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Notes on prismatic cohomology

Kiran S. Kedlaya

Section 18 Derived prismatic cohomology

Reference.

[18], lecture VII.
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In this section, we discuss how to adapt our previous statements about smooth algebras to the singular case. The idea is to use simplicial resolutions of singular algebras by smooth ones, so that all the heavy lifting gets done by the smooth case.
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Subsection 18.1 Derived prismatic cohomology

Definition 18.1.1.

Let (A,I) be a bounded prism with slice A―. The derived prismatic cohomology functor LΔ∙/A:RingA―→Dcomp(A) is the left derived functor of the functor PolyA―→Dcomp(A) given by R0↦ΔR0^/A (where R0^ is the derived p-completion). Note that LΔR/A is a commutative algebra object in Dcomp(A).
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Similarly, the derived Hodge-Tate cohomology functor LΔ―∙/A:RingA―→Dcomp(A―) is the left derived functor of the functor PolyA―→Dcomp(A) given by R0↦Δ―R0^/A. Note that LΔ―R/A is a commutative algebra object in Dcomp(R). There is a natural isomorphism LΔR/A⊗ALA―≅LΔ―R/A in Dcomp(A―).
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Remark 18.1.2.

The object LΔR/A admits a ϕA-semilinear endomorphism ϕR. One can further show that LΔR/A carries the structure of a derived δ-ring once one makes a precise definition of this concept (which we will not do here).
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Remark 18.1.3.

While ordinary prismatic and Hodge-Tate cohomology are concentrated in nonnegative degrees, the same is not true of derived prismatic and Hodge-Tate cohomology. In general, they will not even be bounded below!
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Proof.

This follows from the same argument as in Proposition 17.2.4 upon checking that when R is the p-adic completion of a polynomial ring over A, we have LΔ―R/A≅Δ―R/A; this amounts to an application of Lemma 17.1.3.

Proof.

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Subsection 18.2 Regular semilenses

The statement that derived de Rham cohomology can be computed easily using regular semiperfect rings (Remark 17.3.6) can be adapted as follows.
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Definition 18.2.1.

Let (A,I) be a perfect prism. A semilens over A― is a derived p-complete ring which can be written as the quotient of some lens over A―. (This corresponds to a semiperfectoid ring in [18] and [25].) If S is a semilens, then S/p is semiperfect and θ:W(S♭)→S is surjective. It will follow from Remark 18.2.3 that LΔS/A∈D≤0(A), but in general it will not be concentrated in degree 0.
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For (A,I) a perfect prism, a regular semilens over (A,I) is a ring S of the form R/J where R is a lens over A― and J is an ideal of R generated by a regular sequence.
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Example 18.2.2.

By analogy with Example 17.3.2, note that for any lens R,
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S=R[x1p−∞,…,xrp−∞](p)∧/(x1,…,xr)
is a regular semilens.
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Remark 18.2.3.

Let (A,I) be a perfect prism and let S be a regular semilens over (A,I). For simplicity, assume also that A― is p-torsion-free and S is p-completely flat over A―. From Proposition 18.1.4, we see that LΔ―S/A admits an increasing exhaustive filtration with graded pieces (⋀iLS/A―{−i})[−i]. By our assumptions on S, each of these graded pieces is a finite projective S-module (compare the proof of Lemma 17.3.3). It follows that LΔ―S/A is concentrated in degree 0, where it is a p-completely flat S-algebra; consequently, LΔS/A is concentrated in degree 0, where it is a (p,I)-completely flat A-algebra.
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It can also be shown (as in Remark 18.1.2) that the Frobenius on prismatic cohomology provides LΔS/A with a δ-ring structure, so (LΔS/A,I) is in fact a prism over (A,I)! This can even be made explicit: if we write S=R/J with R a lens and J generated by a regular sequence, then
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LΔS/A≅W(R♭){J/d}(p)∧
(compare Lemma 14.4.2).
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Exercises 18.3 Exercises

1.

Let R be a lens and let x1,…,xr be a regular sequence in R. Prove that the regular semilens S=R/(x1,…,xr) is a lens if and only if S is a direct factor of R.
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Hint.
Argue by analogy with Exercise 17.5.3, using Exercise 17.5.4.
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