September 24, 2025
Let \(R\) be a real smooth affine domain of dimension \(3\) such that \(R\) has either no real maximal ideals or the intersection of all real maximal
ideals in \(R\) has height at least \(1\). Then we prove that all stably free \(R\)-modules of rank \(2\) are free if and
only if the Hermitian \(K\)-theory group \(W_{SL}(R)\) is trivial.
2020 Mathematics Subject Classification: 13C10, 19A13, 19G38.
Keywords: stably free module, projective module, cancellation.
A finitely generated module \(P\) over a commutative ring \(R\) is called stably free if there exists an isomorphism of the form \(P \oplus R^m \cong
R^n\) for some integers \(m,n \geq 0\). A fundamental question in algebraic \(K\)-theory and commutative algebra is the question under which conditions stably free modules are
actually free. If \(R\) is a Noetherian commutative ring of Krull dimension \(d\), then classical results imply that a stably free \(R\)-module of rank \(\geq d+1\) is always free (cf. [1]). Improvements of this classical result were proven for affine algebras over algebraically closed
fields: If \(R\) is affine algebra of dimension \(d\) over an algebraically closed field \(k\), then any stably free \(R\)-module of rank \(d\) is free (cf. [2]); furthermore, if \(R\) is a
smooth affine algebra of dimension \(d \geq 3\) over an algebraically closed field \(k\) with \((d-1)! \in k^{\times}\), then stably free \(R\)-modules of rank \(d-1\) are free (cf. [3]).
For affine algebras over \(\mathbb{R}\), analogues of the results over algebraically closed fields mentioned above do not exist in general. For example, if we let \(R = \mathbb{R}[x,y,z]/\langle
x^2 + y^2 + z^2 -1 \rangle\) be the real algebraic \(2\)-sphere, then it is well-known that the kernel of the \(R\)-linear homomorphism \((x,y,z): R^3
\rightarrow R, (\lambda_{1},\lambda_{2},\lambda_{3}) \mapsto \lambda_{1} x + \lambda_{2} y + \lambda_{3} z\) is a stably free module of rank \(2\) which is not free. As shown in S. Banerjee’s beautiful work (cf.
[4]), the situation for affine \(\mathbb{R}\)-algebras changes substantially when one focuses on specific \(\mathbb{R}\)-algebras:
Theorem 1 ([4]). Let \(R\) be a commutative ring satisfying the following condition P: The ring \(R\) is an affine \(\mathbb{R}\)-algebra of dimension \(d\) such that
it has no real maximal ideals or
the intersection of all real maximal ideals in \(R\) has height at least \(\geq 1\).
Then all stably free \(R\)-modules of rank \(d\) are free.
The purpose of this article is prove a result on stably free modules of rank \(d-1\) over regular integral domains of dimension \(d\) satisfying condition P in Theorem 1 in the first interesting case, namely in dimension \(3\) (in dimensions \(d=1\) and \(d=2\), stably free modules of rank \(d-1\) are easily seen to be free). We prove the following cohomological criterion for all stably free modules of rank \(2\) over a regular integral domain of dimension \(3\) satisfying condition P to be free (cf. Theorem 11):
Theorem 2. Let \(R\) be a regular integral domain of dimension \(d=3\) satisfying condition \(\boldsymbol{P}\) from Theorem 1. Then all stably free \(R\)-modules of rank \(2\) are free if and only if \(W_{SL}(R) = 0\).
The abelian group \(W_{SL}(R)\) is a Hermitian \(K\)-theory group and was introduced in [5].
Theorem 2 is proven by analyzing the generalized Vaserstein symbol modulo SL introduced in [6];
as a matter of fact, we prove that this map is bijective in Theorem 10 and hence induces a group structure on the set of isomorphism classes of oriented stably free \(R\)-modules of rank \(2\) (Remark 12). The bijectivity of the Vaserstein symbol is essentially a consequence of a
transitivity statement on the actions of certain symplectic groups on the set of unimodular rows of length \(4\) over \(R\), which is proven in Theorem 8.
The paper is structured as follows: In Section [2] we give a brief introduction unimodular rows, the groups \(W_E (R)\) and \(W_{SL}(R)\) of a commutative ring \(R\) as needed for this paper and recall some statements proven in [4]
and [7] which will be used in the proof of the main results of this paper. Then we prove the main results of this paper in Section [3].
Let \(R\) be a commutative ring with unit and \(n \geq 1\) an integer. A unimodular row of length \(n\) over \(R\) is a
row vector \((v_{1},...,v_{n})\) of length \(n\) with \(v_{i} \in R\), \(1 \leq i \leq n\), such that \(\langle v_{1},...,v_{n} \rangle = R\). We denote the set of unimodular rows of length \(n\) over \(R\) by \(Um_{n}(R)\). The
group \(GL_{n}(R)\) of invertible \(n \times n\)-matrices acts on the right on \(Um_{n}(R)\) by matrix multiplication. The same holds automatically for any
subgroup of \(GL_{n}(R)\); in particular, the subgroup \(SL_{n}(R)\) of matrices with determinant \(1\), the subgroup \(E_{n}(R)\) generated by elementary matrices act on the right on \(Um_{n}(R)\). If \(\chi\) is an alternating invertible matrix of rank \(2n\), then the subgroup \(Sp(\chi)\) of \(GL_{2n}(R)\) consisting of matrices which are symplectic with respect to \(\chi\) act
on the right on \(Um_{2n}(R)\). For integers \(n,i\) with \(1 \leq i \leq n\), we denote by \(\pi_{i,n}\) the unimodular row
\((0,...,0,1,0,...,0)\) of length \(n\) over \(R\) with \(1\) in the \(i\)th slot and \(0\)’s elsewhere and \(\pi_{n}=\pi_{n,n}\).
Similarly, a unimodular column of length \(n\) over \(R\) is a column vector \({(v_{1},...,v_{n})}^{t}\) of length \(n\)
such that \((v_{1},...,v_{n}) \in Um_{n}(R)\). We denote the set of unimodular columns of length \(n\) over \(R\) by \(Um_{n}^{t}(R)\). The group \(GL_{n}(R)\) and hence all its subgroups act on the left on \(Um_{n}^{t}(R)\) by matrix multiplication. If \(\chi\) is an alternating invertible matrix of rank \(2n\), then the group \(Sp(\chi)\) acts on the left on \(Um_{2n}^{t}(R)\).
For integers \(n,i\) with \(1 \leq i \leq n\), we let \(e_{i,n} = \pi_{i,n}^{t}\) and \(e_{n}=e_{n,n}\).
For any integer \(n \geq 1\), we let \(A_{2n}(R)\) denote the set of invertible alternating matrices of rank \(2n\). We denote by \(\psi_{2n} \in A_{2n} (R)\) the matrix which is defined inductively by
\(\psi_2 = \begin{pmatrix} 0 & 1 \\ - 1 & 0 \end{pmatrix}\)
and \(\psi_{2n+2} = \psi_{2n} \perp \psi_2\). Then we obtain embeddings \(A_{2m} (R) \rightarrow A_{2n} (R)\), \(M \mapsto M \perp \psi_{2n-2m}\) for any \(m < n\) and we let \(A (R)\) denote the direct limit of the sets \(A_{2n} (R)\) under these embeddings. If \(M \in A_{2m} (R)\) and \(N \in A_{2n} (R)\), we call them equivalent, \(M \sim N\), if there is an integer \(l \geq 1\) and a matrix \(E \in E_{2n+2m+2l}(R)\) such that
\(M \perp \psi_{2n+2l} = E^{t} (N \perp \psi_{2m+2l}) E\).
It is easy to see that this defines an equivalence relation on \(A(R)\) and the corresponding set of equivalence classes \(A(R)/{\sim}\) is denoted \(W'_E
(R)\). It follows from [5] that the orthogonal sum of matrices induces the structure of an abelian group on \(W'_E (R)\).
The subgroup generated by alternating invertible matrices with Pfaffian \(1\) will be denoted \(W_E (R)\) and is called the elementary symplectic Witt group of \(R\).
Following [7], we have an exact Karoubi periodicity sequence
\(K_{1}{Sp} (R) \xrightarrow{f} K_{1} (R) \xrightarrow{H} W'_{E} (R) \xrightarrow{\eta} K_{0}{Sp} (R) \xrightarrow{f'} K_{0} (R)\)
involving the group \(W'_E (R)\) and classical algebraic \(K\)-theory and symplectic \(K\)-theory groups. Here the homomorphisms \(f\) and \(f'\) are both the usual forgetful homomorphisms. Furthermore, the map \(K_{1} (R) \xrightarrow{H} W'_{E} (R)\) is given by the assignments \(M \mapsto M^{t} \psi_{2n} M\) for all \(M \in GL_{2n} (R)\), while the homomorphism \(W'_{E} (R) \xrightarrow{\eta} K_{0}{Sp} (R)\) is given by the assignments \(M \mapsto [R^{2n}, M] - [R^{2n},\psi_{2n}]\) for all \(M \in A_{2n} (R)\). Again following [7], the sequence above can be rewritten as
\(K_{1}{Sp} (R) \xrightarrow{f} SK_{1} (R) \xrightarrow{H} W_{E} (R) \xrightarrow{\eta} K_{0}{Sp} (R) \xrightarrow{f'} K_{0} (R)\).
We let \(W_{SL}(R)\) denote the cokernel of the homomorphism \(SK_{1} (R) \xrightarrow{H} W_E (R)\). We refer the reader to [7] for details on the groups \(W'_E (R)\), \(W_E (R)\) and \(W_{SL}(R)\). The following lemma characterizes matrices which lie in the kernel of the map \(H\) above and will be used in the proof of Theorem 8:
Lemma 3 ([7]). Let \(R\) be a commutative ring and let \(\chi \in A_{2n} (R)\). If \(\varphi \in GL_{2n} (R)\) such that its class \([\varphi] \in K_{1} (R)\) lies in \(\ker (H)\), then there are \(m \in \mathbb{N}\) and \(\varphi' \in SL_{2n+2m} (R)\) such that \([\varphi] = [\varphi'] \in K_1 (R)\) and \(\varphi'\) is symplectic with respect to \(\chi \perp \psi_{2m}\).
The following lemma is another important ingredient for the proof of Theorem 8:
Lemma 4 ([7]). Let \(R\) be a commutative ring and let \(\chi_{1}\) and \(\chi_{2}\) be invertible alternating matrices of rank \(2n\) over \(R\) such that \({\varphi}^{t} (\chi_{1} \perp \psi_{2}) \varphi = \chi_{2} \perp \psi_{2}\) holds for some \(\varphi \in SL_{2n+2}(R)\). Furthermore, let \(\chi = \chi_{1} \perp \psi_{2}\). If the equality \(Um^{t}_{2n+2}(R) = (E_{2n+2}(R) \cap {Sp} (\chi)) e_{2n+2}\) holds, then one has \({\psi}^{t} \chi_{2} \psi = \chi_{1}\) for some \(\psi \in SL_{2n}(R)\) such that \([\psi] = [\varphi] \in K_{1}(R)\).
Now let \(n \geq 1\) and let \(v = (v_{1},...,v_{n}) \in Um_{n}(R)\) be a unimodular row and \(w={(w_{1},...,w_{n})}^{t} \in Um^{t}_{n}(R)\) a unimodular column such that \(\sum_{i=1}^{n} v_{i} w_{i} = 1\) (i.e., \(w\) defines a section of \(v\)). Then Suslin defined matrices \(\alpha_{n} (v,w) \in SL_{2^{n-1}} (R)\) called Suslin matrices in [8]; he then showed that for any such \(v\) with section \(w\) there exists an invertible \(n \times n\)-matrix \(\beta (v,w)\) whose first row is \((v_{1},...,v_{n}^{(n-1)!})\) such that the classes of \(\beta (v,w)\) and \(\alpha_{n} (v,w)\) coincide in \(K_{1} (R)\) (cf. [9]).
Lemma 5 ([7]). Let \(R\) be an affine algebra over a perfect field \(k\) and let \(l \geq 1\) and \(n \geq 3\). Furthermore, let \(v = (v_{1},...,v_{n}) \in Um_{n} (R)\) be a unimodular row with a section \(w\) and let \(v' = (v_{1},...,v_{n}^{l}) \in Um_{n} (R)\) with any section \(w'\). If \(l\) is even, then \([\alpha_{n}(v',w')] \in SK_{1} (R)\) is an \(\dfrac{l}{2}\)-fold multiple of an element in \(SK_{1}(R)\).
For all \(n \geq 1\) and any field \(k\), let \(S_{2n-1} = (k[x_{1},...,x_{n},y_{1},...,y_{n}]/\langle \sum_{i=1}^{n} x_{i}y_{i} - 1 \rangle\). Note that \(x=(x_{1},...,x_{n})\) is a unimodular row of length \(n\) over \(S_{2n-1}\) with section given by \(y=(y_{1},...,y_{n})\). Now assume that \(R\) is an affine \(k\)-algebra. Then it is easy to see that one has a correspondence
\(\{(a,b)|a,b \in \mathit{Um}_{n} (R), a b^{t} = 1\} = Hom_{{k}\textit{-Alg}} (S_{2n-1},R)\),
where \(k\)-Alg is the category of \(k\)-algebras. The rings \(S_{2n-1}\) will be considered in the proof of Theorem 8.
We now conclude this section by recalling two important results from [4] on certain commutative rings satisfying the condition \(\boldsymbol{P}\) from the introduction.
Theorem 6 ([4]). Let \(R\) be a regular integral domain satisfying condition \(\boldsymbol{P}\) from Theorem 1. Then the map \(SL_{d+1}(R)/E_{d+1}(R) \rightarrow SK_{1}(R)\) is bijective.
Theorem 7 ([4]). Let \(R\) be a ring satisfying condition \(\boldsymbol{P}\) from Theorem 1. Then, for any unimodular row \(v \in Um_{d+1}(R)\) and integer \(n \geq 1\), there are \(w = (w_{1},...,w_{d+1}) \in Um_{d+1}(R)\) and \(\varphi \in E_{d+1}(R)\) such that \(v \varphi = (w_{1},...,w_{d+1}^{n})\).
Theorem 8. Let \(R\) be a regular integral domain satisfying condition \(\boldsymbol{P}\) from Theorem 1. Let \(\chi \in A_{d+1}(R)\) be an alternating invertible matrix of rank \(d+1\). Then \(Sp (\chi)\) acts transitively on \(Um_{d+1}(R)\).
Proof. First we realize that it suffices to show the following statement: If \(v = (v_{1},...,v_{d+1}) \in Um_{d+1}(R)\), then there exists \(\varphi \in SL_{d+1}(R)\) with first
row \(v\) whose class \([\varphi] \in K_{1} (R)\) lies in the image of \(K_{1}{Sp} (R) \xrightarrow{f} K_{1} (R)\).
Indeed, assume that there is \(\varphi\) as above; then Lemma 3 shows that, for some \(m \geq 0\), there is \(\psi \in Sp(\chi \perp \psi_{2m})\) such that \([\psi]=[\varphi] \in K_1 (R)\). As a matter of fact, we can actually assume that \(m = 0\) by Lemma 4 because \(E_{d+1+2n}(R) \cap Sp(\chi \perp \psi_{2n})\) acts transitively on \(Um_{d+1+2n} (R)\) for \(n \geq 1\) by [1] and [5]. By Theorem 6, the map \(SL_{d+1} (R)/E_{d+1} (R) \rightarrow SK_{1} (R)\) is injective. Therefore \(\varphi {\psi}^{-1} \in E_{d+1}
(R)\). By [5] the equality \(\pi_{1,d+1} E_{d+1} (R) = \pi_{1,d+1} ({E}_{d+1} (R) \cap Sp(\chi))\) holds, so there exists \(\psi' \in {E}_{d+1} (R) \cap Sp (\chi)\) such that \(\pi_{1,d+1}\varphi {\psi}^{-1} = \pi_{1,d+1} \psi'\). As a consequence, \(v = \pi_{1,d+1} \varphi =
\pi_{1,d+1} \psi' \psi\) is indeed the first row of a matrix in \(Sp (\chi)\).
So let us now show that for \(v = (v_{1},...,v_{d+1}) \in Um_{d+1}(R)\) there exists \(\varphi \in SL_{d+1}(R)\) with first row \(v\) whose class \([\varphi] \in K_{1} (R)\) lies in the image of \(f: K_{1}{Sp} (R) \rightarrow K_{1} (R)\) or, equivalently, in the kernel of \(H: SK_{1}(R) \rightarrow W_E
(R)\). First of all, we notice that by Theorem 7 any unimodular row of length \(d+1\) over \(R\) can actually be
transformed via elementary matrices to a row of the form \(v = (v_{1},...,v_{d+1}^{2{d!}^{2}})\). Therefore we only need to consider rows of the form \(v =
(v_{1},...,v_{d+1}^{2{d!}^{2}})\).
For this purpose, we also consider the row \(v' = (v_{1},...,v_{d+1}^{2d!})\) with a chosen section \(w'\); then the first row of the matrix \(\beta_{d+1}(v',w') \in SL_{d+1}(R)\) is precisely the row \(v\). It is well-known that \([\beta_{d+1}(v',w')]\) lies in the image of the
forgetful map \(K_{1}{Sp} (R) \xrightarrow{f} K_{1} (R)\) if \(d+1 \equiv 2~mod~4\) (cf. [10]), which
finishes the proof of the theorem in this case.
It remains to prove the theorem for \(d+1 \equiv 0~mod~4\). In this case, by [6], we have that \(W_E
(S_{2(d+1)-1}) = \mathbb{Z}/2\mathbb{Z}\) and therefore \(W_E (S_{2(d+1)-1})\) is \(2\)-torsion. If we now let \(x'=(x_{1},...,x_{d},x_{d+1}^{2d!}) \in
Um_{d+1}(S_{2(d+1)-1})\) with a chosen section \(y'\), then Lemma 5 shows that \([\beta_{d+1} (x',y')] \in
SK_{1}(S_{2(d+1)-1})\) is a \(d!\)-fold multiple of an element in \(SK_{1}(S_{2(d+1)-1})\). Consequently, we must have \(H([\beta_{d+1}(x',y')]) =
0\) as \(W_E (S_{2(d+1)-1})\) is \(2\)-torsion.
This implies that \(H([\beta_{d+1}(v',w')]) = 0\) as follows: We let \(v''=(v_{1},...,v_{d+1}) \in Um_{d+1}(R)\) with some chosen section \(w''\). We obtain an induced ring homomorphism \(\varphi: S_{2(d+1)-1} \rightarrow R, (x,y) \mapsto (v'',w'')\) fitting into a commutative diagram
\(\begin{xy} \xymatrix{ SK_1 (S_{2(d+1)-1}) \ar[r]^{H} \ar[d]_{\varphi_{\ast}} & W_E (S_{2(d+1)-1}) \ar[d]^{\varphi_{\ast}} \\ SK_{1} (R) \ar[r]_{H} & W_E (R). } \end{xy}\)
Since \(\varphi_{\ast}: SK_{1} (S_{2(d+1)-1}) \rightarrow SK_{1}(R)\) sends \([\beta_{d+1} (x',y')]\) to \([\beta_{d+1} (v',w')]\), the commutative diagram implies that indeed \(H([\beta_{d+1}(v',w')]) = 0 \in W_E (R)\). This finishes the proof of the theorem in case \(d+1 \equiv 0~mod~4\). ◻
Remark 9. Theorem 8 is essentially a consequence of the results obtained in [4] and [7]. It was already observed in [4] that the proof of [7] together with Theorem 6 and Theorem 7 imply that \(Sp_{d+1}(R)\) acts transitively on \(Um_{d+1}(R)\). Lemma 3 and Lemma 4 enable us to prove a more general statement in Theorem 8. For the convenience of the reader, we have given a full proof of this statement.
It follows from [6] that there is a well-defined map
\(V_{\theta_{0}}: Um_3 (R)/SL_3 (R) \rightarrow \tilde{V}_{SL}(R)\)
called Vaserstein symbol modulo SL associated to any fixed isomorphism \(\theta_{0}: \det(R^2) \xrightarrow{\cong} R\), where \(\tilde{V}_{SL}(R)\) is an abelian group introduced in [6]; as explained in [6], the abelian group \(\tilde{V}_{SL}(R)\) is canonically isomorphic to the group \(W_{SL}(R)\) from Section [2].
Theorem 10. Let \(R\) be a regular integral domain of dimension \(d=3\) satisfying condition \(\boldsymbol{P}\) from Theorem 1. Let \(\theta_{0}: R \xrightarrow{\cong} \det (R^2)\) be a fixed isomorphism. Then \(V_{\theta_{0}}: Um_{3}(R)/SL_{3}(R) \xrightarrow{\cong} \tilde{V}_{SL}(R)\) is a bijection.
Proof. It follows from [6] that the generalized Vaserstein symbol modulo SL is bijective if \(SL_{5}(R)\) acts transitively on \(Um_{5}(R)\) and \(Sp (\chi)\) acts transitively on \(Um_{4}^{t}(R)\) for any invertible alternating \(4\times4\)-matrix \(\chi\) over \(R\). It follows directly from [1] that \(SL_{5}(R)\) acts transitively on \(Um_{5}(R)\). Theorem 8 shows that \(Sp (\chi^{-1})\) acts transitively on the right on \(Um_{4}(R)\) for any invertible alternating \(4 \times 4\)-matrix \(\chi\); the fact that \(Sp(\chi)\) acts transitively on the left on \(Um_{4}^{t}(R)\) then follows by transposition. ◻
Theorem 11. Let \(R\) be a regular integral domain of dimension \(d=3\) satisfying condition \(\boldsymbol{P}\) from Theorem 1. Then all stably free \(R\)-modules of rank \(2\) are free if and only if \(W_{SL}(R) = 0\).
Proof. Assume \(P\) is a stably free \(R\)-module of rank \(2\) and \(P \oplus R^n \cong R^{n+2}\) for some integer \(n \geq 1\). Then, by [4], we obtain an isomorphism \(P \oplus R \cong R^3\). In particular, \(P\) is defined via a unimodular row \(v\) of length \(3\) over \(R\) and \(P\) is free if and only if there is \(\varphi \in SL_{3}(R)\) with first row \(v\). The theorem now follows from Theorem 10 and the fact that \(\tilde{V}_{SL}(R)\) is isomorphic to \(W_{SL}(R)\). ◻
Remark 12. Theorem 10 actually implies a slightly stronger statement than the statement of Theorem 11: It follows from the discussion in [6] that the set \(Um_{3}(R)/SL_{3}(R)\) in Theorem 10 corresponds precisely to the set of isomorphism classes of oriented stably free \(R\)-modules of rank \(2\). This set turns out to be in bijection with the abelian group \(W_{SL}(R)\) by Theorem 10 and therefore inherits an abelian group structure.