A geometric generalization of Kaplansky's direct finiteness conjecture

11/15/2021

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by   Xuan Kien Phung, et al.

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Let G be a group and let k be a field. Kaplansky's direct finiteness conjecture states that every one-sided unit of the group ring k[G] must be a two-sided unit. In this paper, we establish a geometric direct finiteness theorem for endomorphisms of symbolic algebraic varieties. Whenever G is a sofic group or more generally a surjunctive group, our result implies a generalization of Kaplansky's direct finiteness conjecture for the near ring R(k, G) which is k[X_g g ∈ G] as a group and which contains naturally k[G] as the subring of homogeneous polynomials of degree one. We also prove that Kaplansky's stable finiteness conjecture is a consequence of Gottschalk's Surjunctivity conjecture.