Prismatic cohomology

In mathematics, specifically arithmetic geometry, prismatic cohomology is a cohomology introduced by Bhargav Bhatt and Peter Scholze for [[p-adic number|Template:Math-adic]] formal schemes.Template:Sfnp It generalizes various Template:Math-adic cohomology theories.

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

See also

• Crystalline cohomology
• Motivic cohomology
• de Rham cohomology

References

• Template:Cite journal
• Template:Cite web