The classes of the graded projectives Af in K₀^gr(proj A)

Let A be a k-algebra graded by 𝒜 : ℤ → Submodule k A, and let f be an idempotent of A for which the left ideal Af is homogeneous. The graded module TauCeti.GradedModuleCat.ofIdeal 𝒜 (Ideal.span {f}) hI is generated by f, and its underlying module is projective, being a direct summand of A. It is therefore an object of the full subcategory TauCeti.gradedFiniteProjectiveModules 𝒜 on which the graded Cartan map is defined, and so it has a class [Af] in K₀^gr(proj A).

This file records that membership only; it needs neither a base field nor finite-dimensionality of A, which enter later when the graded Cartan map is read in idempotent coordinates (TauCeti.Algebra.Category.GradedModuleCat.CartanMap.IdempotentCoordinate).

Main results

• TauCeti.gradedFiniteProjectiveModules_ofIdeal_span_singleton: the left ideal Af generated by an idempotent is a finite graded projective.

References

• C. Năstăsescu and F. Van Oystaeyen, Methods of Graded Rings, Section 2.3, for graded modules and projectivity in the graded category.
• I. Assem, D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras, Vol. 1, Section I.4, for the projectives Af.

theorem TauCeti.gradedFiniteProjectiveModules_ofIdeal_span_singleton {k : Type uk} [CommRing k] {A : Type uA} [Ring A] [Algebra k A] {𝒜 : ℤ → Submodule k A} [GradedAlgebra 𝒜] {f : A} (hf : IsIdempotentElem f) (hI : Ideal.IsHomogeneous 𝒜 (Ideal.span {f})) : gradedFiniteProjectiveModules 𝒜 (GradedModuleCat.ofIdeal 𝒜 (Ideal.span {f}) hI)

The left ideal Af generated by an idempotent is a finite graded projective, so it has a class in K₀^gr(proj A).