Grading algebraic cycles

Stream: maths

[Raphael Douglas Giles]

Hello!

Following some discussion on the PR #26304, I'm wondering what peoples' opinions are on the best way to define gradings of algebraic cycles in the right generality. I think we want API which can accommodate grading by dimension and codimension (defined in terms of height and coheight in the specialization order), and by dimension functions in the sense of the stacks project. We can't think of a good common generalisation of these kinds of function, and we're also not sure if there are other kinds of gradings on algebraic cycles people might want to consider.

Currently, things are defined such that Grading X N is just a wrapper around functions X \to N. For example, using this notion we have the following definition of homogeneous cycles of "dimension" d:

def Grading.homogeneousCycles {N : Type*} (g : Grading X N) (R : Type*) [AddMonoid R] (d : N) :
    AddSubmonoid (AlgebraicCycle X R) where
  carrier := { c | ∀ x ∈ c.support, g x = d }
  add_mem' {c₁} c₂ hc₁ hc₂ := by
    simp at hc₁ hc₂ ⊢
    grind
  zero_mem' := by simp

It feels a little bit strange/like there's not enough structure to just have gradings be a wrapper around plain functions. We would be interested to know if there are any natural properties we should expect these functions to satisfy.

Thanks,
Raphael

[Andrew Yang]

For this definition alone, it probably should just take a s : Set X and defined to be { c | c.support ⊆ s }.

[Raphael Douglas Giles]

Right, that's true, you can certainly have a grading of cycles be a function N \to Set X for some indexing type N, and I suppose this allows you the flexibility that the sets of "homogeneous cycles" can overlap. Do you think this definition will be as convenient to use as gradings defined by functions X \to N, and are there cases where we want the increased generality your definition provides?

[Antoine Chambert-Loir]

Intersection theory on more general schemes than proper varieties over fields sometimes requires different notions of dimension/codimension. For example, over a discrete valuation ring, Fulton (Intersection theory, chapter 20) considers the relative dimension. In their Non-archimedean Arakelov theory, Bloch, Gillet and Soulé consider the dimension of a proper scheme containing the given scheme. Thorup (1990, Rational equivalence on arbitrary Noetherian schemes) describes various dimension/codimension functions. And all have their interest.