11.1 Graded rings and modules

Graded rings and modules are powerful tools in algebraic geometry. They provide a way to study geometric objects like projective varieties using algebraic techniques. This connection allows us to translate complex geometric problems into more manageable algebraic ones.

The concept of grading adds structure to rings and modules, making them easier to work with. It's particularly useful for understanding projective spaces, their subvarieties, and associated sheaves. This algebraic approach opens up new ways to tackle geometric questions.

Graded rings and modules

Definition and basic properties

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• A graded ring is a ring R together with a direct sum decomposition R = ⨁ₙ₌₀ Rₙ (as abelian groups) such that Rₘ·Rₙ ⊆ Rₘ₊ₙ for all m, n ≥ 0
  • The elements of Rₙ are called homogeneous elements of degree n
• A graded R-module is an R-module M together with a direct sum decomposition M = ⨁ₙ₌₀ Mₙ (as abelian groups) satisfying Rₘ·Mₙ ⊆ Mₘ₊ₙ for all m, n ≥ 0
  • The elements of Mₙ are called homogeneous elements of degree n
• A homomorphism of graded rings (resp. modules) maps homogeneous elements to homogeneous elements of the same degree, preserving the grading structure
• The direct sum, tensor product, and Hom of graded modules inherit a natural grading
  • Similarly, the kernel, image, and cokernel of a homomorphism of graded modules are also graded

Noetherian property

• A graded ring (resp. module) is called Noetherian if it satisfies the ascending chain condition on graded ideals (resp. submodules)
  • This means that any ascending chain of graded ideals (resp. submodules) stabilizes after finitely many steps
• A graded ring is Noetherian if and only if it is finitely generated as an R₀-algebra
  • In other words, there exist finitely many homogeneous elements that generate the ring over R₀

Examples from algebraic geometry

Polynomial rings and projective varieties

• The polynomial ring R = k[x₀, ..., xₙ] over a field k is a graded ring with the standard grading deg(xᵢ) = 1 for all i
  • The homogeneous elements of degree d are the homogeneous polynomials of degree d
• The homogeneous coordinate ring of a projective variety X ⊆ Pⁿ is the graded ring S(X) = k[x₀, ..., xₙ]/I(X), where I(X) is the homogeneous ideal of X
  • The degree d part of S(X) corresponds to the global sections of the line bundle O_X(d) on X

Twisted sheaves and graded modules

• The twisted sheaf O_X(d) on a projective variety X is a graded S(X)-module, where the degree n part is given by the global sections of O_X(d+n)
• More generally, any coherent sheaf F on X gives rise to a graded S(X)-module ⨁ₙ H⁰(X, F(n))
  • The graded module encodes information about the sheaf cohomology of F

Weighted projective spaces

• The homogeneous coordinate ring of a weighted projective space P(a₀, ..., aₙ) is the graded ring k[x₀, ..., xₙ] with the weighted grading deg(xᵢ) = aᵢ
  • Weighted projective spaces and their subvarieties provide examples of graded rings with non-standard gradings
  • They arise naturally in the study of certain singular varieties and orbifolds

Applications in algebraic geometry

Hilbert functions and polynomials

• The Hilbert function of a graded module M is the function HM(n) = dimₖ Mₙ, measuring the dimension of each graded component
  • For a projective variety X, the Hilbert function of S(X) encodes important geometric information (dimension, degree)
• The Hilbert polynomial of a graded module M is the unique polynomial PM(t) such that PM(n) = HM(n) for all sufficiently large n
  • The Hilbert polynomial of a projective variety X is a key invariant used to classify X and study its properties
• The Hilbert series of a graded module M is the generating function HilbM(t) = Σₙ (dimₖ Mₙ)tⁿ, encoding the same information as the Hilbert function and polynomial in a compact form

Sheaf cohomology and regularity

• Graded modules can be used to study the sheaf cohomology of projective varieties
  • For example, the graded module ⨁ₙ H⁰(X, O_X(n)) determines the cohomology groups Hⁱ(X, O_X(n)) for all i and n
• The Castelnuovo-Mumford regularity of a graded module M is the smallest integer r such that Mₙ is spanned by its elements of degree ≤ n+r for all n
  • It measures the complexity of M and has applications to the computation of syzygies and free resolutions
  • Regularity is a key invariant in the study of projective embeddings and the geometry of subvarieties

Graded rings vs projective varieties

Correspondence between graded rings and projective varieties

• The homogeneous coordinate ring S(X) of a projective variety X ⊆ Pⁿ determines X up to isomorphism
  • Conversely, every finitely generated graded k-algebra R = ⨁ₙ₌₀ Rₙ with R₀ = k gives rise to a projective variety Proj(R)
• The functor Proj establishes a contravariant equivalence between the category of finitely generated graded k-algebras (modulo nilpotents) and the category of projective k-varieties
  • This correspondence is a fundamental tool in algebraic geometry, allowing to study geometric properties using algebraic techniques

Subvarieties and graded ideals

• The closed subvarieties of Proj(R) correspond to the graded prime ideals of R (except the irrelevant ideal ⨁ₙ₌₁ Rₙ)
  • The dimension of Proj(R) is equal to the Krull dimension of R minus one
• The twisting sheaf O_Proj(R)(n) on Proj(R) corresponds to the graded R-module R(n), where R(n)ₘ = Rₙ₊ₘ
  • More generally, coherent sheaves on Proj(R) correspond to finitely generated graded R-modules (modulo torsion)

Serre-Grothendieck correspondence

• The Serre-Grothendieck correspondence establishes an equivalence between the category of coherent sheaves on Proj(R) and the quotient category of finitely generated graded R-modules modulo torsion modules
  • It allows to translate between sheaf-theoretic and module-theoretic properties
  • Many geometric constructions (tensor products, pullbacks, pushforwards) can be studied algebraically using this correspondence