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We examine the shuffle algebra defined over the ring $\mathbf{R} =\mathbb{C}[q_1^{\pm 1}, q_2^{\pm 1}]$, also called the integral shufflealgebra, which was found by Schiffmann and Vasserot to act on the equivariant$K$-theory of the Hilbert scheme of points in the plane. We find that themodules of 2 and 3 variable elements of the integral shuffle algebra arefinitely generated and prove a necessary condition for an element to be in theintegral shuffle algebra for arbitrarily many variables.

The integral shuffle algebra and the $K$-theory of the Hilbert scheme of points in $\mathbb{A}^2$