Homogeneous Localization #

Notation #

• ι is a commutative monoid;
• R is a commutative semiring;
• A is a commutative ring and an R-algebra;
• 𝒜 : ι → submodule R A is the grading of A;
• x : submonoid A is a submonoid

Main definitions and results #

This file constructs the subring of Aₓ where the numerator and denominator have the same grading, i.e. {a/b ∈ Aₓ | ∃ (i : ι), a ∈ 𝒜ᵢ ∧ b ∈ 𝒜ᵢ}.

• homogeneous_localization.num_denom_same_deg: a structure with a numerator and denominator field where they are required to have the same grading.

However num_denom_same_deg 𝒜 x cannot have a ring structure for many reasons, for example if c is a num_denom_same_deg, then generally, c + (-c) is not necessarily 0 for degree reasons --- 0 is considered to have grade zero (see deg_zero) but c + (-c) has the same degree as c. To circumvent this, we quotient num_denom_same_deg 𝒜 x by the kernel of c ↦ c.num / c.denom.

• homogeneous_localization.num_denom_same_deg.embedding : for x : submonoid A and any c : num_denom_same_deg 𝒜 x, or equivalent a numerator and a denominator of the same degree, we get an element c.num / c.denom of Aₓ.

• homogeneous_localization: num_denom_same_deg 𝒜 x quotiented by kernel of embedding 𝒜 x.

• homogeneous_localization.val: if f : homogeneous_localization 𝒜 x, then f.val is an element of Aₓ. In another word, one can view homogeneous_localization 𝒜 x as a subring of Aₓ through homogeneous_localization.val.

• homogeneous_localization.num: if f : homogeneous_localization 𝒜 x, then f.num : A is the numerator of f.

• homogeneous_localization.denom: if f : homogeneous_localization 𝒜 x, then f.denom : A is the denominator of f.

• homogeneous_localization.deg: if f : homogeneous_localization 𝒜 x, then f.deg : ι is the degree of f such that f.num ∈ 𝒜 f.deg and f.denom ∈ 𝒜 f.deg (see homogeneous_localization.num_mem_deg and homogeneous_localization.denom_mem_deg).

• homogeneous_localization.num_mem_deg: if f : homogeneous_localization 𝒜 x, then f.num_mem_deg is a proof that f.num ∈ 𝒜 f.deg.

• homogeneous_localization.denom_mem_deg: if f : homogeneous_localization 𝒜 x, then f.denom_mem_deg is a proof that f.denom ∈ 𝒜 f.deg.

• homogeneous_localization.eq_num_div_denom: if f : homogeneous_localization 𝒜 x, then f.val : Aₓ is equal to f.num / f.denom.

• homogeneous_localization.local_ring: homogeneous_localization 𝒜 x is a local ring when x is the complement of some prime ideals.

References #

• Robin Hartshorne, Algebraic Geometry