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.
Subsection18.1Derived prismatic cohomology
Definition18.1.1.
Let \((A,I)\) be a bounded prism with slice \(\overline{A}\text{.}\) The derived prismatic cohomology functor \(L\Prism_{\bullet/A}\colon \Ring_{\overline{A}} \to D_{\comp}(A)\) is the left derived functor of the functor \(\Poly_{\overline{A}} \to D_{\comp}(A)\) given by \(R_0 \mapsto \Prism_{\widehat{R_0}/A}\) (where \(\widehat{R_0}\) is the derived \(p\)-completion). Note that \(L\Prism_{R/A}\) is a commutative algebra object in \(D_{\comp}(A)\text{.}\)
Similarly, the derived Hodge-Tate cohomology functor \(L\overline{\Prism}_{\bullet/A}\colon \Ring_{\overline{A}} \to D_{\comp}(\overline{A})\) is the left derived functor of the functor \(\Poly_{\overline{A}} \to D_{\comp}(A)\) given by \(R_0 \mapsto \overline{\Prism}_{\widehat{R_0}/A}\text{.}\) Note that \(L\overline{\Prism}_{R/A}\) is a commutative algebra object in \(D_{\comp}(R)\text{.}\) There is a natural isomorphism \(L\Prism_{R/A} \otimes_A^L \overline{A} \cong L \overline{\Prism}_{R/A}\) in \(D_{\comp}(\overline{A})\text{.}\)
Remark18.1.2.
The object \(L\Prism_{R/A}\) admits a \(\phi_A\)-semilinear endomorphism \(\phi_R\text{.}\) One can further show that \(L\Prism_{R/A}\) carries the structure of a derived \(\delta\)-ring once one makes a precise definition of this concept (which we will not do here).
Remark18.1.3.
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!
Proposition18.1.4.Derived Hodge-Tate comparison.
Let \((A,I)\) be a bounded prism. For any \(R \in \Ring_{\overline{A}}\text{,}\) the complex \(L\overline{\Prism}_{R/A}\) admits a functorial (in \(R\)) multiplicative exhaustive increasing filtration \(\Fil_\bullet^{\HT}\) in \(D_{\comp}(R)\) for which we have canonical identifications
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 \(A\text{,}\) we have \(L\overline{\Prism}_{R/A} \cong \overline{\Prism}_{R/A}\text{;}\) this amounts to an application of Lemma 17.1.3.
Corollary18.1.5.Comparison with the smooth case.
Let \((A,I)\) be a bounded prism. For any \(p\)-completely smooth \(A/I\)-algebra, the natural map \(L\Prism_{R/A} \to \Prism_{R/A}\) is an isomorphism.
The statement that derived de Rham cohomology can be computed easily using regular semiperfect rings (Remark 17.3.6) can be adapted as follows.
Definition18.2.1.
Let \((A,I)\) be a perfect prism. A semilens over \(\overline{A}\) is a derived \(p\)-complete ring which can be written as the quotient of some lens over \(\overline{A}\text{.}\) (This corresponds to a semiperfectoid ring in [18] and [25].) If \(S\) is a semilens, then \(S/p\) is semiperfect and \(\theta\colon W(S^\flat) \to S\) is surjective. It will follow from Remark 18.2.3 that \(L\Prism_{S/A} \in D^{\leq 0}(A)\text{,}\) but in general it will not be concentrated in degree 0.
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 \(\overline{A}\) and \(J\) is an ideal of \(R\) generated by a regular sequence.
Example18.2.2.
By analogy with Example 17.3.2, note that for any lens \(R\text{,}\)
\begin{equation*}
S = R[x_1^{p^{-\infty}}, \dots, x_r^{p^{-\infty}}]^\wedge_{(p)}/(x_1,\dots,x_r)
\end{equation*}
is a regular semilens.
Remark18.2.3.
Let \((A,I)\) be a perfect prism and let \(S\) be a regular semilens over \((A,I)\text{.}\) For simplicity, assume also that \(\overline{A}\) is \(p\)-torsion-free and \(S\) is \(p\)-completely flat over \(\overline{A}\text{.}\) From Proposition 18.1.4, we see that \(L\overline{\Prism}_{S/A}\) admits an increasing exhaustive filtration with graded pieces \(\left(\bigwedge^i L_{S/\overline{A}}\{-i\}\right)[-i]\text{.}\) By our assumptions on \(S\text{,}\) each of these graded pieces is a finite projective \(S\)-module (compare the proof of Lemma 17.3.3). It follows that \(L\overline{\Prism}_{S/A}\) is concentrated in degree 0, where it is a \(p\)-completely flat \(S\)-algebra; consequently, \(L\Prism_{S/A}\) is concentrated in degree 0, where it is a \((p,I)\)-completely flat \(A\)-algebra.
It can also be shown (as in Remark 18.1.2) that the Frobenius on prismatic cohomology provides \(L\Prism_{S/A}\) with a \(\delta\)-ring structure, so \((L\Prism_{S/A}, I)\) is in fact a prism over \((A,I)\text{!}\) This can even be made explicit: if we write \(S = R/J\) with \(R\) a lens and \(J\) generated by a regular sequence, then
Let \(R\) be a lens and let \(x_1,\dots,x_r\) be a regular sequence in \(R\text{.}\) Prove that the regular semilens \(S = R/(x_1,\dots,x_r)\) is a lens if and only if \(S\) is a direct factor of \(R\text{.}\)
