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 homological shift algebras \({\text{HS}}_i(\mathcal {R}(I))=\bigoplus _{k\ge 1}{\text{HS}}_i(I^k)\) of I. When I has linear powers, these K-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 \({\rm v} \) -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 \(k\gg 0\) .
