5

Here (at the bottom of page 13) it's stated that for a smooth map $f:X\to S$, the relative Spencer complex $$\Omega_{X/S}^\bullet \otimes_{f^{-1}\mathcal{O}_S} D_X$$ is a resolution of the transfer module $$D_{S\leftarrow X} \ = \ f^{-1}(D_S \otimes_{f^{-1}\mathcal{O}_S} \Omega_S^{-1}) \otimes_{f^{-1}\mathcal{O}_S} \Omega_X$$ by left $f^{-1}D_S$-modules.

  1. Why is this a resolution?
  2. Is there a similar resolution in the case where some fibres of $f$ are singular? If so, what can we say about it?

$$\text{}$$

It's easy to show that $\Omega_{X/S}^\bullet \otimes D_X$ is a complex, but I'm not sure exactly how to get a map $$\Omega_{X/S} \otimes_{f^{-1}\mathcal{O}_S} D_X\longrightarrow D_{S\leftarrow X}$$ of left $f^{-1}D_S$-modules. I know it should be a trivial consequence of $\Omega_X\otimes_{f^{-1}\mathcal{O}_S} f^{-1}\Omega_S\simeq \Omega_{X/S}$, but for some reason I can't get it to work. I would guess something like $$\omega \otimes P \longrightarrow \phi(P) \otimes \omega$$ might work for an appropriate map $\phi:D_X\to f^{-1}D_S$.

I'm also curious about what happens when the map $f$ is smooth except at a point where the fibre is singular.

Meow
  • 6,083
  • 6
  • 39
  • 60
  • Aren't you looking for a map going the other way ? That is a map $D_{S\leftarrow X}\to\Omega_{X/S}\otimes_{f^{-1}\mathcal{O}_S}D_X$. – Roland Apr 23 '18 at 20:14
  • @Roland I'm trying to compute $Rf_*\mathcal{O}_X$, for which one first needs to compute $D_{S\leftarrow X}\otimes^L\mathcal{O}_X$, so I definitely do need a locally free (or flat) resolution of $D_{S\leftarrow X}$ (see for instance the bottom of p.13 in https://www.math.purdue.edu/~dvb/preprints/dmod.pdf). – Meow Apr 23 '18 at 20:38
  • I am not an expert in algebraic geometry nor I am an expert in the language of $D$-modules: however I remember an old paper of Francesco Succi (1966), "[Alcune osservazioni sui teoremi di de Rham (Some observations on de Rham's theorems)](http://rivista.math.unipr.it/fulltext/1966-7/1966-7-035.pdf)", Rivista di Matematica della Universitá di Parma, II Serie, vol. 7, 35-46 (1966), MR0231305, Zbl 0178.57302, which deals with the relative de Rham cohomology. It is written in Italian ad perhaps the symbology is outdated: however I hope it could be of some help. – Daniele Tampieri Apr 29 '18 at 15:24

0 Answers0