50.15 Log poles along a divisor
Let X \to S be a morphism of schemes. Let Y \subset X be an effective Cartier divisor. If X étale locally along Y looks like Y \times \mathbf{A}^1, then there is a canonical short exact sequence of complexes
0 \to \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}(\log Y) \to \Omega ^\bullet _{Y/S}[-1] \to 0
having many good properties we will discuss in this section. There is a variant of this construction where one starts with a normal crossings divisor (Étale Morphisms, Definition 41.21.1) which we will discuss elsewhere (insert future reference here).
Definition 50.15.1. Let X \to S be a morphism of schemes. Let Y \subset X be an effective Cartier divisor. We say the de Rham complex of log poles is defined for Y \subset X over S if for all y \in Y and local equation f \in \mathcal{O}_{X, y} of Y we have
\mathcal{O}_{X, y} \to \Omega _{X/S, y}, g \mapsto g \text{d}f is a split injection, and
\Omega ^ p_{X/S, y} is f-torsion free for all p.
An easy local calculation shows that it suffices for every y \in Y to find one local equation f for which conditions (1) and (2) hold.
Lemma 50.15.2. Let X \to S be a morphism of schemes. Let Y \subset X be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for Y \subset X over S. There is a canonical short exact sequence of complexes
0 \to \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}(\log Y) \to \Omega ^\bullet _{Y/S}[-1] \to 0
Proof.
Our assumption is that for every y \in Y and local equation f \in \mathcal{O}_{X, y} of Y we have
\Omega _{X/S, y} = \mathcal{O}_{X, y}\text{d}f \oplus M \quad \text{and}\quad \Omega ^ p_{X/S, y} = \wedge ^{p - 1}(M)\text{d}f \oplus \wedge ^ p(M)
for some module M with f-torsion free exterior powers \wedge ^ p(M). It follows that
\Omega ^ p_{Y/S, y} = \wedge ^ p(M/fM) = \wedge ^ p(M)/f\wedge ^ p(M)
Below we will tacitly use these facts. In particular the sheaves \Omega ^ p_{X/S} have no nonzero local sections supported on Y and we have a canonical inclusion
\Omega ^ p_{X/S} \subset \Omega ^ p_{X/S}(Y)
see More on Flatness, Section 38.42. Let U = \mathop{\mathrm{Spec}}(A) be an affine open subscheme such that Y \cap U = V(f) for some nonzerodivisor f \in A. Let us consider the \mathcal{O}_ U-submodule of \Omega ^ p_{X/S}(Y)|_ U generated by \Omega ^ p_{X/S}|_ U and \text{d}\log (f) \wedge \Omega ^{p - 1}_{X/S} where \text{d}\log (f) = f^{-1}\text{d}(f). This is independent of the choice of f as another generator of the ideal of Y on U is equal to uf for a unit u \in A and we get
\text{d}\log (uf) - \text{d}\log (f) = \text{d}\log (u) = u^{-1}\text{d}u
which is a section of \Omega _{X/S} over U. These local sheaves glue to give a quasi-coherent submodule
\Omega ^ p_{X/S} \subset \Omega ^ p_{X/S}(\log Y) \subset \Omega ^ p_{X/S}(Y)
Let us agree to think of \Omega ^ p_{Y/S} as a quasi-coherent \mathcal{O}_ X-module. There is a unique surjective \mathcal{O}_ X-linear map
\text{Res} : \Omega ^ p_{X/S}(\log Y) \to \Omega ^{p - 1}_{Y/S}
defined by the rule
\text{Res}(\eta ' + \text{d}\log (f) \wedge \eta ) = \eta |_{Y \cap U}
for all opens U as above and all \eta ' \in \Omega ^ p_{X/S}(U) and \eta \in \Omega ^{p - 1}_{X/S}(U). If a form \eta over U restricts to zero on Y \cap U, then \eta = \text{d}f \wedge \eta ' + f\eta '' for some forms \eta ' and \eta '' over U. We conclude that we have a short exact sequence
0 \to \Omega ^ p_{X/S} \to \Omega ^ p_{X/S}(\log Y) \to \Omega ^{p - 1}_{Y/S} \to 0
for all p. We still have to define the differentials \Omega ^ p_{X/S}(\log Y) \to \Omega ^{p + 1}_{X/S}(\log Y). On the subsheaf \Omega ^ p_{X/S} we use the differential of the de Rham complex of X over S. Finally, we define \text{d}(\text{d}\log (f) \wedge \eta ) = -\text{d}\log (f) \wedge \text{d}\eta . The sign is forced on us by the Leibniz rule (on \Omega ^\bullet _{X/S}) and it is compatible with the differential on \Omega ^\bullet _{Y/S}[-1] which is after all -\text{d}_{Y/S} by our sign convention in Homology, Definition 12.14.7. In this way we obtain a short exact sequence of complexes as stated in the lemma.
\square
Definition 50.15.3. Let X \to S be a morphism of schemes. Let Y \subset X be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for Y \subset X over S. Then the complex
\Omega ^\bullet _{X/S}(\log Y)
constructed in Lemma 50.15.2 is the de Rham complex of log poles for Y \subset X over S.
This complex has many good properties.
Lemma 50.15.4. Let p : X \to S be a morphism of schemes. Let Y \subset X be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for Y \subset X over S.
The maps \wedge : \Omega ^ p_{X/S} \times \Omega ^ q_{X/S} \to \Omega ^{p + q}_{X/S} extend uniquely to \mathcal{O}_ X-bilinear maps
\wedge : \Omega ^ p_{X/S}(\log Y) \times \Omega ^ q_{X/S}(\log Y) \to \Omega ^{p + q}_{X/S}(\log Y)
satisfying the Leibniz rule \text{d}(\omega \wedge \eta ) = \text{d}(\omega ) \wedge \eta + (-1)^{\deg (\omega )} \omega \wedge \text{d}(\eta ),
with multiplication as in (1) the map \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}(\log (Y) is a homomorphism of differential graded \mathcal{O}_ S-algebras,
via the maps in (1) we have \Omega ^ p_{X/S}(\log Y) = \wedge ^ p(\Omega ^1_{X/S}(\log Y)), and
the map \text{Res} : \Omega ^\bullet _{X/S}(\log Y) \to \Omega ^\bullet _{Y/S}[-1] satisfies
\text{Res}(\omega \wedge \eta ) = \text{Res}(\omega ) \wedge \eta |_ Y
for \omega a local section of \Omega ^ p_{X/S}(\log Y) and \eta a local section of \Omega ^ q_{X/S}.
Proof.
This follows by direct calculation from the local construction of the complex in the proof of Lemma 50.15.2. Details omitted.
\square
Consider a commutative diagram
\xymatrix{ X' \ar[r]_ f \ar[d] & X \ar[d] \\ S' \ar[r] & S }
of schemes. Let Y \subset X be an effective Cartier divisor whose pullback Y' = f^*Y is defined (Divisors, Definition 31.13.12). Assume the de Rham complex of log poles is defined for Y \subset X over S and the de Rham complex of log poles is defined for Y' \subset X' over S'. In this case we obtain a map of short exact sequences of complexes
\xymatrix{ 0 \ar[r] & f^{-1}\Omega ^\bullet _{X/S} \ar[r] \ar[d] & f^{-1}\Omega ^\bullet _{X/S}(\log Y) \ar[r] \ar[d] & f^{-1}\Omega ^\bullet _{Y/S}[-1] \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \Omega ^\bullet _{X'/S'} \ar[r] & \Omega ^\bullet _{X'/S'}(\log Y') \ar[r] & \Omega ^\bullet _{Y'/S'}[-1] \ar[r] & 0 }
Linearizing, for every p we obtain a linear map f^*\Omega ^ p_{X/S}(\log Y) \to \Omega ^ p_{X'/S'}(\log Y').
Lemma 50.15.5. Let f : X \to S be a morphism of schemes. Let Y \subset X be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for Y \subset X over S. Denote
\delta : \Omega ^\bullet _{Y/S} \to \Omega ^\bullet _{X/S}[2]
in D(X, f^{-1}\mathcal{O}_ S) the “boundary” map coming from the short exact sequence in Lemma 50.15.2. Denote
\xi ' : \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}[2]
in D(X, f^{-1}\mathcal{O}_ S) the map of Remark 50.4.3 corresponding to \xi = c_1^{dR}(\mathcal{O}_ X(-Y)). Denote
\zeta ' : \Omega ^\bullet _{Y/S} \to \Omega ^\bullet _{Y/S}[2]
in D(Y, f|_ Y^{-1}\mathcal{O}_ S) the map of Remark 50.4.3 corresponding to \zeta = c_1^{dR}(\mathcal{O}_ X(-Y)|_ Y). Then the diagram
\xymatrix{ \Omega ^\bullet _{X/S} \ar[d]_{\xi '} \ar[r] & \Omega ^\bullet _{Y/S} \ar[d]^{\zeta '} \ar[ld]_\delta \\ \Omega ^\bullet _{X/S}[2] \ar[r] & \Omega ^\bullet _{Y/S}[2] }
is commutative in D(X, f^{-1}\mathcal{O}_ S).
Proof.
More precisely, we define \delta as the boundary map corresponding to the shifted short exact sequence
0 \to \Omega ^\bullet _{X/S}[1] \to \Omega ^\bullet _{X/S}(\log Y)[1] \to \Omega ^\bullet _{Y/S} \to 0
It suffices to prove each triangle commutes. Set \mathcal{L} = \mathcal{O}_ X(-Y). Denote \pi : L \to X the line bundle with \pi _*\mathcal{O}_ L = \bigoplus _{n \geq 0} \mathcal{L}^{\otimes n}.
Commutativity of the upper left triangle. By Lemma 50.10.3 the map \xi ' is the boundary map of the triangle given in Lemma 50.10.2. By functoriality it suffices to prove there exists a morphism of short exact sequences
\xymatrix{ 0 \ar[r] & \Omega ^\bullet _{X/S}[1] \ar[r] \ar[d] & \Omega ^\bullet _{L^\star /S, 0}[1] \ar[r] \ar[d] & \Omega ^\bullet _{X/S} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \Omega ^\bullet _{X/S}[1] \ar[r] & \Omega ^\bullet _{X/S}(\log Y)[1] \ar[r] & \Omega ^\bullet _{Y/S} \ar[r] & 0 }
where the left and right vertical arrows are the obvious ones. We can define the middle vertical arrow by the rule
\omega ' + \text{d}\log (s) \wedge \omega \longmapsto \omega ' + \text{d}\log (f) \wedge \omega
where \omega ', \omega are local sections of \Omega ^\bullet _{X/S} and where s is a local generator of \mathcal{L} and f \in \mathcal{O}_ X(-Y) is the corresponding section of the ideal sheaf of Y in X. Since the constructions of the maps in Lemmas 50.10.2 and 50.15.2 match exactly, this works.
Commutativity of the lower right triangle. Denote \overline{L} the restriction of L to Y. By Lemma 50.10.3 the map \zeta ' is the boundary map of the triangle given in Lemma 50.10.2 using the line bundle \overline{L} on Y. By functoriality it suffices to prove there exists a morphism of short exact sequences
\xymatrix{ 0 \ar[r] & \Omega ^\bullet _{X/S}[1] \ar[r] \ar[d] & \Omega ^\bullet _{X/S}(\log Y)[1] \ar[r] \ar[d] & \Omega ^\bullet _{Y/S} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \Omega ^\bullet _{Y/S}[1] \ar[r] & \Omega ^\bullet _{\overline{L}^\star /S, 0}[1] \ar[r] & \Omega ^\bullet _{Y/S} \ar[r] & 0 \\ }
where the left and right vertical arrows are the obvious ones. We can define the middle vertical arrow by the rule
\omega ' + \text{d}\log (f) \wedge \omega \longmapsto \omega '|_ Y + \text{d}\log (\overline{s}) \wedge \omega |_ Y
where \omega ', \omega are local sections of \Omega ^\bullet _{X/S} and where f is a local generator of \mathcal{O}_ X(-Y) viewed as a function on X and where \overline{s} is f|_ Y viewed as a section of \mathcal{L}|_ Y = \mathcal{O}_ X(-Y)|_ Y. Since the constructions of the maps in Lemmas 50.10.2 and 50.15.2 match exactly, this works.
\square
Lemma 50.15.6. Let X \to S be a morphism of schemes. Let Y \subset X be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for Y \subset X over S. Let b \in H^ m_{dR}(X/S) be a de Rham cohomology class whose restriction to Y is zero. Then c_1^{dR}(\mathcal{O}_ X(Y)) \cup b = 0 in H^{m + 2}_{dR}(X/S).
Proof.
This follows immediately from Lemma 50.15.5. Namely, we have
c_1^{dR}(\mathcal{O}_ X(Y)) \cup b = -c_1^{dR}(\mathcal{O}_ X(-Y)) \cup b = -\xi '(b) = -\delta (b|_ Y) = 0
as desired. For the second equality, see Remark 50.4.3.
\square
Lemma 50.15.7. Let X \to T \to S be morphisms of schemes. Let Y \subset X be an effective Cartier divisor. If both X \to T and Y \to T are smooth, then the de Rham complex of log poles is defined for Y \subset X over S.
Proof.
Let y \in Y be a point. By More on Morphisms, Lemma 37.17.1 there exists an integer 0 \geq m and a commutative diagram
\xymatrix{ Y \ar[d] & V \ar[l] \ar[d] \ar[r] & \mathbf{A}^ m_ T \ar[d]^{(a_1, \ldots , a_ m) \mapsto (a_1, \ldots , a_ m, 0)} \\ X & U \ar[l] \ar[r]^-\pi & \mathbf{A}^{m + 1}_ T }
where U \subset X is open, V = Y \cap U, \pi is étale, V = \pi ^{-1}(\mathbf{A}^ m_ T), and y \in V. Denote z \in \mathbf{A}^ m_ T the image of y. Then we have
\Omega ^ p_{X/S, y} = \Omega ^ p_{\mathbf{A}^{m + 1}_ T/S, z} \otimes _{\mathcal{O}_{\mathbf{A}^{m + 1}_ T, z}} \mathcal{O}_{X, x}
by Lemma 50.2.2. Denote x_1, \ldots , x_{m + 1} the coordinate functions on \mathbf{A}^{m + 1}_ T. Since the conditions (1) and (2) in Definition 50.15.1 do not depend on the choice of the local coordinate, it suffices to check the conditions (1) and (2) when f is the image of x_{m + 1} by the flat local ring homomorphism \mathcal{O}_{\mathbf{A}^{m + 1}_ T, z} \to \mathcal{O}_{X, x}. In this way we see that it suffices to check conditions (1) and (2) for \mathbf{A}^ m_ T \subset \mathbf{A}^{m + 1}_ T and the point z. To prove this case we may assume S = \mathop{\mathrm{Spec}}(A) and T = \mathop{\mathrm{Spec}}(B) are affine. Let A \to B be the ring map corresponding to the morphism T \to S and set P = B[x_1, \ldots , x_{m + 1}] so that \mathbf{A}^{m + 1}_ T = \mathop{\mathrm{Spec}}(P). We have
\Omega _{P/A} = \Omega _{B/A} \otimes _ B P \oplus \bigoplus \nolimits _{j = 1, \ldots , m} P \text{d}x_ j \oplus P \text{d}x_{m + 1}
Hence the map P \to \Omega _{P/A}, g \mapsto g \text{d}x_{m + 1} is a split injection and x_{m + 1} is a nonzerodivisor on \Omega ^ p_{P/A} for all p \geq 0. Localizing at the prime ideal corresponding to z finishes the proof.
\square
Comments (1)
Comment #9868 by Fiasco on