We introduce semi-infinite homology of non-graded algebras due to Positselski and discuss the setup of semi-algebras, semi-modules and semi-contramodules. Our main example is the square of the small quantum group at a root of unity. Using a result of Bezrukavnikov and Lachowska we express the nilpotent part of semi-infinite homology for this algebra in terms of a certain completion functor on the category of coherent sheaves on the square of the Springer variety.