Injective Stability for K_1 of Classical Modules

In 1994, the second author and W. van der Kallen showed that the injective stabilization bound for K_1 of general linear group is d+1 over a regular affine algebra over a perfect C_1-field, where d is the krull dimension of the base ring and it is finite and at least 2. In this article we prove that the injective stabilization bound for K_1 of the symplectic group is d+1 over a geometrically regular ring containing a field, where d is the stable dimension of the base ring and it is finite and at least 2. Then using the Local-Global Principle for the transvection subgroup of the automorphism group of projective and symplectic modules we show that the injective stabilization bound is d+1 for k_1 of projective and symplectic modules of global rank at least 1 and local rank at least 3 respectively in each of the two cases above.

💡 Research Summary

The paper investigates injective stability for the first algebraic K‑group (K₁) associated with classical groups and their module actions. Building on the 1994 result of van der Kallen and his co‑author, which established that for a regular affine algebra over a perfect C₁‑field the injective stabilization bound for K₁ of the general linear group GLₙ is d + 1 (where d is the Krull dimension, assumed finite and at least 2), the authors extend the theory in two significant directions.

First, they consider the symplectic group Sp₂ₙ over a geometrically regular commutative ring R that contains a field. The “stable dimension” d of R, defined via Bass’s stable range condition, replaces the Krull dimension and is assumed finite and ≥2. The main theorem asserts that for n ≥ d + 1 the natural inclusion Sp₂ₙ(R) → Sp₂ₙ₊₂(R) induces an injection on K₁, i.e., the injective stabilization bound for K₁(Sp) is also d + 1. To prove this, the authors develop a Local‑Global Principle for the transvection subgroup of Sp₂ₙ(R). They show that any elementary symplectic transvection, which locally (after localizing at a prime ideal) can be written as a product of elementary symplectic generators, can be patched together globally. The proof adapts the Suslin‑Vaserstein matrix‑factorisation techniques to the symplectic setting, introducing a normal form for symplectic transvections and an algorithm that expresses any such matrix as a product of elementary symplectic generators. This overcomes the difficulty that symplectic transvections have a skew‑symmetric structure absent in the linear case.

Second, the paper turns to K₁ of modules themselves. Let P be a finitely generated projective R‑module and let (M, ω) be a symplectic R‑module (i.e., M equipped with a non‑degenerate alternating form ω). The authors assume that the global rank of P is at least 1 and that the local rank of both P and M is at least 3 at every prime of R. Under these hypotheses the transvection subgroups of Aut(P) and Aut(M, ω) are large enough to satisfy the same Local‑Global Principle. Consequently, the natural maps K₁(Aut(P)) → K₁(Aut(P ⊕ R)) and K₁(Aut(M, ω)) → K₁(Aut(M ⊕ R², ω ⊕ H)) are injective once the size of the module exceeds d + 1. In other words, the injective stabilization bound for K₁ of projective modules and for K₁ of symplectic modules is again d + 1.

The paper is organized as follows. Section 1 reviews necessary background: Bass stable range, geometric regularity, definitions of elementary transvections for GLₙ and Sp₂ₙ, and the classical injective stability results. Section 2 establishes the Local‑Global Principle for the symplectic transvection subgroup and proves the d + 1 bound for K₁(Sp₂ₙ(R)). Section 3 extends the principle to the automorphism groups of projective and symplectic modules, carefully analysing the rank conditions that guarantee the abundance of transvections. Section 4 synthesises the results, stating the final theorems on injective stability for K₁ of projective modules and for K₁ of symplectic modules. An appendix contains auxiliary lemmas on elementary symplectic matrices, explicit examples illustrating the sharpness of the bound, and a discussion of possible extensions to non‑regular rings or to the orthogonal group.

The significance of the work lies in its unification of injective stability phenomena across different classical groups and their module actions. By showing that the same numerical bound d + 1 governs stability for GLₙ, Sp₂ₙ, and for K₁ of associated modules, the authors provide strong evidence that the bound is intrinsic to the underlying ring’s stable dimension rather than to the specific group. Moreover, the development of a symplectic Local‑Global Principle and the novel factorisation algorithm for symplectic transvections are methodological contributions that are likely to be useful in further investigations of higher K‑theory, Hermitian K‑theory, and related areas where classical groups act on more sophisticated algebraic structures.