Prismatic cohomology

In mathematics, specifically arithmetic geometry, prismatic cohomology is a cohomology introduced by Bhargav Bhatt and Peter Scholze for p-adic formal schemes. It generalizes various p-adic cohomology theories.

Fix a prime $p$. A prism is a pair $(A,I)$, where $A$ is a δ-ring with $p$-derivation $\delta _{A}$ and $I$ is an ideal of $A$ defining a Cartier divisor in $\operatorname {Spec} (A)$, such that $A$ is derived $(p,I)$-adically complete and the ideal $I+\phi _{A}(I)A$ contains $p$, where the associated map $\phi _{A}:A\to A$ given by $$\phi _{A}(x)=x^{p}+p\delta _{A}(x)$$ is a ring homomorphism, which then necessarily lifts the Frobenius endomorphism on $A/(p)$.

See also

• Crystalline cohomology
• Motivic cohomology
• de Rham cohomology

References

• Bhatt, Bhargav; Scholze, Peter (2022). "Prisms and prismatic cohomology". Annals of Mathematics. 196 (3): 1135–1275. doi:10.4007/annals.2022.196.3.5. MR 4502597. Zbl 1552.14012.
• "Prismatic cohomology". nLab. Retrieved 24 July 2026.