Based on the table located at stacky.net based on Table 1 from Bjorn Poonen's Rational points on varieties we had the idea of doing something analogous for the functors which are associated to a morphism of schemes (or where applicable ringed spaces).
This would serve as a good way to
- solidify our knowledge about this subject
- learn more about all the associated functors we only vaguely know the existence or properties of
- do something awesome based the Stacks project (hoping to inspire others by this)
By "we" Mauro Porta and Pieter Belmans are implied.
- direct image
- inverse image (various incarnations, get terminology straight)
- higher direct images
- sections with support [Har, Ex. II.1.20]
- direct image with compact support
- ...
- exactness
- adjointness
- natural interpreations (which categories etc.)
- preservation of (quasi-)coherent sheaves, locally free sheaves, ...
- behaviour for algebraic cycles
- ...
Hence we would consider both what happens for (derived) categories of sheaves, and what happens in an intersection theory context. The emphasis will be first on sheaves, later on (after our course which treats intersection theory) we could work out the facts for algebraic cycles and related objects.
-
HTML
- good styling
- dynamic view (think toggles: see Remarks)
- ...
-
TeX
- both a A4 printable and poster version
- ...
-
having both would require some metaformat I'm afraid, or some Python magic to process things
- refer to Stacks project and EGA wherever possible, also Hartshorne, Liu, Fulton (but make the visibility of these, or all references, optional?)
- also provide counterexamples where applicable (again refer to Stacks project, possibly adding them?)