交换代数：2025–2026 年度综述

An annual survey of the math.AC arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 120 papers and organised into 44 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Finite Fields and Polynomial Invariants (24.0%) — Investigates the structure of polynomial rings and invariants under finite group actions, with applications to coding theory and finite field arithmetic.
• Gröbner Bases and Cluster Algebras (11.9%) — Develops computational methods using Gröbner bases for commutative algebras and explores cluster structures in geometric and algebraic contexts.
• Local Rings and Singularity Theory (11.1%) — Studies homological properties and singularity types (e.g., Gorenstein, regular) in local and Noetherian rings, including F-purity and Briançon-Skoda theorems.
• Monomial Ideals and Combinatorial Commutative Algebra (10.8%) — Analyzes asymptotic properties, resolutions, and arithmetic ranks of monomial and lattice ideals arising from combinatorial structures.
• Jacobian Conjecture and Algebraic Varieties (9.1%) — Explores the Jacobian conjecture, counterexamples, and related conjectures concerning the geometry of varieties and vector bundles.
• Commutative Algebra Algorithms and AI (8.9%) — Applies automated reasoning, reinforcement learning, and computational frameworks to solve problems in commutative algebra and hypercomplex tensor algebra.
• Local Cohomology and Module Theory (7.2%) — Examines local cohomology modules, Bass numbers, and vanishing theorems, with formalizations in proof assistants like Lean4.
• Binomial Edge Ideals and Graph Theory (6.3%) — Investigates homological invariants, regularity, and purity of binomial edge ideals associated with graphs.
• Betti Numbers and Simplicial Complexes (5.8%) — Computes graded Betti numbers and homological invariants for simplicial complexes, skeletons, and multigraded structures.
• Cohen-Macaulay Rings and Perfectoid Geometry (4.8%) — Studies Cohen-Macaulayness, Frobenius actions, and perfectoid towers in equicharacteristic and mixed-characteristic settings.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.