. Rao
Tata Institute of Fundamental Research, 1, Dr. Homi Bhabha Road, Navy Nagar, Mumbai 400 005, India
ravi@math.tifr.res.in R.G. Swan
University of Chicago, Chicago, IL, 60637, USA swan@math.uchicago.edu
Manuscrit reçu le 13 novembre 2011 Manuscrit accepté le 7 juin 2012 publié en ligne le 30 juin 2012.