Historically, developments in Algebraic Geometry have always been enabled by progress in algebra. At first, Italian mathematicians started the study of algebraic curves with simple algebraic tools, but the foundations were insufficient. Subsequently, Emmy Noether's development of Commutative Algebra lead to substantial progress in algebraic geometry.
Later, Grothendieck made huge progress, finding that in order to study varieties, which are the fundamental objects in algebraic geometry, one must endow spaces with more structure. The additional structure and flexibility of schemes enabled the usage of powerful tools from homological algebra.
More recently, homotopical algebra has become increasingly popular, suggesting another powerful tool with which one might study algebraic geometry. However, schemes possess nearly no homotopical data. The solution is to study schemes enriched with even more structure, prompting the study of ``derived algebraic geometry".