50.10 de Rham cohomology of a line bundle
A line bundle is a special case of a vector bundle, which in turn is a cone endowed with some extra structure. To intelligently talk about the de Rham complex of these, it makes sense to discuss the de Rham complex of a graded ring.
Let $f : X \to S$ be a morphism of schemes. Let $\pi : C \to X$ be a cone, see Constructions, Definition 27.7.2. Recall that this means $\pi $ is affine and we have a grading $\pi _*\mathcal{O}_ C = \bigoplus _{n \geq 0} \mathcal{A}_ n$ with $\mathcal{A}_0 = \mathcal{O}_ X$. Using the discussion in Remark 50.10.1 over affine opens we find that1
\[ \pi _*(\Omega ^\bullet _{C/S}) = \bigoplus \nolimits _{n \geq 0} \Omega ^\bullet _{C/S, n} \]
is canonically a direct sum of subcomplexes. Moreover, we have a factorization
\[ \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{C/S, 0} \to \pi _*(\Omega ^\bullet _{C/S}) \]
and we know that $\omega \wedge \eta \in \Omega ^{p + q}_{C/S, n + m}$ if $\omega \in \Omega ^ p_{C/S, n}$ and $\eta \in \Omega ^ q_{C/S, m}$.
Let $f : X \to S$ be a morphism of schemes. Let $\pi : L \to X$ be the line bundle associated to the invertible $\mathcal{O}_ X$-module $\mathcal{L}$. This means that $\pi $ is the unique affine morphism such that
\[ \pi _*\mathcal{O}_ L = \bigoplus \nolimits _{n \geq 0} \mathcal{L}^{\otimes n} \]
as $\mathcal{O}_ X$-algebras. Thus $L$ is a cone over $X$. By the discussion above we find a canonical direct sum decomposition
\[ \pi _*(\Omega ^\bullet _{L/S}) = \bigoplus \nolimits _{n \geq 0} \Omega ^\bullet _{L/S, n} \]
compatible with wedge product, compatible with the decomposition of $\pi _*\mathcal{O}_ L$ above, and such that $\Omega _{X/S}$ maps into the part $\Omega _{L/S, 0}$ of degree $0$.
There is another case which will be useful to us. Namely, consider the complement2 $L^\star \subset L$ of the zero section $o : X \to L$ in our line bundle $L$. A local computation shows we have a canonical isomorphism
\[ (L^\star \to X)_*\mathcal{O}_{L^\star } = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{L}^{\otimes n} \]
of $\mathcal{O}_ X$-algebras. The right hand side is a $\mathbf{Z}$-graded quasi-coherent $\mathcal{O}_ X$-algebra. Using the discussion in Remark 50.10.1 over affine opens we find that
\[ (L^\star \to X)_*(\Omega ^\bullet _{L^\star /S}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \Omega ^\bullet _{L^\star /S, n} \]
compatible with wedge product, compatible with the decomposition of $(L^\star \to X)_*\mathcal{O}_{L^\star }$ above, and such that $\Omega _{X/S}$ maps into the part $\Omega _{L^\star /S, 0}$ of degree $0$. The complex $\Omega ^\bullet _{L^\star /S, 0}$ will be of particular interest to us.
Lemma 50.10.2. With notation as above, there is a short exact sequence of complexes
\[ 0 \to \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{L^\star /S, 0} \to \Omega ^\bullet _{X/S}[-1] \to 0 \]
Proof.
We have constructed the map $\Omega ^\bullet _{X/S} \to \Omega ^\bullet _{L^\star /S, 0}$ above.
Construction of $\text{Res} : \Omega ^\bullet _{L^\star /S, 0} \to \Omega ^\bullet _{X/S}[-1]$. Let $U \subset X$ be an open and let $s \in \mathcal{L}(U)$ and $s' \in \mathcal{L}^{\otimes -1}(U)$ be sections such that $s' s = 1$. Then $s$ gives an invertible section of the sheaf of algebras $(L^\star \to X)_*\mathcal{O}_{L^\star }$ over $U$ with inverse $s' = s^{-1}$. Then we can consider the $1$-form $\text{d}\log (s) = s' \text{d}(s)$ which is an element of $\Omega ^1_{L^\star /S, 0}(U)$ by our construction of the grading on $\Omega ^1_{L^\star /S}$. Our computations on affines given below will show that $1$ and $\text{d}\log (s)$ freely generate $\Omega ^\bullet _{L^\star /S, 0}|_ U$ as a right module over $\Omega ^\bullet _{X/S}|_ U$. Thus we can define $\text{Res}$ over $U$ by the rule
\[ \text{Res}(\omega ' + \text{d}\log (s) \wedge \omega ) = \omega \]
for all $\omega ', \omega \in \Omega ^\bullet _{X/S}(U)$. This map is independent of the choice of local generator $s$ and hence glues to give a global map. Namely, another choice of $s$ would be of the form $gs$ for some invertible $g \in \mathcal{O}_ X(U)$ and we would get $\text{d}\log (gs) = g^{-1}\text{d}(g) + \text{d}\log (s)$ from which the independence easily follows. Finally, observe that our rule for $\text{Res}$ is compatible with differentials as $\text{d}(\omega ' + \text{d}\log (s) \wedge \omega ) = \text{d}(\omega ') - \text{d}\log (s) \wedge \text{d}(\omega )$ and because the differential on $\Omega ^\bullet _{X/S}[-1]$ sends $\omega '$ to $-\text{d}(\omega ')$ by our sign convention in Homology, Definition 12.14.7.
Local computation. We can cover $X$ by affine opens $U \subset X$ such that $\mathcal{L}|_ U \cong \mathcal{O}_ U$ which moreover map into an affine open $V \subset S$. Write $U = \mathop{\mathrm{Spec}}(A)$, $V = \mathop{\mathrm{Spec}}(R)$ and choose a generator $s$ of $\mathcal{L}$. We find that we have
\[ L^\star \times _ X U = \mathop{\mathrm{Spec}}(A[s, s^{-1}]) \]
Computing differentials we see that
\[ \Omega ^1_{A[s, s^{-1}]/R} = A[s, s^{-1}] \otimes _ A \Omega ^1_{A/R} \oplus A[s, s^{-1}] \text{d}\log (s) \]
and therefore taking exterior powers we obtain
\[ \Omega ^ p_{A[s, s^{-1}]/R} = A[s, s^{-1}] \otimes _ A \Omega ^ p_{A/R} \oplus A[s, s^{-1}] \text{d}\log (s) \otimes _ A \Omega ^{p - 1}_{A/R} \]
Taking degree $0$ parts we find
\[ \Omega ^ p_{A[s, s^{-1}]/R, 0} = \Omega ^ p_{A/R} \oplus \text{d}\log (s) \otimes _ A \Omega ^{p - 1}_{A/R} \]
and the proof of the lemma is complete.
$\square$
Lemma 50.10.3. The “boundary” map $\delta : \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}[2]$ in $D(X, f^{-1}\mathcal{O}_ S)$ coming from the short exact sequence in Lemma 50.10.2 is the map of Remark 50.4.3 for $\xi = c_1^{dR}(\mathcal{L})$.
Proof.
To be precise we consider the shift
\[ 0 \to \Omega ^\bullet _{X/S}[1] \to \Omega ^\bullet _{L^\star /S, 0}[1] \to \Omega ^\bullet _{X/S} \to 0 \]
of the short exact sequence of Lemma 50.10.2. As the degree zero part of a grading on $(L^\star \to X)_*\Omega ^\bullet _{L^\star /S}$ we see that $\Omega ^\bullet _{L^\star /S, 0}$ is a differential graded $\mathcal{O}_ X$-algebra and that the map $\Omega ^\bullet _{X/S} \to \Omega ^\bullet _{L^\star /S, 0}$ is a homomorphism of differential graded $\mathcal{O}_ X$-algebras. Hence we may view $\Omega ^\bullet _{X/S}[1] \to \Omega ^\bullet _{L^\star /S, 0}[1]$ as a map of right differential graded $\Omega ^\bullet _{X/S}$-modules on $X$. The map $\text{Res} : \Omega ^\bullet _{L^\star /S, 0}[1] \to \Omega ^\bullet _{X/S}$ is a map of right differential graded $\Omega ^\bullet _{X/S}$-modules since it is locally defined by the rule $\text{Res}(\omega ' + \text{d}\log (s) \wedge \omega ) = \omega $, see proof of Lemma 50.10.2. Thus by the discussion in Differential Graded Sheaves, Section 24.32 we see that $\delta $ comes from a map $\delta ' : \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}[2]$ in the derived category $D(\Omega ^\bullet _{X/S}, \text{d})$ of right differential graded modules over the de Rham complex. The uniqueness averted in Remark 50.4.3 shows it suffices to prove that $\delta (1) = c_1^{dR}(\mathcal{L})$.
We claim that there is a commutative diagram
\[ \xymatrix{ 0 \ar[r] & \mathcal{O}_ X^* \ar[r] \ar[d]_{\text{d}\log } & E \ar[r] \ar[d] & \underline{\mathbf{Z}} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \Omega ^\bullet _{X/S}[1] \ar[r] & \Omega ^\bullet _{L^\star /S, 0}[1] \ar[r] & \Omega ^\bullet _{X/S} \ar[r] & 0 } \]
where the top row is a short exact sequence of abelian sheaves whose boundary map sends $1$ to the class of $\mathcal{L}$ in $H^1(X, \mathcal{O}_ X^*)$. It suffices to prove the claim by the compatibility of boundary maps with maps between short exact sequences. We define $E$ as the sheafification of the rule
\[ U \longmapsto \{ (s, n) \mid n \in \mathbf{Z},\ s \in \mathcal{L}^{\otimes n}(U)\text{ generator}\} \]
with group structure given by $(s, n) \cdot (t, m) = (s \otimes t, n + m)$. The middle vertical map sends $(s, n)$ to $\text{d}\log (s)$. This produces a map of short exact sequences because the map $Res : \Omega ^1_{L^\star /S, 0} \to \mathcal{O}_ X$ constructed in the proof of Lemma 50.10.2 sends $\text{d}\log (s)$ to $1$ if $s$ is a local generator of $\mathcal{L}$. To calculate the boundary of $1$ in the top row, choose local trivializations $s_ i$ of $\mathcal{L}$ over opens $U_ i$ as in Section 50.9. On the overlaps $U_{i_0i_1} = U_{i_0} \cap U_{i_1}$ we have an invertible function $f_{i_0i_1}$ such that $f_{i_0i_1} = s_{i_1}|_{U_{i_0i_1}} s_{i_0}|_{U_{i_0i_1}}^{-1}$ and the cohomology class of $\mathcal{L}$ is given by the Čech cocycle $\{ f_{i_0i_1}\} $. Then of course we have
\[ (f_{i_0i_1}, 0) = (s_{i_1}, 1)|_{U_{i_0i_1}} \cdot (s_{i_0}, 1)|_{U_{i_0i_1}}^{-1} \]
as sections of $E$ which finishes the proof.
$\square$
Lemma 50.10.4. With notation as above we have
$\Omega ^ p_{L^\star /S, n} = \Omega ^ p_{L^\star /S, 0} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}$ for all $n \in \mathbf{Z}$ as quasi-coherent $\mathcal{O}_ X$-modules,
$\Omega ^\bullet _{X/S} = \Omega ^\bullet _{L/X, 0}$ as complexes, and
for $n > 0$ and $p \geq 0$ we have $\Omega ^ p_{L/X, n} = \Omega ^ p_{L^\star /S, n}$.
Proof.
In each case there is a globally defined canonical map which is an isomorphism by local calculations which we omit.
$\square$
Lemma 50.10.5. In the situation above, assume there is a morphism $S \to \mathop{\mathrm{Spec}}(\mathbf{Q})$. Then $\Omega ^\bullet _{X/S} \to \pi _*\Omega ^\bullet _{L/S}$ is a quasi-isomorphism and $H_{dR}^*(X/S) = H_{dR}^*(L/S)$.
Proof.
Let $R$ be a $\mathbf{Q}$-algebra. Let $A$ be an $R$-algebra. The affine local statement is that the map
\[ \Omega ^\bullet _{A/R} \longrightarrow \Omega ^\bullet _{A[t]/R} \]
is a quasi-isomorphism of complexes of $R$-modules. In fact it is a homotopy equivalence with homotopy inverse given by the map sending $g \omega + g' \text{d}t \wedge \omega '$ to $g(0)\omega $ for $g, g' \in A[t]$ and $\omega , \omega ' \in \Omega ^\bullet _{A/R}$. The homotopy sends $g \omega + g' \text{d}t \wedge \omega '$ to $(\int g') \omega '$ were $\int g' \in A[t]$ is the polynomial with vanishing constant term whose derivative with respect to $t$ is $g'$. Of course, here we use that $R$ contains $\mathbf{Q}$ as $\int t^ n = (1/n)t^{n + 1}$.
$\square$
Example 50.10.6. Lemma 50.10.5 is false in positive characteristic. The de Rham complex of $\mathbf{A}^1_ k = \mathop{\mathrm{Spec}}(k[x])$ over a field $k$ looks like a direct sum
\[ k \oplus \bigoplus \nolimits _{n \geq 1} (k \cdot t^ n \xrightarrow {n} k \cdot t^{n - 1} \text{d}t) \]
Hence if the characteristic of $k$ is $p > 0$, then we see that both $H^0_{dR}(\mathbf{A}^1_ k/k)$ and $H^1_{dR}(\mathbf{A}^1_ k/k)$ are infinite dimensional over $k$.
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