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Martin Sleziak
Martin Sleziak
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Certainly the fact that the ring of differential operators is non-Noetherian is an inconvenience but it is not clear if it is more than that. For instance one can define the notion of holonomic module. It is not a direct translation of the characteristic zero definition (and this is certainly related to this inconvenience) but once given it seems to work as well as in characteristic zero:

MR1918185MR1918185 (2003h:14030) Bögvad, Rikard(S-STOC) An analogue of holonomic D-modules on smooth varieties in positive characteristics. (English summary) The Roos Festschrift volume, 1. Homology Homotopy Appl. 4 (2002), no. 2, part 1, 83–116. 14F10 (16S32 32C38)

Certainly the fact that the ring of differential operators is non-Noetherian is an inconvenience but it is not clear if it is more than that. For instance one can define the notion of holonomic module. It is not a direct translation of the characteristic zero definition (and this is certainly related to this inconvenience) but once given it seems to work as well as in characteristic zero:

MR1918185 (2003h:14030) Bögvad, Rikard(S-STOC) An analogue of holonomic D-modules on smooth varieties in positive characteristics. (English summary) The Roos Festschrift volume, 1. Homology Homotopy Appl. 4 (2002), no. 2, part 1, 83–116. 14F10 (16S32 32C38)

Certainly the fact that the ring of differential operators is non-Noetherian is an inconvenience but it is not clear if it is more than that. For instance one can define the notion of holonomic module. It is not a direct translation of the characteristic zero definition (and this is certainly related to this inconvenience) but once given it seems to work as well as in characteristic zero:

MR1918185 (2003h:14030) Bögvad, Rikard(S-STOC) An analogue of holonomic D-modules on smooth varieties in positive characteristics. (English summary) The Roos Festschrift volume, 1. Homology Homotopy Appl. 4 (2002), no. 2, part 1, 83–116. 14F10 (16S32 32C38)

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Torsten Ekedahl
Torsten Ekedahl
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Certainly the fact that the ring of differential operators is non-Noetherian is an inconvenience but it is not clear if it is more than that. For instance one can define the notion of holonomic module. It is not a direct translation of the characteristic zero definition (and this is certainly related to this inconvenience) but once given it seems to work as well as in characteristic zero:

MR1918185 (2003h:14030) Bögvad, Rikard(S-STOC) An analogue of holonomic D-modules on smooth varieties in positive characteristics. (English summary) The Roos Festschrift volume, 1. Homology Homotopy Appl. 4 (2002), no. 2, part 1, 83–116. 14F10 (16S32 32C38)