Efficient generation of projective modules: a motivic view

2025/10/15 by Aravind Asok, Asok, Aravind, Morgan Opie +5
Mathematics · #13C10 #14F42 #19E15 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2510.13687

openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

Abstract

Assume k is a field and R is a smooth k-algebra of dimension d. If P is a projective module of rank r, then it is well-known that P can be generated by r+d-elements (Forster--Swan). Under suitable assumptions on r and d, we investigate obstructions to generation of P by fewer than r+d elements using motivic homotopy theory. For example, we observe that a quadratic enhancement of the classical Segre class obstructs generation by r+d-1 elements, whether or not k is algebraically closed, generalizing old results of M.P. Murthy. Along the way, we also establish efficient generation results for symplectic modules.

Citations

• Splitting vector bundles over real algebraic varieties
• Strongly A¹-invariant sheaves (after F. Morel)
• On P¹-stabilization in unstable motivic homotopy theory
• Efficient generation, unimodular element in a geometric subring of a polynomial ring
• On the Whitehead theorem for nilpotent motivic spaces
• Suslin's cancellation conjecture in the smooth case
• Algebraic vs. topological vector bundles on spheres
• Comparing Euler classes
• Splitting vector bundles outside the stable range and A¹-homotopy sheaves of punctured affine spaces

Related

• On the number of generators of an algebra over a commutative ring
• Projective generation for equivariant D-modules
• Splitting criteria for projective modules over polynomial algebras
• Reconstructing projective modules from its trace ideal
• Irreducible components of module varieties: projective equations and rationality