In this section, we apply the formalism of nonabelian derived functors (Section 16) to the cohomology of differential forms, starting with the cotangent complex and then moving to derived de Rham cohomology. This will set up a paradigm of leveraging our knowledge about polynomial rings (or their completions) which will persist in the discussion of derived prismatic cohomology in Section 18.
Subsection17.1The cotangent complex
We illustrate the formalism with Illusie's construction of the “derived cotangent bundle” [68], [69].
Definition17.1.1.
For \(A \in \Ring\text{,}\) the cotangent complex is the functor \(L_{\bullet/A}\colon \Ring_A \to D(A)\) taking obtained by taking the left derived functor of the functor \(\Poly_A \to D(A)\) given by \(B \mapsto \Omega^1_{B/A}[0]\text{.}\) It is straightforward to check that in fact \(L_{B/A} \in D^{\leq 0}(B)\text{.}\)
Note that it also makes sense to talk about \(L_{B/A}\) when \(B\) is itself a simplicial object in \(\Ring_A\text{.}\) This will be useful when stating the base change property in Proposition 17.1.2.
Proposition17.1.2.
For \(A \to B\) a morphism in \(\Ring\text{,}\) the cotangent complex \(L_{B/A} \in D(B)\) has the following properties. (These also hold when \(B\) is a simplicial object in \(\Ring_A\text{.}\))
We have a natural (in \(B\)) isomorphism \(H^0(L_{B/A}) \cong \Omega^1_{B/A}\text{.}\)
If \(A \to B\) is smooth, then \(L_{B/A} \cong \Omega^1_{B/A}[0]\text{.}\) In particular, if \(A \to B\) is étale then \(L_{B/A} \cong 0\) (but not conversely; see for example Exercise 17.5.1).
For any morphisms \(A \to B \to C\text{,}\) we have a distinguished triangle in \(D(C)\)
\begin{equation}
L_{B/A} \otimes^L_B C \to L_{C/A} \to L_{C/B} \to.\tag{17.1}
\end{equation}
This extends the usual low-degree exact sequence for differentials.
For any surjective morphism \(A \to B\) with kernel \(I\text{,}\) we have \(H^{-1}(L_{B/A}) \cong I/I^2\text{.}\) Moreover, if \(I\) is generated by a regular sequence, then \(H^i(L_{B/A}) = 0\) for \(i \neq -1\text{.}\) (This generalizes the assertion that a closed immersion of schemes is unramified.)
For any morphisms \(A \to B, A \to C\text{,}\) we have a natural base change isomorphism
\begin{equation*}
L_{C/A} \otimes_A^L B \cong L_{C \otimes_A^L B/B}
\end{equation*}
where \(C \otimes_A^L B\) is to be interpreted as a simplicial ring as per Example 16.2.9. We can replace \(C \otimes_A^L B\) with \(C \otimes_A B\) if one of \(A \to B\) or \(A \to C\) is flat, or more generally if \(A \to B\) and \(A \to C\) are Tor independent (meaning that \(\Tor_i^A(B,C) = 0\) for all \(i \gt 0\)).
The following is an analogue of the flatness of completion for noetherian rings, but without a noetherian hypothesis.
Lemma17.1.3.
Let \(A \in \Ring\) be classically \(p\)-complete with bounded \(p\)-power torsion. Let \(B \in \Ring_A\) be flat (so that \(B\) also has bounded \(p\)-power torsion). Let \(\widehat{B}\) be the classical \(p\)-completion of \(B\text{.}\) Then the derived \(p\)-completion of the cotangent complex of \(B \to \widehat{B}\) is zero in \(D(\widehat{B})\text{.}\)
The complex in question vanishes after applying \(\otimes_{\ZZ}^L \ZZ/p\) by the base change formula (Proposition 17.1.2), and then derived Nakayama (Proposition 6.6.2) yields the claim.
Subsection17.2Derived de Rham cohomology
We first prepare for the Hodge-Tate comparison by introducing derived de Rham cohomology, picking up the thread from our previous discussion of the cotangent complex (Subsection 17.1).
Definition17.2.1.
For \(k \in \Ring_{\FF_p}\text{,}\) the derived de Rham cohomology functor \(\dR_{\bullet/k}\colon \Ring_k \to D(k)\) is the left derived functor of the functor \(\Poly_k \to D(k)\) given by \(R \mapsto \Omega^\bullet_{R/k}\text{.}\)
Let us unwind this for a given ring \(R \in k\text{.}\) Let \(P_\bullet \to R\) be the standard simplicial resolution of \(R\) (Example 16.3.4) Then \(\dR_{R/k}\) is the totalization of the double complex \(\Omega^\bullet_{P_\bullet/k}\text{.}\) Note that this double complex is bounded below in one direction (coming from the de Rham complex) and bounded above in the other (coming from the simplicial resolution), so we must handle this totalization with some care (see Remark 17.2.8).
To state the derived analogue of the Cartier isomorphism, we need to explain what we mean by a “filtration” in a derived category. (This is also the correct context in which to construct the spectral sequence associated to a filtered complex, as arose in the proof of Proposition 13.3.1.)
Definition17.2.2.
For \(k \in \Ring\text{,}\) by an increasing exhaustive filtration of an object \(K \in D(k)\text{,}\) we will mean a sequence \(\Fil_0 \to \Fil_1 \to \cdots\) in \(D(k)\) with colimit \(K\text{.}\) The associated graded quotients are the mapping cones \(\gr_i(\Fil_\bullet) = \Cone(\Fil_{i-1} \to \Fil_i)\text{.}\)
Remark17.2.3.
You might initially find it confusing that Definition 17.2.2 does not specify that the maps \(\Fil_i \to \Fil_{i+1}\) are injections. The point is that this is not a meaningful concept in \(D(k)\text{!}\)
Take note of the level of generality in the following proposition; there is no restriction on \(R\) at all!
Proposition17.2.4.Derived Cartier isomorphism.
For \(k \in \Ring_{\FF_p}\) and \(R \in \Ring_k\text{,}\) there is a functorial (in \(R\)) increasing exhaustive filtration on \(\dR_{R/k}\) in \(D(R^{(1)})\text{,}\) called the conjugate filtration, equipped with canonical identifications
For \(R\) a polynomial ring, we take the filtration on \(\Omega^{\bullet}_{R/k}\) where \(\Fil_i\) is the canonical truncation \(\tau^{\leq i} \Omega^\bullet_{R/k}\) (Definition 10.4.1). The desired identifications in this case are just a reformulation of the Cartier isomorphism (Lemma 14.1.6). To deduce the general case, just take left derived functors.
Remark17.2.5.
The conjugate filtration derives its name from the fact that it goes in the opposite direction from the usual Hodge filtration; its relationship with the Cartier isomorphism seems to have been observed first by Katz [78]. The Hodge filtration and the conjugate filtration give rise to the usual Hodge-de Rham spectral sequence and the conjugate spectral sequence, respectively; the latter is unnamed in [78], the modern terminology appearing first in [92].
Note that the following corollary is not automatic, because we don't currently have any way to control the effect of étale localization on derived de Rham cohomology; rather, we must prove this first and then deduce étale localization as a further corollary (Corollary 17.2.7).
Corollary17.2.6.
For \(k \in \Ring_{\FF_p}\) and \(R \in \Ring_k\) smooth, there is a canonical isomorphism \(\dR_{R/k} \cong \Omega^\bullet_{R/k}\) which respects the conjugate filtrations on both sides.
The map in question comes from the construction using the universal property of left Kan extensions. By Proposition 17.2.4, it respects the conjugate filtration on both sides.
The map on graded pieces can be written as \(\left(\bigwedge^i L_{R^{(1)}/k} \right)[-i] \to \left(\Omega^i_{R^{(1)}/k} \right)[-i]\) using Proposition 17.2.4 and the usual Cartier isomorphism (Lemma 14.1.6 in the case of affine space; the general case follows by étale localization). This map is an isomorphism for \(i=1\) by Proposition 17.1.2, and hence is an isomorphism for general \(i\) as well.
Corollary17.2.7.
For \(k \in \Ring_{\FF_p}\) and \(R \to S\) an étale morphism in \(\Ring_k\text{,}\) there is a natural isomorphism \(\dR_{R/k} \otimes_R^L S \cong \dR_{S/k}\text{.}\)
This formally reduces to the case where \(R\) is a polynomial ring in finitely many variables over \(k\text{,}\) in which case \(S\) is smooth over \(k\) and Corollary 17.2.6 applies.
Remark17.2.8.
Note that Corollary 17.2.6 fails when \(k\) is not a ring of characteristic \(p\text{.}\) For example, if \(k\) is a \(\QQ\)-algebra, then \(k \cong \Omega^*_{A/k}\) by the Poincare lemma for any \(A \in \Poly_k\text{,}\) so \(k \cong \dR_{A/k}\) for all \(A \in \Ring_k\text{.}\)
What is going on here is that the definition of derived de Rham cohomology we are using is a shortcut taking advantage of the Cartier isomorphism. The correct construction for general \(k\) requires an extra completion step that correctly accounts for the Hodge filtration, plus some care for the difference between the direct sum totalization and the direct product totalization of a double complex which is bounded above in one direction and bounded below in the other direction (see Definition 13.1.4). See [18], Lecture VII, Remark 3.8 for additional discussion and onward references.
Subsection17.3Regular semiperfect rings
We next describe a large class of rings for which derived de Rham cohomology can be described explicitly. These can then be used in the terms of a simplicial resolution to compute derived de Rham cohomology more generally.
Definition17.3.1.
Let \(k \in \Ring_{\FF_p}\) be perfect. A regular semiperfect \(k\)-algebra is an object \(S \in \Ring_k\) of the form \(R/I\) where \(R \in \Ring_{\FF_p}\) is perfect and \(I\) is an ideal of \(R\) generated by a regular sequence. Note that any such ring is semiperfect, that is, the Frobenius map is surjective; this can be used to partially recover \(R\) from \(S\text{,}\) as per Exercise 17.5.5.
Example17.3.2.
A typical example of a regular semiperfect \(k\)-algebra is
\begin{equation*}
S = k[x_1^{p^{-\infty}}, \dots, x_r^{p^{-\infty}}]/(x_1,\dots,x_r).
\end{equation*}
Note that there are many ways to write \(S\) as \(R/I\text{;}\) for instance, we may take \(R = k[x_1^{p^{-\infty}}, \dots, x_r^{p^{-\infty}}]\) and \(I = (x_1,\dots,x_r)\text{,}\) but we can also replace \(R\) with its classical \(I\)-completion (or anything between). One can recover the completion from \(S\text{;}\) see Exercise 17.5.5.
Lemma17.3.3.
Let \(k \in \Ring_{\FF_p}\) be perfect and let \(S \in \Ring_k\) be regular semiperfect. Then \(\dR_{S/k}\) is concentrated in degree \(0\) (and thus can be viewed as an object in \(\Ring_k\)).
Set notation as in Definition 17.3.1. By Exercise 17.5.1, \(L_{R/k}\) vanishes in \(D(R)\text{.}\) From the distinguished triangle (17.1) associated to the morphisms \(k \to R \to S\text{,}\) we deduce that \(L_{S/k} \to L_{S/R}\) is an isomorphism in \(D(S)\text{.}\) Since \(I\) is generated by a regular sequence, Proposition 17.1.2 asserts that \(L_{S/R} \cong I/I^2[1]\) where \(I/I^2\) is a finite projective \(S\)-module. Consequently, the derived exterior power
In general, we get the divided power envelope of \(I\) in \(R\) (in the sense of [15]; we cannot apply Definition 14.2.1 as we are not in the \(\ZZ_p\)-flat case).
As an illustration of the technique we have in mind, let us apply this logic to ordinary de Rham cohomology in the smooth case.
Lemma17.3.5.
Let \(k \in \Ring_{\FF_p}\) be perfect and let \(R \in \Ring_k\) be smooth over \(k\text{.}\) Let \(S\) be the coperfection of \(R\text{.}\) Let \(S^\bullet\) be the Čech nerve of \(R \to S\text{.}\)
The map \(R \to S\) is flat.
For each \(n \geq 0\text{,}\) the ring \(S^n\) is regular semiperfect.
Since \(R\) is a smooth \(k\)-algebra and \(k\) is perfect, the Frobenius map \(\phi_R\colon R \to R\) is flat (see Remark 19.1.2). It follows that \(R \to S\) is flat.
To see that \(S^n\) is regular semiperfect, we may work étale locally to reduce to the case where \(R = k[x_1,\dots,x_r]\text{.}\) In this case, we may write
With notation as in Lemma 17.3.5, Corollary 17.2.6 implies that we can identify \(\Omega^\bullet_{R/k}\) with \(\dR_{R/k}\) in \(D(k)\text{.}\) For each \(n\text{,}\)\(\dR_{S^n/k}\) is concentrated in degree 0 by Lemma 17.3.3. It can be shown further that the functor \(\dR_{\bullet/k}\) satisfies descent with respect to the fpqc cover \(R \to S\) (see for instance [23], section 3); consequently, \(\dR_{R/k}\) can be computed by the complex \(\dR_{S^\bullet/k}\text{.}\)
Subsection17.4Derived crystalline cohomology
Definition17.4.1.Derived crystalline cohomology.
Let \(k \in \Ring_{\FF_p}\) be perfect and let \(A \in \Ring\) be a classically \(p\)-complete ring with \(A/p \cong k\text{.}\) We wish to derive a crystalline cohomology functor \(R\Gamma_{\crys}\colon \Poly_k \to D(A)\) taking \(R = k[x_1,\dots,x_r]\) to \(\widehat{\Omega}^{\bullet}_{P/A}\) for \(P = A[x_1,\dots,x_r]\text{;}\) however, we need to make sure that this construction does not depend on the choice of coordinates on \(R\text{.}\) Fortunately, we can deduce this from our proof of the Hodge-Tate comparison, which gives us a canonical isomorphism of \(\widehat{\Omega}^{\bullet}_{P/A}\) with \(\phi_A^* \Prism_{R/k}\) (Corollary 14.4.10).
We define the derived crystalline cohomology functor \(R\Gamma_{\dcrys}\colon \Ring_k \to D(A)\) by taking the left derived functor of the ordinary crystalline cohomology \(R\Gamma_{\crys}\text{.}\) From the comparison of crystalline cohomology with de Rham cohomology (Proposition 14.2.8), we obtain a natural isomorphism
\begin{equation*}
R\Gamma_{\dcrys}(\bullet/A) \otimes_A^L k \cong \dR_{\bullet/k}.
\end{equation*}
Remark17.4.2.
Using Lemma 17.3.5, we can carry out the analogue of Remark 17.3.6 to construct an explicit functorial complex computing the (derived) crystalline cohomology of a smooth algebra over a perfect ring. We omit the details here.
Exercises17.5Exercises
1.
Let \(A \to B\) be a morphism of perfect \(\FF_p\)-algebras. Show that \(L_{A/B} \cong 0\text{.}\)
On one hand, \(\phi_B\) evidently induces an isomorphism on \(L_{A/B}\text{.}\) On the other hand, the induced map is zero when \(B\) is a polynomial ring over \(A\text{.}\)
2.
Let \(f\colon A \to B\) be a morphism of finite presentation between perfect \(\FF_p\)-algebras. Show that \(f\) is étale.
Let \(R\) be a perfect \(\FF_p\)-algebra and let \(x_1,\dots,x_r \in R\) be a regular sequence. Prove that the regular semiperfect ring \(S = R/(x_1,\dots,x_r)\) is perfect if and only if \(S\) is a direct factor of \(R\text{.}\)
