Symplectic K-theory and a problem of Murthy (2311.03077v1)

Abstract: We compute low-dimensional K-groups of certain rings associated with the study of the Hermite ring conjecture. This includes a monoid ring whose low-dimensional K-groups were recently computed by Krishna and Sarwar in the case where the base ring is a regular ring containing the rationals. We are able to extend their result to an arbitrary regular base ring, thereby completing an answer to a question of Gubeladze. Our computation only relies on certain conveniently chosen analytic patching diagrams. These patching diagrams also allow us to investigate stably free modules appearing in a problem posed by Murthy. They make it possible to relate Murthy's problem to a stable question (about the relationship between symplectic and ordinary K-theory). More precisely, we show that Murthy's problem has a solution if the stable question has an affirmative answer, while a negative answer to the stable question implies that there are counterexamples to the Hermite ring conjecture. Since the module appearing in Murthy's problem has rank 2, the above mentioned argument relies on some theorems about patching with 2-by-2 matrices. These use so-called pseudoelementary 2-by-2 matrices, a notion we introduce which makes it possible to extend certain results valid for larger matrices to the case of 2-by-2 matrices (if we use pseudoelementary instead of elementary matrices). For example, we prove that the analogue of Vorst's theorem about matrices over polynomial extensions over a regular ring containing a field holds for 2-by-2 matrices.