Current tasks

Stream: toric

[Yaël Dillies]

Here is what I gathered each person was doing:

• @Patrick Luo: Show that affine monoids embed in ℤⁿ for some n
• @Michał Mrugała: Define toric ideals
• @Yaël Dillies: Refactor docs#Submonoid.LocalizationMap to implement docs#FunLike

[Yaël Dillies]

It would be great if we could acquire a blueprint covering at least the first few chapters of Cox-Little-Schenck. Would anyone like to write that?

[Yaël Dillies]

Please create one topic per task, to avoid flooding this one.

[Yaël Dillies]

• @Michał Mrugała, @Andrew Yang, @Yaël Dillies: Group object API

[Yaël Dillies]

• @Michał Mrugała, @Andrew Yang: The torus is a commutative group object. Discussion: #toric > The torus is a commutative group object

[Yaël Dillies]

• @Patrick Luo: Write the blueprint for chapter 1.1 of CLS

[Yaël Dillies]

• @Yaël Dillies: Write the blueprint for chapter 1.2 of CLS

[Michał Mrugała]

• @Paul Lezeau: Develop rudimentary monoid object action API. Discussion: #toric > Monoid object action API

[Yaël Dillies]

• @Michał Mrugała, @Yaël Dillies: Group-like elements in Hopf algebras
• @Yaël Dillies: Characterise group-like elements in group algebras
• @Yaël Dillies, @Michał Mrugała: Hopf algebra homs preserve group-like elements

[Yaël Dillies]

• @Matthew Johnson: define polytopes
• @Paul Reichert: define polyhedral cones

[Yaël Dillies]

• @Matthew Johnson: Prove that the Minkowski sum of two polytopes is a polytope
• @Matthew Johnson : Rescue the content of https://github.com/Jun2M/Main-theorem-of-polytopes

[Yaël Dillies]

• 0-tensor-lin-indep: The tensor product of two linearly independent families of vectors is linearly independent. @Paul Lezeau, @Andrew Yang

[Yaël Dillies]

• @Aaron Liu: State the blueprint items from Chapter 1.2 of CLS

[Michał Mrugała]

These are tasks which are either not necessary yet, but will be necessary to do all of 1.1. Listed roughly in order of priority.

• No one yet: Define representations of algebraic groups.
• No. one yet: Define eigenspaces of characters.
• No one yet: Define scheme-theoretic image.
• No one yet: Prove that if f : X ⟶ Y is quasi-compact and Z is the scheme-theoretic image of f, then f : X ⟶ Z is dominant.

[Andrew Yang]

Scheme theoretic image is probably docs#AlgebraicGeometry.Scheme.Hom.ker (or f.ker.glueData.glued) and the last point is on the way too.

[Yaël Dillies]

• 0-mv-laurent-poly-domain: Multivariate Laurent polynomials form a domain. @Paul Lezeau
• 0-irred-subset-gen: Irreducible elements lie in all sets generating a salient monoid. @Patrick Luo
• 0-irred-gen: Salient affine monoids are generated by their irreducible elements. @Patrick Luo

[Yaël Dillies]

• Yoneda for monoid object actions. @Paul Lezeau
• 0-hopf-cogrp-alg: Hopf algebras are cogroup objects in the category of algebras. @Michał Mrugała, @Andrew Yang, @Christian Merten
• 0-spec-alg: Spec as a functor on algebras. @Yaël Dillies
• 0-spec-hopf: Spec as a functor on Hopf algebras. @Yaël Dillies
• 0-ess-image-spec-hopf: Essential image of Spec on Hopf algebras. @Yaël Dillies

[Yaël Dillies]

• 0-diag. Define diagonalisable groups. @Sophie Morel
• 1-1-14-aff-tor-var-spec-aff-mon-alg: The spectrum of an affine monoid algebra is an affine toric variety. @Patrick Luo

[Yaël Dillies]

• 1-2-6-face-polyhedral-cone. A face of a polyhedral cone is polyhedral. @Paul Reichert

[Yaël Dillies]

• 0-grp-equiv: Equivalences lift to group object categories. @Yaël Dillies

[Yaël Dillies]

• 0-over-lim: Limit-preserving functors lift to over categories. @Moisés Herradón Cueto
• 0-ess-image-grp: Essential image of a functor on group objects. @Andrew Yang

[Yaël Dillies]

• 1-2-4-dual-polyhedral-cone: The dual of a polyhedral cone is polyhedral. @Justus Springer
• 1-2-dual-cone-add: Dual cone of a sumset. @Justus Springer
• 1-2-4-double-dual-polyhedral-cone: The double dual of a polyhedral cone is the original cone. @Justus Springer

[Yaël Dillies]

• 0-slice-adj: Sliced adjoint functors. @Yaël Dillies, @Michał Mrugała
• 0-full-grp-hopf-alg. The group algebra functor is fully faithful. @Michał Mrugała
• 1-1-torus-spec: The torus over Spec R. @Raphael Douglas Giles

[Yaël Dillies]

• 0-ess-image-over: Essential image of a sliced functor. @Yaël Dillies
• 0-ess-image-spec-alg: Essential image of Spec on algebras. @Yaël Dillies

Unclaimed

Scheme stuff

• 1-1-char-torus: The character lattice of the torus.
• 1-1-group-hom-torus: The image of a torus is a torus.
• 1-1-subgroup-subtorus: A subgroup of a torus is a torus.
• 1-1-char-eigenspace: Define the character eigenspace.
• 1-1-phiA: Define φₐ.
• 1-1-14-char-spec-aff-mon-alg: The character lattice of the spectrum of an affine monoid algebra.

Convex stuff

• 1-2-5-facet: Define facets.
• 1-2-5-edge: Define edges.
• 1-2-6-inter-faces: The intersection of two faces of a polyhedral cone is a face.
• 1-2-6-face-face: The face of a face of a polyhedral cone is a face.
• 1-2-6-face-mem-of-add: Membership criterion for a face.
• 1-2-8-dual-cone-inter-halfspaces: Dual cone of the intersection of halfspaces.
• 1-2-rel-interior-inner: The relative interior in terms of the inner product.
• 1-2-min-face: Minimal face of a cone
• 1-2-12-salient-cone-tfae: Alternative definitions of salient cones.
• 1-2-14-rat-cone: Define rational cones.
• 1-2-17-dual-lat-cone: Define the dual lattice of a cone.