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Bruhat Intervals in the Infinite Symmetric Group are Cohen-Macaulay

We show that the (non-Noetherian) Stanley-Reisner ring of the order complex of certain intervals in the Bruhat order on the infinite symmetric group $S_\infty$ of all auto-bijections of $\mathbb{N}$ is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. This gives an infinite-dimensional version of results due to Edelman, Björner, and Kind and Kleinschmidt for finite symmetric groups $S_n$.