  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.
