The homological shift algebra of a monomial ideal

Let $S=K[x_1,\dots ,x_n]$ be the polynomial ring over a field $K$, and let $I\subset S$ be a monomial ideal. In this paper, we introduce the $i$th $\mathit{\text{homological}}\mathit{\text{shift}}\mathit{\text{algebras}}$ ${\text{HS}}_i(\mathcal{R}(I))={⨁}_{k\ge 1}{\text{HS}}_i(I^k)$ of $I$. If $I$ has linear powers, these algebras have the structure of a finitely generated bigraded module over the Rees algebra $\mathcal{R}(I)$ of $I$. Hence, many invariants of ${\text{HS}}_i(I^k)$, such as depth, associated primes, regularity, and the $\text{v}$-number, exhibit well behaved asymptotic behavior. We determine several families of monomial ideals $I$ for which ${\text{HS}}_i(I^k)$ has linear resolution for all $k\gg 0$. Finally, we show that ${\text{HS}}_i(I^k)$ is Golod for all monomial ideals $I\subset S$ with linear powers and all $k\gg 0$.