HODGE-TATE COMPARISON // DERIVED PRISMS // CARTIER DIVISOR DESCENT
The prismatic cohomology H^n_△(X) reduces to various classical cohomologies via base change. The Hodge-Tate specialization gives: H^n_△(X) ⊗_A A/I ≈ ⊕ H^i(X, Ω^j_{X/k}){-j} where the right side is Hodge cohomology with Breuil-Kisin twists.
The derived prismatic cohomology △_{X/A} is a commutative algebra object in D(A) equipped with a Frobenius φ. For smooth X, it recovers crystalline, de Rham, and etale cohomology via different specializations.
A Cartier divisor on the prismatic site is an effective Cartier divisor I ⊆ A. The Cartier-Witt divisor stack WCart parametrizes all prisms and admits a Frobenius-equivariant structure governing the descent of prismatic F-gauges.