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 \textit{ homological shift algebras} $\text{ HS}_i(\mathcal{ R}(I))=\bigoplus_{ k\ge1}\text{ HS}_i(I^k)$ 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 $k\gg0$. Finally, we show that $\text{ HS}_i(I^k)$ is Golod for all monomial ideals $I\subset S$ with linear powers and all $k\gg0$.
Categories: math.AC math.CO