This is a gathering place for requests and suggestions for algebraic geometry in Sage.
Requests
Critical
- fast R[x_1,...,x_n], also with fractional exponents
- GB's, flexible gradings, term orders
- rings (not necessarily commutative)
- modules (not just ideals, not just free)
- sheaves
- homological algebra (free resolutions)
- linear algebra with basis an arbitrary index set I
- very very fast sparse and dense linear algebra over many types of rings and fields
Dream features
- representation theory for finite groups (char 0 and modular, not just GAP, compare to what MAGMA can do -- and how fast it can do it)
- rings of representations (Grothendieck rings, etc.)
- local rings and global rings: localization, really working (not just M2)
- GB over all rings (e.g. field extensions), even noncommutative when possible
- full functoriality (e.g. preservation of GL_n-actions, functors, operations on functors, Yoneda product, tensor products)
- full homological algebra (spectral sequences, derived categories, etc.)
- parallelize everything
- a "good clean" programming language (not just M2, e.g. Maple -- having to put things into rings before being able to use them is annoying; work easily with general expressions)
- super fast GB's and syzygies (speed and low memory usage)
- super fast and low memory primary decomposition (e.g. numerical) and integral closure
- sheaves, Chern classes, intersection theory on singular spaces
- algebraic topology on complex and real points on a variety
- etale cohomology
- usable resolution of singularities