2
$\begingroup$

Let $X$ be a complex manifold and $D$ a divisor on $X$. Let $f : Z \rightarrow X$ be a morphism of complex manifolds and assume that $f^{-1}D$ is a divisor on $Z$. A meromorphic connection on $X$ along $D$ is a coherent $\mathcal{O}_X[D]$-module endowed with a $\mathbb{C}$-linear morphism $\nabla : M \rightarrow \Omega^{1}_X \otimes_{\mathcal{O}_X} M$ that satisfies the Liebnitz rule and that is flat. Hence, a meromorphic connection is a $\mathcal{D}_X$-module such that $M \vert_Y$, $Y = X \backslash D$, is locally free over $\mathcal{O}_Y$. In the book $\mathcal{D}$-modules, Perverse Sheaves and Representation Theory the inverse image for $\mathcal{D}_X$-modules is used to define the inverse image in the category of meromorphic connections along $D$. The authors point out that $$ \mathcal{O}_Z[f^{-1}D] \simeq \mathcal{O}_Z \otimes_{f^{-1}\mathcal{O}_X} f^{-1} \mathcal{O}_X[D] \simeq \mathcal{O}_Z \otimes^{L}_{f^{-1}\mathcal{O}_X} f^{-1} \mathcal{O}_X[D] $$ because $\mathcal{O}_X[D]$ is flat over $\mathcal{O}_X$. Then, they write $$ Lf^{*}M \simeq \mathcal{O}_Z[f^{-1}D] \otimes^L_{f^{-1}\mathcal{O}_X[D]} f^{-1}M \simeq \mathcal{O}_Z[f^{-1}D] \otimes_{f^{-1}\mathcal{O}_X[D]} f^{-1}M. $$ I think the last isomorphism can be proven using the (non-trivial) fact that the category of meromorphic connections is in fact a subcategory of holonomic $\mathcal{D}_X$-modules, and therefore any map of complex manifolds is non-characteristic with respect to $M$. However, the authors say that in the book the above fact is not used. How can the last isomorphism be proved?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.