Commuting matrices, and modules over Artinian local rings

Abstract

Gerstenhaber showed in 1961 that any commuting pair of n x n matrices over a field k generates a k-algebra A of k-dimension \leq n. A well-known example shows that the corresponding statement for 4 matrices is false. The question for 3 matrices is open. Gerstenhaber's result can be looked at as a statement about the relation between the length of a 2-generator finite-dimensional commutative k-algebra A and the lengths of faithful A-modules. Wadsworth generalized this result to a larger class of commutative rings than those generated by two elements over a field. We recover his generalization, using a slightly improved argument. We then explore some examples, raise further questions, and make a bit of progress toward answering some of them. An appendix gives some results on generation and subdirect decompositions of modules over not necessarily commutative Artinian rings, generalizing a special case used in the paper. What I originally thought of as my main result turned out to have been anticipated by Wadsworth, so I probably won't submit this for publication unless I find further strong results to add. However, others may find interesting the observations, partial results, and questions noted, and perhaps make some progress on them.