In
[10] one only briefly touches on the definition of the pushforward and pullback functors for logarithmic D-modules. In this section we will go into more detail on the definition of these functors, as well as giving an explicit formula for duality and showing some standard identities for these three operations.
Pushforward
Let \(f:X \rightarrow Y\) be a morphism of smooth idealized log varieties. If \(\mathcal N\) is a (left) \(\mathcal {D}_Y\)-module, then as in the classical setting the map \(\mathcal {T}_X \rightarrow f^*\mathcal {T}_Y\) gives the pullback \(f^*\mathcal N = \mathcal {O}_X \otimes _{f^{-1}\mathcal {O}_Y} f^{-1}\mathcal N\) a canonical \(\mathcal {D}_X\)-module structure (see
[10, Section 3.1.4]). In particular, one obtains the transfer module\(\mathcal {D}_{X \rightarrow Y} = f^*\mathcal {D}_Y\). As \(\mathcal {D}_Y\) is a locally free \(\mathcal {O}_Y\)-module, \(\mathcal {D}_{X \rightarrow Y}\) is a locally free \(\mathcal {O}_X\)-module concentrated in cohomological degree 0.
Lemma 3.1
Let \(f:X \rightarrow Y\) be a morphism of smooth idealized log varieties. Then \(\mathcal {D}_{X \rightarrow Y}\) has weak dimension at most \({\text {logdim}}X\) as left \(\mathcal {D}_X\)-module.
Proof
Exactness can be checked locally, so we may assume that X is affine and \(\mathcal {D}_{X \rightarrow Y}\) is a free \(\mathcal {O}_X\)-module, as is \({\text {gr}}\mathcal {D}_{X \rightarrow Y}\). Since locally \({\text {gr}}\mathcal {D}_X\) is isomorphic to \(\mathcal {O}_X[t_1,\dotsc ,t_{{\text {logdim}}X}]\), taking a Koszul resolution of the free \(\mathcal {O}_X\)-module \({\text {gr}}_{\mathcal {D}_{X \rightarrow Y}}\) shows that \({\text {gr}}\mathcal {D}_{X \rightarrow Y}\) has weak dimension at most \({\text {logdim}}X\) as a \({\text {gr}}\mathcal {D}_X\)-module (alternatively, we could apply the more general
[1, Lemma 2.3.5]). Thus, by
[1, Proposition 2.3.12], \(\mathcal {D}_{X \rightarrow Y}\) also has weak dimension at most \({\text {logdim}}X\) as a left \(\mathcal {D}_X\)-module. \(\square \)
Concretely, as each \(F_i\mathcal {D}_{X \rightarrow Y}\) is a locally free \(\mathcal {O}_X\)-module, the Spencer complex \({\text {Sp}}(\mathcal {D}_{X \rightarrow Y})\) is a locally free resolution of \(\mathcal {D}_{X \rightarrow Y}\).
As in the classical setting, one defines the \((f^{-1}\mathcal {D}_Y,\, \mathcal {D}_X)\)-bimodule
$$\begin{aligned} \mathcal {D}_{Y \leftarrow X} = \omega _X \otimes _{\mathcal {O}_X} \mathcal {D}_{X \rightarrow Y} \otimes _{f^{-1}\mathcal {O}_Y} f^{-1}\omega _Y^\vee , \end{aligned}$$
which by Lemma 3.1 has finite weak dimension as a right \(\mathcal {D}_X\)-module. Thus the pushforward functor
$$\begin{aligned} f_{\bullet } :\mathbf {D}^{+}(\mathcal {D}_{X}) \rightarrow \mathbf {D}^{+}(\mathcal {D}_{Y}),\quad \mathcal M \mapsto f_*(\mathcal {D}_{Y \leftarrow X} \otimes _{\mathcal {D}_X} \mathcal M) \end{aligned}$$
is well defined. If \(f:X \rightarrow Y\) and \(g:Y \rightarrow Z\) are two morphisms of smooth idealized log varieties, one can copy the proof of
[5, Proposition 1.5.21] (which is essentially an application of the projection formula for \(f_*:\mathbf {D}^{}(f^{-1}\mathcal {D}_Y) \rightarrow \mathbf {D}^{}(\mathcal {D}_Y)\)) to show that
$$\begin{aligned} (g \circ f)_\bullet \cong g_\bullet \circ f_\bullet :\mathbf {D}^{\mathrm {b}}(\mathcal {D}_{X}) \rightarrow \mathbf {D}^{\mathrm {b}}(\mathcal {D}_{Z}). \end{aligned}$$
For a projection \(f:X \times Y \rightarrow Y\) one sets \(\Omega _f^k = (\bigwedge ^k \Omega _X) \boxtimes \mathcal {O}_Y\) and defines the relative de Rham complex\({\text {DR}}_f(\mathcal M)\) for \(\mathcal M \in \mathbf {QCoh}(\mathcal {D}_{X \times Y})\) by
$$\begin{aligned} ({\text {DR}}_f(\mathcal M))^i = {\left\{ \begin{array}{ll} \Omega _f^{i + {\text {logdim}}Y} \otimes _{\mathcal {O}_{X \times Y}} \mathcal M &{} \text {if } -{\text {logdim}}Y \le i \le 0 \\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$
(5)
with the differential as usual induced by the map \(\mathcal M \rightarrow \mathcal M \otimes \Omega ^1_{X \times Y}\) defining the \(\mathcal {D}_{X \times Y}\)-module structure on \(\mathcal M\). Since \(\mathcal {D}_{Y \leftarrow X \times Y} = \omega _X \boxtimes \mathcal {D}_Y\) as right \(\mathcal {D}_{X \times Y}\)-modules, it follows from Corollary 2.8 that
$$\begin{aligned} f_{\bullet }\mathcal M = f_*{\text {DR}}_f(\mathcal M). \end{aligned}$$
Any morphism \(f:X \rightarrow Y\) of smooth idealized log varieties can be factored into a closed immersion \(X \rightarrow X \times Y\) followed by the smooth projection \(X \times Y \rightarrow Y\). For both of these the pushforward of \(\mathcal {O}\)-modules preserves boundedness of complexes. Thus, by Lemma 3.1, \(f_{\bullet }\) also preserves boundedness, i.e. restricts to a functor \(f_{\bullet }:\mathbf {D}^{\mathrm {b}}(\mathcal {D}_{X}) \rightarrow \mathbf {D}^{\mathrm {b}}(\mathcal {D}_{Y})\).
Lemma 3.2
Let \(f:X \rightarrow Y\) be a proper map of smooth idealized log varieties. Let \(\mathcal M \in \mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{X})\) and assume that locally on Y the cohomology modules of \(\mathcal M\) admit a good filtration. Then \(f_\bullet \mathcal M\) is contained in \(\mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{Y})\).
Proof
Using the resolution of Lemma 2.7 one reduces to the fact that pushforward along proper maps preserves coherence of \(\mathcal {O}\)-modules, see the first part of
[13, Théorème I.5.4.1]. \(\square \)
Let us remark that here, as in many of the following statements, the condition on the existence of a global good filtration is always satisfied in the algebraic setting.
We will call a morphism \(i:X \rightarrow Y\) a closed immersion if the underlying morphism of varieties \(i:\underline{X} \rightarrow \underline{Y}\) is a closed immersion (note that this is different from the usual use of the term in logarithmic geometry). For a closed immersion i, we can give the right adjoint to \(i_\bullet \) in the usual way:
Definition 3.3
Let \(i:Z \hookrightarrow X\) be a closed immersion of smooth idealized log varieties. Define \(i^!:\mathbf {D}^{+}(\mathcal {D}_{X}) \rightarrow \mathbf {D}^{+}(\mathcal {D}_{X})\) by
$$\begin{aligned} \mathcal M \mapsto \underline{{\text {Hom}}}_{i^{-1}\mathcal {D}_X}(\mathcal {D}_{X \leftarrow Z},\, i^{-1}\mathcal M), \end{aligned}$$
where the left \(\mathcal {D}_Z\)-module structure is given by the right action of \(\mathcal {D}_Z\) on \(\mathcal {D}_{X \leftarrow Z}\).
Proposition 3.4
Let \(i:Z \hookrightarrow X\) be a closed immersion of smooth idealized log varieties. Then \(i^!\) is the right adjoint to \(i_\bullet :\mathbf {D}^{+}(\mathcal {D}_{Z}) \rightarrow \mathbf {D}^{+}(\mathcal {D}_{X})\).
Lemma 3.5
Let \(i:Z \hookrightarrow X\) be a closed immersion of smooth idealized log varieties. Then for any \(\mathcal M \in \mathbf {D}^{+}(\mathcal {D}_{X})\) there exists a canonical equivalence
$$\begin{aligned} {\text {Hom}}_{i^{-1}\mathcal {D}_X}(\mathcal {D}_{X \leftarrow Z},\, i^{-1}\mathcal M) \cong {\text {Hom}}_{i^{-1}\mathcal {D}_X}(\mathcal {D}_{X \leftarrow Z},\, i^{-1}\Gamma _Z\mathcal M). \end{aligned}$$
Proof
We first show the statement on the level of abelian categories, i.e. for \(\mathcal M \in \mathbf {Mod}_{}(\mathcal {D}_{X})\). Let \(\mathcal J\) be the sheaf of ideals defining the closed subvariety Z. Then \(i^{-1}\mathcal J\) annihilates \(\mathcal {D}_{X \leftarrow Z}\) and for any \(i^{-1}\mathcal {D}_X\)-module map \(\psi :\mathcal {D}_{X \leftarrow Z} \rightarrow \mathcal M\) one has \((i^{-1}\mathcal J)\psi (m) = \psi ((i^{-1}\mathcal J)m) = 0\), i.e. \(\psi \) factors through \(i^{-1}R^0\Gamma _Z(\mathcal M)\).
To deduce the derived statement, it is now sufficient to show that if \(\mathcal I\) is an injective \(\mathcal {D}_Y\)-module, then \(i^{-1}R^0\Gamma _Z \mathcal I\) is an injective \(i^{-1}\mathcal {D}_X\)-module. This follows from
$$\begin{aligned} R^0{\text {Hom}}_{i^{-1}\mathcal {D}_X}(\mathcal N,\, R^0i^{-1}\Gamma _Z \mathcal I)&\cong R^0{\text {Hom}}_{i^{-1}\mathcal {D}_X}(i^{-1}i_*\mathcal N,\, i^{-1}R^0\Gamma _Z \mathcal I) \\&\cong R^0{\text {Hom}}_{\mathcal {D}_X}(i_*\mathcal N,\, R^0\Gamma _Z \mathcal I) \\&\cong R^0{\text {Hom}}_{\mathcal {D}_X}(i_*\mathcal N,\, \mathcal I) \end{aligned}$$
for any \(\mathcal {D}_Z\)-module \(\mathcal N\). \(\square \)
Proof of Proposition 3.4
By Lemma 3.5 and tensor-hom adjunction we have
$$\begin{aligned} {\text {Hom}}_{\mathcal {D}_Z}(\mathcal M,\, i^!\mathcal N)&= {\text {Hom}}_{\mathcal {D}_Z}\bigr (\mathcal M,\, \underline{{\text {Hom}}}_{i^{-1}\mathcal {D}_X}(\mathcal {D}_{X \leftarrow Z},\, i^{-1}\mathcal N)\bigl ) \\&= {\text {Hom}}_{\mathcal {D}_Z}\bigr (\mathcal M,\, \underline{{\text {Hom}}}_{i^{-1}\mathcal {D}_X}(\mathcal {D}_{X \leftarrow Z},\, i^{-1}\Gamma _Z\mathcal N)\bigl ) \\&= {\text {Hom}}_{i^{-1}\mathcal {D}_X}\bigr (\mathcal {D}_{X \leftarrow Z} \otimes _{\mathcal {D}_Z} \mathcal M,\, i^{-1}\Gamma _Z\mathcal N\bigl ) \\&= {\text {Hom}}_{\mathcal {D}_X}\bigr (i_*(\mathcal {D}_{X \leftarrow Z} \otimes _{\mathcal {D}_Z} \mathcal M),\, \mathcal N\bigl ) \\&= {\text {Hom}}_{\mathcal {D}_X}(i_\bullet \mathcal M,\, \mathcal N). \end{aligned}$$
\(\square \)
Duality
Let X be a (algebraic of analytic) scheme together with an \(\mathcal {O}_X\)-ring \(\mathcal A\), that is, a sheaf of (not necessarily commutative) noetherian rings \(\mathcal A\) on X endowed with a ring homomorphism \(\mathcal {O}_X \rightarrow \mathcal A\). A dualizing complex over \(\mathcal A\) is a bounded complex \(\mathcal R\) of \(\mathcal A\)-bimodules such that the following conditions hold:
-
(i)
The functors
$$\begin{aligned} D:\mathbf {D}^{\mathrm {b}}_{}(\mathcal A)^{{\mathrm {op}}} \rightarrow \mathbf {D}^{}(\mathcal A^{{\mathrm {op}}}), \quad \mathcal M \mapsto \underline{{\text {Hom}}}_{\mathcal A}(\mathcal M, \mathcal R) \end{aligned}$$
and
$$\begin{aligned} D^{{\mathrm {op}}}:\mathbf {D}^{\mathrm {b}}_{}(\mathcal A^{\mathrm {op}})^{\mathrm {op}}\rightarrow \mathbf {D}^{}(\mathcal A), \quad \mathcal M \mapsto \underline{{\text {Hom}}}_{\mathcal A^{{\mathrm {op}}}}(\mathcal M, \mathcal R) \end{aligned}$$
have finite cohomological dimension when restricted to \(\mathbf {Coh}(\mathcal A)\) and \(\mathbf {Coh}(\mathcal A^{\mathrm {op}})\) respectively.
-
(ii)
The functors D and \(D'\) preserve coherence.
-
(iii)
The adjunction morphisms \({\text {id}}\rightarrow D^{\mathrm {op}}D\) in \(\mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal A)\) and \({\text {id}}\rightarrow D D^{\mathrm {op}}\) in \(\mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal A^{\mathrm {op}})\) are both isomorphisms.
Unfortunately, dualizing complexes are not necessarily unique. For example, in the commutative case \(\mathcal A = \mathcal {O}_X\), they are only determined up to shift and tensoring with a line bundle. To restore uniqueness and obtain a duality theory in the non-commutative setting, van den Bergh
[21] introduced the concept of a rigid dualizing complex. This is a dualizing complex \(\mathcal R\) together with a rigidifying isomorphismFootnote 1
$$\begin{aligned} \rho :\mathcal R \xrightarrow {\sim }\underline{{\text {Hom}}}_{\mathcal A \otimes \mathcal A^{\mathrm {op}}}( \mathcal A,\, \mathcal R \otimes \mathcal R ). \end{aligned}$$
If it exists, a rigid dualizing complex is unique up to unique isomorphism.
Let us now return to the case that X is a smooth idealized log variety and \(\mathcal A = \mathcal {D}_X\). As was discussed in
[10], if X is algebraic, then \(\mathcal {D}_X\) always admits a rigid dualizing complex \(\mathcal R\). As we will see below, if the log structure of X is non-trivial, \(\mathcal R\) is not simply a shift of \(\mathcal {D}_X\).
To show that the de Rham functor commutes with duality, we need an explicit computation of \(\mathcal R\). In the case that \(\underline{X}\) is smooth this is a special case of
[2] where the rigid dualizing complex of the enveloping algebra of a Lie algebroid is computed. In this case the canonical bundle \(\omega _{\underline{X}}\) is a right \(\mathcal {D}_{\underline{X}}\)-module and hence by restriction also a right \(\mathcal {D}_X\)-module. The rigid dualizing complex of \(\mathcal {D}_X\) is then given by \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \underline{{\text {Hom}}}_{\mathcal {O}_X}(\omega _X,\, \omega _{\underline{X}})[\dim X + {\text {logdim}}X]\).
We will show that this formula holds for general smooth idealized log varieties. To do so, one replaces the canonical bundle \(\omega _{\underline{X}}\) by the Grothendieck dualizing complex \(\omega '_{\underline{X}}\), i.e. if \(p:\underline{X} \rightarrow \mathrm {pt}\) is the structure map, then \(\omega '_{\underline{X}} = p_{\mathcal {O}\text {-mod}}^!\mathbb {C}\). Hence, if \(\underline{X}\) is smooth, one has \(\omega '_{\underline{X}} = \omega _{\underline{X}}[\dim X]\).
To give \(\omega '_{\underline{X}}\) the structure of a right \(\mathcal {D}_X\)-module we can use a result of Tsuji
[20, Theorem 2.21]. For comparison with duality on the Kato–Nakayama space, an explicit description of \(\omega '_{\underline{X}}\) given by Ishida is particularly useful. We will sketch the construction of Ishida’s complex. A detailed translation of Ishida’s construction into the language of logarithmic geometry is given in
[3, Section 1], while the local (toric) situation is discussed in
[15, Section 3.2]
Any smooth idealized log variety X is étale locally isomorphic to a union of orbits of a normal toric variety. Thus the filtration \(X^k\) turns X into a filtered semi-toroidal variety on the sense of
[6, Definiton 5.2]. Let \(\phi _k:\tilde{X}^k \rightarrow X^k \) be the normalization of the closed log stratum \(X^k\). Since X is étale locally isomorphic to a normal toric variety with its toric boundary divisor, \(\tilde{X}^k\) is the disjoint union of the irreducible components of \(X^k\). As each component is again étale locally isomorphic to a toric variety, \(\tilde{X}^k\) has a canonical induced smooth log structure (note that this log structure is not idealized). Set
$$\begin{aligned} \mathcal C^i = \phi _{i,*}\omega _{\tilde{X}^{i}}, \quad 0 \le i \le \dim X. \end{aligned}$$
The local Poincaré residue maps (see Sect. 2.2) induce a differential \(d:\mathcal C^i \rightarrow \mathcal C^{i+1}\), making \((\mathcal C^\bullet ,\, d)\) into a complex of \(\mathcal {O}_X\)-modules.
Proposition 3.6
[6, Theorem 5.4] The complex \((\mathcal C^\bullet ,\, d)\) is isomorphic to the Grothendieck dualizing complex \(\omega '_{\underline{X}}[-\dim X]\).
Remark 3.7
Any smooth log variety is étale locally isomorphic to an affine toric variety and hence is Cohen–Macaulay. Thus despite the above description of \(\omega '_{\underline{X}}\) as a complex, in this case it is actually a sheaf concentrated in the single degree \(-\dim \underline{X}\). In other words the complex \(\mathcal C^\bullet \) only has non-trivial cohomology at \(\mathcal C^0\).
Example 3.8
Let us again consider the example of the affine space \(X = \mathbb {A}^{n}\) with the log structure given by the coordinate hyperplanes. Set \(Z_i = \{z_i = 0\}\). The first differential in the complex is then
$$\begin{aligned} \omega _X \rightarrow \bigoplus _{i=1}^n \omega _{Z_i}, \quad f\frac{dz_1}{z_1} \wedge \dots \wedge \frac{dz_n}{z_n} \mapsto \sum _{i=1}^n \mathchoice{\left. f\right| _{Z_i}}{f|_{Z_i}}{f|_{Z_i}}{f|_{Z_i}} \frac{dz_1}{z_1} \wedge \dots \wedge \widehat{\frac{dz_i}{z_i}} \wedge \cdots \wedge \frac{dz_n}{z_n}. \end{aligned}$$
The kernel of this morphism consists of all sections of the form \(z_1\cdots z_n f\frac{dz_1}{z_1} \wedge \dots \wedge \frac{dz_n}{z_n}\), agreeing with Example 2.3.
Each \(\omega _{\tilde{X}^k}\) is canonically a right \(\mathcal {D}_{\tilde{X}^i}\)-module and this structure trivially extends to a right \(\mathcal {D}_X\)-module structure on \(\mathcal C^i\), which one checks to be compatible with the differential d. Thus we have defined a right \(\mathcal {D}_X\)-module structure on \(\omega '_{\underline{X}}\). If \(\underline{X}\) is smooth then \(\omega _{\underline{X}}\) is a submodule of \(\omega _X\) and hence this definition agrees with the \(\mathcal {D}_X\)-module structure on \(\omega _{\underline{X}}\) given by restriction of the Lie derivative. We let \(\mu _X\) be the left module corresponding by (2) to \(\omega '_{\underline{X}}\), that is,
$$\begin{aligned} \mu _X = \underline{{\text {Hom}}}_{\mathcal {O}_X}(\omega _X,\,\omega '_{\underline{X}}). \end{aligned}$$
From Proposition 3.6 one obtains the following corollary.
Corollary 3.9
$$\begin{aligned} \mu _X \cong \bigl (\phi _{0,*}\mathcal {O}_{\tilde{X}^0} \rightarrow \dotsc \rightarrow \phi _{\dim X,*}\mathcal {O}_{\tilde{X}^{\dim X}}\bigr )[\dim X]. \end{aligned}$$
Example 3.10
Continuing Examples 2.3 and 3.8, \(\mu _{\mathbb {A}^{n}}\) is the left \(\mathcal {D}_{\mathbb {A}^{n}}\)-module \(z_1\cdots z_n \mathcal {O}_{\mathbb {A}^{n}}[n]\) and the above corollary presents this as the kernel of the restriction morphism \(\mathcal {O}_{\mathbb {A}^{n}} \mapsto \bigoplus _i \mathcal {O}_{Z_i}\).
Remark 3.11
One notes that if \(X = \underline{X}\) is a smooth variety, then \(\mu _X = \mathcal {O}_X[\dim X]\). The general idea is that all occurrences of this shifted structure sheaf in the classical theory should be replaced by \(\mu _X\) is the logarithmic theory. In particular, all shifts by \(\dim X\) should replaced by tensoring with \(\mu _X\).
Following this idea, the dualizing complex for \(\mathcal {D}_X\) should be given by \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\). There are two possible left \(\mathcal {D}_X\)-module structures on this complex: one by left multiplication and one by Proposition 2.5 as the tensor product of two left modules. Further, \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\) has a right module structure induced from the right module \(\mathcal {D}_X\)-module structure on \(\mathcal {D}_X\) and Proposition 2.5. Unless otherwise stated, we will view \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\) as a \(\mathcal {D}_X \otimes \mathcal {D}_X^{\mathrm {op}}\)-module via left multiplication and the tensor product right module structure.
Lemma 3.12
Let Z be a component of some closed log stratum \(X^k\) with the induced idealized log structure and let \(\mathcal M\) be a left \(\mathcal {D}_Z\)-module which is locally free as an \(\mathcal {O}_Z\)-module. Then there exists an involution of \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mathcal M\) that interchanges the \(\mathcal {D}_X\)-module structures given by left multiplication with that of Proposition 2.5, and fixes the submodule \(\mathcal M\). In particular there exists such an involution of \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\). A similar statement holds for right modules.
Proof
As \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mathcal M = \mathcal {D}_Z \otimes _{\mathcal {O}_Z} \mathcal M\), we can assume that \(Z = X\). Let \(\mathcal N\) and \(\mathcal N'\) be \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mathcal M\) with the two different left \(\mathcal {D}_X\)-actions. Then \(\mathcal N\) (resp. \(\mathcal N'\)) have good filtrations \(F_\bullet \) (resp. \(F'_\bullet \)) such that \(F_0\mathcal N = F'_0\mathcal N'\) and \({\text {gr}}_{F}\mathcal N = {\text {gr}}_{F'}\mathcal N'\). Lifting the identity morphism of this associated graded we obtain the desired involution (see
[5, Lemma D.2.3]).
The involution of \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\) follows from the description of \(\omega '_{\underline{X}}\) given in Proposition 3.6 and the fact that the involutions fix \({\text {Hom}}_{\mathcal {O}_X}(\omega _X,\ \mathcal C^i)\). \(\square \)
Lemma 3.13
The complex \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\) has finite injective dimension.
Proof
It suffices to show that for sufficiently large i one has \({\text {Ext}}^i(\mathcal M,\, \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X) = 0\) for all \(\mathcal {D}_X\)-modules \(\mathcal M\). One can further restrict to \(\mathcal M = \mathcal {D}_X / I\) for all left ideals I of \(\mathcal {D}_X\). Such \(\mathcal M\) are clearly coherent and endowed with a global good filtration. The filtration on \(\mathcal {D}_X\) induces a good filtration on \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\) such that its associated graded is given by \(\pi ^*(\omega _X^\vee \otimes _{\mathcal {O}_X} \omega '_{\underline{X}})\), where \(\pi :T^*X \rightarrow X\) is the projection from the log cotangent bundle. As this is a dualizing complex, it follows that \({\text {Ext}}^i\bigl ({\text {gr}}\mathcal M,\, {\text {gr}}(\mathcal {D}_X \otimes _{\mathcal {D}_X} \mu _X)\bigr )\) vanishes for all sufficiently large i independently of \(\mathcal M\). Thus also \({\text {Ext}}^i(\mathcal M,\, \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X)\) vanishes. \(\square \)
For any left \(\mathcal {D}_X\)-module \(\mathcal M\) the \(\mathcal {D}_X^{\mathrm {op}}\)-structure on \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\) induces a right module structure on \(\underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal M,\, \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X)\). Thus we define the duality functor
$$\begin{aligned} \mathbb {D}_X :&\mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{X})^{\mathrm {op}}\rightarrow \mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{X}) \\&\mathcal M \mapsto \underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal M,\, \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X) \otimes _{\mathcal {O}_X} \omega _X^\vee . \end{aligned}$$
We will show in Theorem 3.17 that, up to a shift, \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\) is indeed the rigid dualizing complex for \(\mathcal {D}_X\). In particular, we will have that \(\mathbb {D}_X \circ \mathbb {D}_X \cong {\text {id}}\). The proof of Theorem 3.17 will be based on the following duality theorem.
Theorem 3.14
Let \(f:X \rightarrow Y\) be a morphism of smooth idealized log varieties such that the induced map \(f:\underline{X} \rightarrow \underline{Y}\) on the underlying varieties is projective. Let \(\mathcal M \in \mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{X})\) and assume that locally on Y the cohomology modules of \(\mathcal M\) admit a good filtration. Then there exists a canonical isomorphism in \(\mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{Y})\)
$$\begin{aligned} f_\bullet \circ \mathbb {D}_X \mathcal M \cong \mathbb {D}_Y \circ f_\bullet \mathcal M. \end{aligned}$$
We note that by Lemma 3.2\(f_\bullet \mathcal M\) is indeed coherent, so that the statement of the theorem makes sense.
Lemma 3.15
Let \(i:Z \rightarrow X\) be a closed immersion of smooth idealized log varieties. Then \(i^!(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X) \cong i^*\mathcal {D}_X \otimes _{\mathcal {O}_Z} \mu _Z\).
Proof
$$\begin{aligned} i^!\mu _X&= \underline{{\text {Hom}}}_{i^{-1}\mathcal {D}_X}(\mathcal {D}_{X \leftarrow Z},\, i^{-1}\mathcal {D}_X \otimes _{i^{-1}\mathcal {O}_X} i^{-1}\mu _X) \\&\cong \underline{{\text {Hom}}}_{i^{-1}\mathcal {D}_X}\bigl (i^{-1}\mathcal {D}_X \otimes _{i^{-1}\mathcal {O}_X} i^{-1}\omega _X^\vee \otimes _{i^{-1}\mathcal {O}_X} \omega _Z,\, i^{-1}\mathcal {D}_X \otimes _{i^{-1}\mathcal {O}_X} i^{-1}\mu _X \bigr ) \\&\cong \underline{{\text {Hom}}}_{i^{-1}\mathcal {O}_X}\bigl (i^{-1}\omega _X^\vee \otimes _{i^{-1}\mathcal {O}_X} \omega _Z,\, i^{-1}\mathcal {D}_X \otimes _{i^{-1}\mathcal {O}_X} i^{-1}\mu _X \bigr ) \\&\cong i^{-1}\mathcal {D}_X \otimes _{i^{-1}\mathcal {O}_X} \underline{{\text {Hom}}}_{i^{-1}\mathcal {O}_X}\bigl (i^{-1}\omega _X^\vee \otimes _{i^{-1}\mathcal {O}_X} \omega _Z,\, i^{-1}\omega _X^\vee \otimes _{i^{-1}\mathcal {O}_X} i^{-1}\omega '_{\underline{X}} \bigr ) \\&\cong i^*\mathcal {D}_X \otimes _{\mathcal {O}_Z} \underline{{\text {Hom}}}_{i^{-1}\mathcal {O}_X}\bigl (\omega _Z,\, i^{-1}\omega '_{\underline{X}} \bigr ) \\&\cong i^*\mathcal {D}_X \otimes _{\mathcal {O}_Z} \underline{{\text {Hom}}}_{i^{-1}\mathcal {O}_X}\bigl (\mathcal {O}_Z \otimes _{\mathcal {O}_Z} \omega _Z,\, i^{-1}\omega '_{\underline{X}} \bigr ) \\&\cong i^*\mathcal {D}_X \otimes _{\mathcal {O}_Z} \underline{{\text {Hom}}}_{\mathcal {O}_Z}\bigl (\omega _Z,\, \underline{{\text {Hom}}}_{i^{-1}\mathcal {O}_X}(\mathcal {O}_Z, i^{-1}\omega '_{\underline{X}}) \bigr ) \\&\cong i^*\mathcal {D}_X \otimes _{\mathcal {O}_Z} \underline{{\text {Hom}}}_{\mathcal {O}_Z}\bigl (\omega _Z,\, \omega '_{\underline{Z}} \bigr ) \cong i^*\mathcal {D}_X \otimes _{\mathcal {O}_Z} \mu _Z. \end{aligned}$$
\(\square \)
Lemma 3.16
Let \(f:X \rightarrow Y\) be a morphism of smooth idealized log varieties such that the induced map \(f:\underline{X} \rightarrow \underline{Y}\) on the underlying varieties is projective. Then there exists a canonical morphism of functors \(\mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{X}) \rightarrow \mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{Y})\)
$$\begin{aligned} f_\bullet \circ \mathbb {D}_X \rightarrow \mathbb {D}_Y \circ f_\bullet . \end{aligned}$$
Proof
For any \(\mathcal M \in \mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{X})\) we have
$$\begin{aligned} f_\bullet \mathbb {D}_X\mathcal M&\cong f_*\bigl (\underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal M,\, \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X) \otimes _{\mathcal {D}_X} \mathcal {D}_{X \rightarrow Y} \bigr ) \otimes _{\mathcal {O}_X} \omega _Y^\vee \\&\cong f_*\bigl (\underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal M,\, \mathcal {D}_{X\rightarrow Y} \otimes _{\mathcal {O}_X} \mu _X) \bigr ) \otimes _{\mathcal {O}_X} \omega _Y^\vee \end{aligned}$$
and
$$\begin{aligned} \mathbb {D}_X f_\bullet \mathcal M&\cong \underline{{\text {Hom}}}_{\mathcal {D}_Y}(f_\bullet \mathcal M,\, \mathcal {D}_Y \otimes _{\mathcal {O}_Y} \mu _Y) \otimes _{\mathcal {O}_Y} \omega _Y^\vee . \end{aligned}$$
Considering the morphism
$$\begin{aligned} f_*\underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal M,&\, \mathcal {D}_{X\rightarrow Y} \otimes _{\mathcal {O}_X} \mu _X) \\&\rightarrow f_*\underline{{\text {Hom}}}_{f^{-1}\mathcal {D}_Y}(\mathcal {D}_{Y \leftarrow X} \otimes _{\mathcal {D}_X} \mathcal M,\, \mathcal {D}_{Y \leftarrow X} \otimes _{\mathcal {D}_X} \mathcal {D}_{X\rightarrow Y} \otimes _{\mathcal {O}_X} \mu _X) \\&\rightarrow \underline{{\text {Hom}}}_{\mathcal {D}_Y}\bigl (f_*(\mathcal {D}_{Y \leftarrow X} \otimes _{\mathcal {D}_X} \mathcal M),\, f_*(\mathcal {D}_{Y \leftarrow X} \otimes _{\mathcal {D}_X} \mathcal {D}_{X\rightarrow Y} \otimes _{\mathcal {O}_X} \mu _X)\bigr ) \\&\cong \underline{{\text {Hom}}}_{\mathcal {D}_Y}\bigl (f_\bullet (\mathcal M),\, f_\bullet (\mathcal {D}_{X\rightarrow Y} \otimes _{\mathcal {O}_X} \mu _X)\bigr ), \end{aligned}$$
it thus suffices to construct a canonical morphism
$$\begin{aligned} f_\bullet (\mathcal {D}_{X \rightarrow Y} \otimes _{\mathcal {O}_X} \mu _X) \rightarrow \mathcal {D}_Y \otimes _{\mathcal {O}_Y} \mu _Y. \end{aligned}$$
We can assume that f is either a closed immersion \(X \rightarrow \underline{\mathbb {P}^{n}} \times Y\) or the projection \(\underline{\mathbb {P}^{n}} \times Y \rightarrow Y\). In the first case, by Proposition 3.4 and Lemma 3.15 we have
$$\begin{aligned} {\text {Hom}}_{\mathcal {D}_Y}(f_\bullet (\mathcal {D}_{X \rightarrow Y}&\otimes _{\mathcal {O}_X} \mu _X),\, \mathcal {D}_Y \otimes _{\mathcal {O}_Y} \mu _Y) \\&\cong {\text {Hom}}_{\mathcal {D}_Y}(f^*\mathcal {D}_Y \otimes _{\mathcal {O}_X} \mu _X,\, f^!(\mathcal {D}_Y \otimes _{\mathcal {O}_Y} \mu _Y)) \\&\cong {\text {Hom}}_{\mathcal {D}_Y}(f^*\mathcal {D}_Y \otimes _{\mathcal {O}_X} \mu _X,\, f^*\mathcal {D}_Y \otimes _{\mathcal {O}_X} \mu _X). \end{aligned}$$
Thus we take the image of the identity morphism for our map.
If f is a projection \(\underline{\mathbb {P}}^{n} \times Y \rightarrow Y\), then \(\mu _{\underline{\mathbb {P}}^{n} \times Y} = \mu _{\underline{\mathbb {P}}^{n}} \boxtimes \mu _Y\). Thus we can reduce to the case that Y is a point. But \(\mu _{\underline{\mathbb {P}}^{n}} = \mathcal {O}_{\mathbb {P}^{n}}[n]\), so that the desired morphism is just the classical trace morphism \(f_{\bullet }\mathcal {O}_{\mathbb {P}^{n}}[n] \rightarrow \mathbb {C}\).
Finally, we need to check that the constructed morphism is independent of the chosen factorization of f. Given two different factorizations as above one can form a commutative diagram
for some \(N \ge n_1, n_2\). For closed immersions, the constructed map is obtained by adjunction. Hence it is compatible with composition. Thus it suffices to show that \(\underline{\mathbb {P}}^{n_1} \times Y \rightarrow Y\) and \(\underline{\mathbb {P}}^{n_1} \times Y \rightarrow \underline{\mathbb {P}}^{N} \times Y \rightarrow Y\) induce the same duality map. As before, we reduce to the case that Y is a point. But then all log structures are trivial, so that the statement is just usual duality for D-modules. \(\square \)
Proof of Theorem 3.14
We have to show that the morphism of Lemma 3.16 is an isomorphism. As usual, it suffices to show this for \(\mathcal M\) an object of the abelian category \(\mathbf {Coh}(\mathcal {D}_{X})\). Then by assumption, locally on Y, \(\mathcal M\) is generated by an \(\mathcal {O}_X\)-coherent \(\mathcal {O}_X\)-submodule, i.e. it is a quotient of a module of the form \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mathcal G\) for some coherent \(\mathcal {O}_X\)-module \(\mathcal G\), with action given by left multiplication. Thus by the Way-out Lemma it suffices to prove the statement for modules of the form \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mathcal G\).
We thus compute
$$\begin{aligned} f_\bullet \mathbb {D}_X(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mathcal G)&= f_\bullet \bigl (\underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mathcal G,\, \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X) \otimes _{\mathcal {O}_X} \omega _X^\vee \bigr ) \\&= f_\bullet \bigl (\mathcal {D}_X \otimes _{\mathcal {O}_X} \underline{{\text {Hom}}}_{\mathcal {O}_X}( \mathcal G,\, \mathcal {O}_X) \otimes _{\mathcal {O}_X} \mu _X \otimes _{\mathcal {O}_X} \omega _X^\vee \bigr ) \\&= f_*\bigl (\mathcal {D}_{Y \leftarrow X} \otimes _{\mathcal {O}_X} \underline{{\text {Hom}}}_{\mathcal {O}_X}( \mathcal G,\, \mu _X) \otimes _{\mathcal {O}_X} \omega _X\bigr ) \\&= f_*\bigl (f^{-1}(\mathcal {D}_Y \otimes _{\mathcal {O}_Y} \omega _Y^\vee ) \otimes _{f^{-1}\mathcal {O}_Y} \underline{{\text {Hom}}}_{\mathcal {O}_X}(\mathcal G,\, \mu _X)\bigr ) \\&= \mathcal {D}_Y \otimes _{\mathcal {O}_Y} \omega _Y^\vee \otimes _{\mathcal {O}_Y} f_*\underline{{\text {Hom}}}_{\mathcal {O}_X}(\mathcal G \otimes _{\mathcal {O}_X} \omega _X,\, \omega '_{\underline{X}}) \end{aligned}$$
and
$$\begin{aligned} \mathbb {D}_Yf_\bullet (\mathcal {D}_X \otimes _{\mathcal {O}_X}&\mathcal G) = \mathbb {D}_Y\bigl (\mathcal {D}_Y \otimes _{\mathcal {O}_Y} \omega _Y^\vee \otimes _{\mathcal {O}_X} f_*(\omega _X \otimes _{\mathcal {O}_X} \mathcal G) \bigr ) \\&= \underline{{\text {Hom}}}_{\mathcal {D}_Y}\bigl (\mathcal {D}_Y \otimes _{\mathcal {O}_Y} \omega _Y^\vee \otimes _{\mathcal {O}_X} f_*(\omega _X \otimes _{\mathcal {O}_X} \mathcal G),\, \mathcal {D}_Y \otimes _{\mathcal {O}_Y} \mu _Y \bigr ) \otimes _{\mathcal {O}_Y} \omega _Y^\vee \\&= \mathcal {D}_Y \otimes _{\mathcal {O}_Y} \underline{{\text {Hom}}}_{\mathcal {O}_Y}( f_*(\omega _X \otimes _{\mathcal {O}_X} \mathcal G),\, \mu _Y) \\&= \mathcal {D}_Y \otimes _{\mathcal {O}_Y} \underline{{\text {Hom}}}_{\mathcal {O}_Y}( f_*(\omega _X \otimes _{\mathcal {O}_X} \mathcal G),\, \omega '_{\underline{Y}}) \otimes _{\mathcal {O}_Y} \omega _{Y}^\vee . \end{aligned}$$
Thus the result follows from Grothendieck duality
$$\begin{aligned} f_*\underline{{\text {Hom}}}_{\mathcal {O}_X}(\mathcal G \otimes _{\mathcal {O}_X} \omega _X,\, \omega '_{\underline{X}}) \cong \underline{{\text {Hom}}}_{\mathcal {O}_Y}( f_*(\omega _X \otimes _{\mathcal {O}_X} \mathcal G),\, \omega '_{\underline{Y}}). \end{aligned}$$
\(\square \)
Theorem 3.17
Let X be a smooth idealized log variety. Then the rigid dualizing complex for \(\mathcal {D}_X\) is given by \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X[{\text {logdim}}X]\).
Proof
We already know that \(\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\) has finite injective dimension. Showing that the canonical map
$$\begin{aligned} \mathcal M \rightarrow \underline{{\text {Hom}}}_{\mathcal {D}_X^{\mathrm {op}}}\bigl (\underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal M,\, \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X),\, \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\bigr ) \end{aligned}$$
is an isomorphism for coherent \(\mathcal M\) is a local question. Hence we can assume that \(\mathcal M\) has a free resolution. Inducting on the length of the resolution, we can further reduce to the case that \(\mathcal M = \mathcal {D}_X\). Then,
$$\begin{aligned}&\underline{{\text {Hom}}}_{\mathcal {D}_X^{\mathrm {op}}}\bigl (\underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal {D}_X,\, \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X),\, \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\bigr ) \cong \\&\quad \underline{{\text {Hom}}}_{\mathcal {D}_X^{\mathrm {op}}}\bigl (\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X,\, \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X\bigr ) \cong \underline{{\text {Hom}}}_{\mathcal {D}_X^{\mathrm {op}}}\bigl (\mathcal {D}_X,\, \mathcal {D}_X\bigr ) \cong \mathcal {D}_X, \end{aligned}$$
where the second isomorphism follows via tensor-hom adjunction from Grothendieck duality \({\text {Hom}}_{\mathcal {O}_X}(\mu _X,\, \mu _X) \cong \mathcal {O}_X\).
Let \(\Delta :X \rightarrow X \times X\) be the diagonal map. Then for any right \(\mathcal {D}_X\)-module \(\mathcal M\) we have \(\mathcal M \otimes _{\mathcal {D}_X} \mathcal {D}_{X \rightarrow X \times X} \cong \mathcal M \otimes _{\mathcal {O}_X} \mathcal {D}_X\) as right \(f^{-1}\mathcal {D}_{X \times X}\)-modules. It follows from the resolution (4) that \(\mathbb {D}_X \mathcal {O}_X = \mu _X [-{\text {logdim}}X]\). Thus \(\Delta _\bullet \mu _X = \mathbb {D}_{X\times X} \Delta _\bullet \mathcal {O}_X[{\text {logdim}}X]\). Hence, with \(d = {\text {logdim}}X\) and applying Lemma 3.12 we have
$$\begin{aligned}&\Delta _*(\omega '_{\underline{X}} \otimes _{\mathcal {O}_X} \mathcal {D}_X) \otimes _{\mathcal {O}_X} \omega _{X \times X}^\vee = \Delta _\bullet \mu _X \\&\quad \cong \underline{{\text {Hom}}}_{\mathcal {D}_{X \times X}}\bigl (\Delta _*(\omega _X \otimes _{\mathcal {O}_X} \mathcal {D}_X) \otimes _{\mathcal {O}_{X \times X}} \omega _{X\times X}^\vee ,\, \mathcal {D}_{X \times X} \otimes _{\mathcal {O}_{X \times X}} \mu _{X\times X} \otimes _{\mathcal {O}_{X \times X}} \omega _{X \times X}^\vee \bigr )[d] \\&\quad \cong \underline{{\text {Hom}}}_{\mathcal {D}_X \otimes \mathcal {D}_X}\bigl (\mathcal {D}_X \otimes _{\mathcal {O}_X} \omega _X^\vee ,\, (\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X \otimes _{\mathcal {O}_X} \omega _X^\vee ) \otimes (\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X \otimes _{\mathcal {O}_X} \omega _X^\vee )\bigr )[d] \\&\quad \cong \underline{{\text {Hom}}}_{\mathcal {D}_X \otimes \mathcal {D}_X^{\mathrm {op}}}\bigl (\mathcal {D}_X,\, (\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X \otimes _{\mathcal {O}_X} \omega _X^\vee ) \otimes (\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X)\bigr )[d] \\&\quad \cong \underline{{\text {Hom}}}_{\mathcal {D}_X \otimes \mathcal {D}_X^{\mathrm {op}}}\bigl (\mathcal {D}_X,\, (\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X) \otimes (\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X)\bigr ) \otimes _{\mathcal {O}_X} \omega _X^\vee [d]. \end{aligned}$$
Twisting by \(\omega _X\) and shifting by d we obtain the desired isomorphism
$$\begin{aligned} \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X[d] \cong \underline{{\text {Hom}}}_{\mathcal {D}_X \otimes \mathcal {D}_X^{\mathrm {op}}}\bigl (\mathcal {D}_X,\, (\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X[d]) \otimes (\mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X[d])\bigr ). \end{aligned}$$
\(\square \)
Lemma 3.18
For any \(\mathcal M \in \mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{X})\) and \(\mathcal N \in \mathbf {D}^{\mathrm {b}}(\mathcal {D}_{X})\) there exists a canonical isomorphism
$$\begin{aligned} \underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal M,\, \mathcal N \otimes _{\mathcal {O}_X} \mu _X)&\cong (\omega _X \otimes _{\mathcal {O}_X} \mathbb {D}_X\mathcal M) \otimes _{\mathcal {D}_X} \mathcal N \\&\cong \omega _X \otimes _{\mathcal {D}_X} (\mathbb {D}_X\mathcal M \otimes _{\mathcal {O}_X} \mathcal N). \end{aligned}$$
Proof
One has
$$\begin{aligned} \underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal M,\, \mathcal N \otimes _{\mathcal {O}_X} \mu _X)&\cong \underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal M,\, \mathcal {D}_X \otimes _{\mathcal {O}_X} \mu _X) \otimes _{\mathcal {D}_X} \mathcal N \\ {}&\cong (\omega _X \otimes _{\mathcal {O}_X} \mathbb {D}_X\mathcal M) \otimes _{\mathcal {D}_X} \mathcal N. \end{aligned}$$
The second equivalence follows from Lemma 2.6. \(\square \)
Pullback
Let \(f:X \rightarrow Y\) be a morphism of smooth idealized log varieties. While the pushforward \(f_\bullet \) is defined as in the classical setting, the definition of the pullback \(f^!\) needs some care in order for the expected adjunctions to exist. Recall that in the classical theory, the functor \(f^!\) differs from the naive pullback \(f^*\) by a shift by \(\dim X - \dim Y\). Thus, following the spirit of Remark 3.11, in the logarithmic theory one must twist by an appropriate relative version of \(\mu _X\).
If \(\mathcal F\) is a (complex of) coherent \(\mathcal {O}_Y\)-module(s), we write \(f^!_{\mathcal {O}\text {-mod}}\) for the \(\mathcal {O}\)-module !-pullback along f. Further we write \(\mathbb {D}^{\mathrm {Se}}_Y(\mathcal F) = \underline{{\text {Hom}}}_{\mathcal {O}_Y}(\mathcal F,\, \omega '_{\underline{Y}})\) for the \(\mathcal {O}\)-module dual of \(\mathcal F\). These two operations are related by \(f^!_{\mathcal {O}\text {-mod}} = \mathbb {D}^{\mathrm {Se}}_X \circ f^* \circ \mathbb {D}^{\mathrm {Se}}_Y\), and if f is Tor-finiteFootnote 2 one has \(f^!_{\mathcal {O}\text {-mod}}\mathcal F = f^*\mathcal F \otimes _{\mathcal {O}_X} f^!_{\mathcal {O}\text {-mod}}\mathcal {O}_Y\). One always has \(f^!_{\mathcal {O}\text {-mod}}\omega '_{\underline{Y}} = \omega '_{\underline{X}}\).
Let now \(\mathcal M \in \mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{Y}^{\mathrm {op}})\) have \(\mathcal {O}_Y\)-coherent cohomology sheaves. Then \(\mathbb {D}^{\mathrm {Se}}_Y\mathcal M\) is canonically a left \(\mathcal {D}_Y\)-module and hence \(f^!_{\mathcal {O}\text {-mod}}\mathcal M = \mathbb {D}^{\mathrm {Se}}_X \circ f^* \circ \mathbb {D}^{\mathrm {Se}}_Y (\mathcal M)\) is canonically a right \(\mathcal {D}_X\)-module. In particular \(\omega '_{\underline{X}/\underline{Y}} = f^!_{\mathcal {O}\text {-mod}}\mathcal {O}_Y\) is canonically a right \(\mathcal {D}_X\)-module.
The right \(\mathcal {D}_Y\)-module structure on \(\mathcal {O}_Y\) induces a left module structure on \(\omega _Y^\vee = \underline{{\text {Hom}}}_{\mathcal {O}_X}(\omega _Y,\, \mathcal {O}_Y)\) and hence a right \(\mathcal {D}_X\)-module structure on \(\omega _{X/Y} = \omega _X \otimes _{\mathcal {O}_X} f^*\omega _Y^\vee \). Set
$$\begin{aligned} \mu _{X/Y} = \underline{{\text {Hom}}}_{\mathcal {O}_X}(\omega _{X/Y},\, \omega '_{\underline{X}/\underline{Y}}) \in \mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{X}). \end{aligned}$$
Definition 3.19
Let \(f:X \rightarrow Y\) be a Tor-finite morphism of smooth idealized log varieties. Define a functor \(f^!:\mathbf {D}^{\mathrm {-}}_{\mathrm {qc}}(\mathcal {D}_{Y}) \rightarrow \mathbf {D}^{\mathrm {-}}_{\mathrm {qc}}(\mathcal {D}_{X})\) by
$$\begin{aligned} f^!(\mathcal M) = f^*\mathcal M \otimes _{\mathcal {O}_X} \mu _{X/Y}. \end{aligned}$$
Recalling that for a closed immersion \(i:Z \rightarrow X\) the sheaf \(i^!_{\mathcal {O}\text {-mod}}\mathcal F\) is isomorphic to \(\underline{{\text {Hom}}}_{i^{-1}\mathcal {O}_X}(\mathcal {O}_Z,\, \mathcal F)\), a straightforward computation shows that for a Tor-finite closed immersion Definitions 3.19 and 3.3 agree.
Any morphism \(f:X \rightarrow Y\) can be canonically factored into a closed immersion \(X \rightarrow X \times Y\) followed by projection \(X \times Y \rightarrow Y\). Having defined the !-pullback for each of those factors, one obtains \(f^!\) in general.
Remark 3.20
This also provides a nice explanation for the definition of \(f^!\) in the classical theory of D-modules. For this one notes that every morphism of classically smooth varieties is Tor-finite. Hence in this case Definition 3.19 suffices to cover the general case. If the log structures on X and Y are trivial, then \(\mu _{X/Y} \cong \mathcal {O}_X[\dim X-\dim Y]\). In other words, even in the classical setting \(f^!\) should be defined via the exceptional \(\mathcal {O}\)-module pullback, but switching between left and right modules reduces this to a shift of the naive pullback.
It might be interesting to see how, for general morphisms f of smooth log varieties, one can obtain a \(\mathcal {D}_X\)-module structure directly on (a suitable modification of) \(f^!_{\mathcal {O}\text {-mod}}\mathcal M\).
Proposition 3.21
Let \(f:X \rightarrow Y\) be a Tor-finite projective morphism of smooth idealized log varieties. Then for any \(\mathcal M \in \mathbf {D}^{\mathrm {b}}_{\mathrm {coh}}(\mathcal {D}_{X})\) and \(\mathcal N \in \mathbf {D}^{\mathrm {b}}(\mathcal {D}_{Y})\) there exists a canonical isomorphism
$$\begin{aligned} \underline{{\text {Hom}}}_{\mathcal {D}_Y}(f_\bullet \mathcal M,\, \mathcal N) \cong f_*\underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal M,\, f^!\mathcal N). \end{aligned}$$
Proof
One has an isomorphism \(\mu _X = \mu _{X/Y} \otimes _{\mathcal {O}_X} f^*\mu _Y\). Set \(\mathcal N' = \underline{{\text {Hom}}}_{\mathcal {O}_Y}(\mu _Y, \mathcal N)\). Then \(\mathcal N = \mathcal N' \otimes _{\mathcal {O}_Y} \mu _Y\) and \(f^!\mathcal N = f^!\mathcal N' \otimes _{\mathcal {O}_X} f^*\mu _Y = f^*\mathcal N' \otimes _{\mathcal {O}_X}\mu _X\). Thus by Lemma 3.18 and Theorem 3.14,
$$\begin{aligned} f_*\underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal M,\, f^!\mathcal N)&\cong f_*\underline{{\text {Hom}}}_{\mathcal {D}_X}(\mathcal M,\, f^*\mathcal N' \otimes _{\mathcal {O}_X} \mu _X) \\&\cong f_*\bigl ( (\omega _X \otimes _{\mathcal {O}_X} \mathbb {D}_X\mathcal M) \otimes _{\mathcal {D}_X} \mathcal {D}_{X \rightarrow Y} \otimes _{f^{-1}\mathcal {D}_Y} f^{-1}\mathcal N'\bigr ) \\&\cong f_*\bigl ( (\omega _X \otimes _{\mathcal {O}_X} \mathbb {D}_X\mathcal M) \otimes _{\mathcal {D}_X} \mathcal {D}_{X \rightarrow Y} \bigr )\otimes _{\mathcal {D}_Y} \mathcal N' \\&\cong (\omega _Y \otimes _{\mathcal {O}_Y} f_\bullet \mathbb {D}_X \mathcal M) \otimes _{\mathcal {D}_Y} \mathcal N' \\&\cong (\omega _Y \otimes _{\mathcal {O}_Y} \mathbb {D}_X f_\bullet \mathcal M) \otimes _{\mathcal {D}_Y} \mathcal N' \\&\cong \underline{{\text {Hom}}}_{\mathcal {D}_Y}(f_\bullet \mathcal M,\, \mathcal N' \otimes _{\mathcal {O}_Y} \mu _Y) = \underline{{\text {Hom}}}_{\mathcal {D}_Y}(f_\bullet \mathcal M,\, \mathcal N). \end{aligned}$$
\(\square \)
Remark 3.22
Let us note that the usual base-change theorem does not generalize to the theory of logarithmic D-modules. For example, if \(\mathrm {pt}\rightarrow \mathbb {A}^{1}\) it the inclusion of the (idealized) log point into the origin, then \(i^!i_\bullet \) is not equal to the identity for the same reason as in the \(\mathcal {O}\)-module case. Indeed,
$$\begin{aligned} i^!i_\bullet \mathbb {C}\cong & {} {\text {Hom}}_{\mathcal {D}_{\mathbb {A}^{1}}(\mathbb {A}^{1})}(\mathbb {C}[\partial ],\, \mathbb {C}) \cong {\text {Hom}}_{\mathcal {D}_{\mathbb {A}^{1}}(\mathbb {A}^{1})}\bigl (\mathcal {D}_{\mathbb {A}^{1}}(\mathbb {A}^{1}) \xrightarrow {x} \mathcal {D}_{\mathbb {A}^{1}}(\mathbb {A}^{1}),\, \mathbb {C}\bigr ) \\\cong & {} \mathbb {C}\oplus \mathbb {C}[-1]. \end{aligned}$$
One would expect this issue to be resolved by a suitable theory of “derived logarithmic geometry.”