Symplectic completion over smooth affine algebras

Abstract: In this article, we prove the following results:\ \noindent \text{(1).} Let $R$ be a smooth affine algebra of dimension $3$ over an algebraically closed field $K$ with $3!\in K$, then we show that $\Um_4(R)=e_1\Sp_4(R)$ and $\Um_4(R [X])=e_1\Sp_4(R[X])$. \noindent \text{(2).} We also show that if $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $K$ with $4!\in K$, and assume that $\W_E(R)$ is divisible, then $\Um_3(R)=e_1\SL_3(R)$. As a consequence it is shown that if $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $K$ with $4!\in K$, and assume that $\W_E(R)$ is divisible, then $\Um_4(R)=e_1\Sp_4(R)$. \noindent \text{(3).} We show that if $R$ is a local ring of dimension $3$ with $\frac{1}{3!}\in R$. Then $\Um_4(R[X])=e_1\Sp_4(R[X])$. \noindent \text{(4).} We also show that if $R=\oplus_{i\geq 0}R_i$ is a graded ring over a local ring of dimension $3$ with $\frac{1}{3!}\in R$. Then $\Um_4(R)=e_1\Sp_4(R)$.

• The paper proves transitivity of Sp4 on unimodular rows (Um4(R)) in smooth affine algebras of dimension 3 and 4 under factorial invertibility conditions.
• It uses techniques such as Vaserstein’s symbol and explicit matrix factorizations to bridge classical algebraic K-theory with modern commutative algebra methods.
• The work extends to polynomial extensions and graded rings, offering new insights into symplectic and relative completion properties.

Symplectic Completion Over Smooth Affine Algebras

Overview and Context

This article investigates the problem of symplectic completion of unimodular rows over smooth affine algebras, extending several classical results in the context of the symplectic group $Sp_{2n}(R)$ and polynomial rings. Specifically, the work provides new transitivity results for the action of $Sp_4(R)$ on the set $Um_4(R)$ under various algebraic conditions. The methodology is rooted in classical algebraic K-theory but also incorporates commutative algebra techniques, $\mathbb{A}^1$-homotopy methods, Swan-Weibel's homotopy trick, and deep structure theorems on unimodular rows in relation to symplectic and special linear groups.

Main Results and Technical Contributions

Transitivity of $Sp_4(R)$ on $Um_4(R)$ in Dimension 3

The article establishes that for $R$ a smooth affine algebra of dimension 3 over an algebraically closed field $K$ with $3!\in K$, the group $Sp_4(R)$ acts transitively on $Sp_4(R)$0; equivalently, $Sp_4(R)$1. The result also holds for $Sp_4(R)$2. This fills a gap in prior literature, where analogous statements were known for higher stability ranges (i.e., for $Sp_4(R)$3 and $Sp_4(R)$4 under further conditions) but had not been established for the case of $Sp_4(R)$5 specifically in the symplectic context.

The proofs proceed by reduction to the case of $Sp_4(R)$6 acting transitively on $Sp_4(R)$7, and then relating this action to the symplectic group via the Vaserstein symbol and explicit matrix factorizations. The approach leverages a detailed analysis of alternating matrices and commutator calculations in $Sp_4(R)$8, as well as an application of results from [suslin1976serre], [suslin1977stably], and the structural results on stably free modules.

The $Sp_4(R)$9 Case with Divisible $Um_4(R)$0

The authors extend their analysis to the case where $Um_4(R)$1 is a smooth affine algebra of dimension 4 over an algebraically closed field $Um_4(R)$2 with $Um_4(R)$3 and divisible $Um_4(R)$4. Under these conditions, they show $Um_4(R)$5 and deduce $Um_4(R)$6 as a corollary. The argument utilizes the group-theoretic structure of the elementary symplectic Witt group $Um_4(R)$7 and its divisibility, coupled with Suslin's results on powers of unimodular rows and the explicit description of the orbit set $Um_4(R)$8 as bijective to $Um_4(R)$9.

Polynomial Extensions and Graded Analogs

The symplectic completion property is further generalized to polynomial extensions and graded rings. For any commutative noetherian local ring $\mathbb{A}^1$0 of dimension 3 with $\mathbb{A}^1$1, the equality $\mathbb{A}^1$2 is proved. This leverages a combination of the Bass-Quillen type results on the stable range of unimodular row completions ([rao1988bass], [rao1991completing]) and intricate commutator relations in polynomial rings.

For graded rings $\mathbb{A}^1$3 of dimension 3 (with $\mathbb{A}^1$4 local and containing inverses of $\mathbb{A}^1$5), Swan-Weibel’s homotopy trick is utilized to transport the symplectic completion property from $\mathbb{A}^1$6 to the full graded ring $\mathbb{A}^1$7, obtaining $\mathbb{A}^1$8. The transfer of algebraic properties between the graded and base components is conducted through explicit homotopy maps and careful use of augmentation ideals.

Relative and Excision Versions

A relative version is also proved: for a commutative noetherian local ring $\mathbb{A}^1$9 of dimension 3, ideal $Sp_4(R)$0, and $Sp_4(R)$1 invertible in $Sp_4(R)$2, one has $Sp_4(R)$3. The proof relies on the excision ring construction and canonical lifts between $Sp_4(R)$4 and $Sp_4(R)$5, combined with the previously established absolute results and reduction to the diagonal and local component.

Significant Claims and Theoretical Advances

• The paper asserts the transitivity of $Sp_4(R)$6 on unimodular rows in cases previously unsettled, thus identifying the precise boundaries for symplectic completion in terms of dimension, invertibility of factorials, and group-theoretic divisibility conditions.
• The results identify a strong form of symplectic cancellation and completion for both affine and graded settings, as well as for polynomial extensions and relative ideal-theoretic contexts.
• The link between orbit sets of unimodular rows, their group structures, and the elementary symplectic Witt group is elucidated with a high degree of technical control, particularly via divisibility properties.
• The adaptation of the Swan-Weibel homotopy trick to the symplectic context over graded rings is a powerful and technically non-trivial contribution.

Methodological Insights

The proofs combine several advanced techniques:

• Detailed structural analysis of alternating and symplectic matrices, leveraging explicit block constructions and Pfaffian calculations.
• The use of Vaserstein's rule and the associated Vaserstein symbol to translate between unimodular rows and $Sp_4(R)$7.
• Inductive arguments on dimension and stability range, invoking the deep theorems of Suslin, Swan, Bass, Gupta, and Fasel regarding cancellation of stably free modules and freeness in the stable range.
• Excision is used for relative statements, ensuring that properties of the base ring transfer to the relevant ideal-related structures.

Numerical Results and Algebraic Constraints

The principal numerical conditions involve the invertibility of $Sp_4(R)$8 in the coefficient ring. For example, the results are contingent on $Sp_4(R)$9 or $Um_4(R)$0 being invertible. The divisible nature of $Um_4(R)$1 is also a key algebraic hypothesis for the results in dimension 4. These constraints are both necessary and optimal given the use of classical commutator calculus, the homotopy principle, and the structure of projective modules.

Implications and Future Directions

From a theoretical standpoint, the paper sharpens our understanding of the interplay between algebraic K-theory, symplectic group actions, and module theory over affine and graded commutative rings. The established results provide tools for tackling finer questions about the structure and classification of projective modules, the existence of free summands, and symplectic group orbits in critical low-rank, low-dimension cases.

Practically, these results impact computations in computational algebra, explicit module decompositions, and may inform further developments in algebraic topology (especially in motivic homotopy theory) through their connections with homotopy invariance and the behavior of vector bundles under symplectic automorphisms.

Potential Future Work

There is scope for extending these completion and transitivity results:

• To cases where the invertibility of $Um_4(R)$2 is replaced by weaker arithmetic conditions, such as in mixed characteristic.
• To singular affine algebras or non-noetherian settings, possibly using $Um_4(R)$3-homotopy or in the presence of torsion phenomena.
• To higher symplectic ranks or to symplectic analogs of the results currently known only for $Um_4(R)$4.
• To studying symplectic orbits for more general classes of graded, filtered, or valuation rings.

Conclusion

This work provides a rigorous and detailed extension of symplectic completion and cancellation theorems for unimodular rows over smooth affine, polynomial, and graded algebras, establishing optimal transitivity results for $Um_4(R)$5 in dimension 3 and 4 under precise algebraic and arithmetic hypotheses. The results not only clarify structural properties of projective modules and their symplectic completions but also open avenues for further research in symplectic K-theory, commutative algebra, and their interaction with algebraic topology.

Reference: "Symplectic completion over smooth affine algebras" (2603.28293)