Admissibility of presentation kernels

An algebra map from a path algebra has admissible kernel if its arrows land in a nilpotent two-sided ideal J and the images of the vertices and arrows are linearly independent modulo J². Nilpotence kills all sufficiently long paths. Independence modulo J² ensures that a relation cannot involve paths of length zero or one.

These are the two checks needed for presentations of finite-dimensional basic algebras: take J to be the Jacobson radical and choose arrows lifting a basis of its first radical layer. The criterion itself needs neither finite-dimensionality nor surjectivity of the map, and allows infinitely many arrows. In particular it can be used before proving surjectivity.

The argument follows Assem--Simson--Skowroński, Elements of the Representation Theory of Associative Algebras I, Chapter II, §3. It uses the short-path basis of the quotient by an arrow-ideal power.

theorem AlgHom.arrowIdeal_pow_le_comap {k : Type w} {Q : Type u} {A : Type z} [Quiver Q] [_root_.Finite Q] [CommSemiring k] [Semiring A] [Algebra k A] (f : TauCeti.pathAlgebra k Q →ₐ[k] A) (J : Ideal A) (harrows : ∀ (i j : Q) (a : i ⟶ j), f (TauCeti.PathAlgebra.ofArrow a) ∈ J) (n : ℕ) : TauCeti.arrowIdeal k Q ^ n ≤ Ideal.comap f (J ^ n)

If every arrow maps into J, the nth power of the arrow ideal maps into J^n.

theorem AlgHom.comap_sq_eq_arrowIdeal_sq {k : Type w} {Q : Type u} {A : Type z} [Quiver Q] [_root_.Finite Q] [CommRing k] [Ring A] [Algebra k A] (f : TauCeti.pathAlgebra k Q →ₐ[k] A) (J : Ideal A) [J.IsTwoSided] (harrows : ∀ (i j : Q) (a : i ⟶ j), f (TauCeti.PathAlgebra.ofArrow a) ∈ J) (hind : LinearIndependent k fun (p : TauCeti.ShortPath Q 2) => (Ideal.Quotient.mk (J ^ 2)) (f (TauCeti.PathAlgebra.ofPath ↑p))) : Ideal.comap f (J ^ 2) = TauCeti.arrowIdeal k Q ^ 2

If arrows map into J and vertices and arrows are independent modulo J², the preimage of J² is exactly the square of the arrow ideal. Thus the map identifies the first two path layers with their images, without requiring surjectivity.

theorem AlgHom.isAdmissibleIdeal_ker {k : Type w} {Q : Type u} {A : Type z} [Quiver Q] [_root_.Finite Q] [CommRing k] [Ring A] [Algebra k A] (f : TauCeti.pathAlgebra k Q →ₐ[k] A) (J : Ideal A) [J.IsTwoSided] (hnil : IsNilpotent J) (harrows : ∀ (i j : Q) (a : i ⟶ j), f (TauCeti.PathAlgebra.ofArrow a) ∈ J) (hind : LinearIndependent k fun (p : TauCeti.ShortPath Q 2) => (Ideal.Quotient.mk (J ^ 2)) (f (TauCeti.PathAlgebra.ofPath ↑p))) : TauCeti.IsAdmissibleIdeal (RingHom.ker f.toRingHom)

A path-algebra map whose arrows lie in a nilpotent two-sided ideal and whose vertices and arrows are independent modulo its square has admissible kernel.