Prismatic cohomology relative to $\delta$-rings

We develop prismatic and syntomic cohomology relative to a $\delta$-ring. This simultaneously generalizes Bhatt and Scholze’s absolute and relative prismatic cohomology and shows that the latter, which was defined relative to a prism, is in fact independent of the prism structure and only depends on the underlying $\delta$-ring. We give several possible definitions of our new version of prismatic cohomology: a site theoretic definition, one using prismatic crystals, and a stack theoretic definition. These are equivalent under mild syntomicity hypotheses. As an application, we note how the theory of prismatic cohomology of filtered rings arises naturally in this context.