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The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide
Two recent constructions independently give bigraded commutative Hopf algebra structures on the homology of $\mathrm{GL}(\mathbb{Z})$ with rational Steinberg coefficients. One, by Ash--Miller--Patzt, uses the sharbly resolution. The other, by Brown--Chan--Galatius--Payne, uses the Quillen spectral s
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