Morita reduction to a basic ring

Every semiperfect ring A -- in particular every finite-dimensional algebra over a field -- is Morita equivalent to a basic ring (TauCeti.IsBasic), one whose quotient by the Jacobson radical is a finite product of division rings. The basic ring is a corner eAe of A:

• by Artin–Wedderburn, A ⧸ J(A) ≅ ∏ᵢ Matₙᵢ(Dᵢ) has a full idempotent ē, the matrix unit E₁₁ of every block, whose corner is ∏ᵢ Dᵢ (IsSemisimpleRing.exists_isIdempotentElem_full_corner_ringEquiv_pi_divisionRing);
• since A is semiperfect, ē lifts to a full idempotent e of A (TauCeti.IsSemiperfectRing.exists_isIdempotentElem_full_eq);
• the radical quotient of the corner is the corner of the radical quotient, eAe ⧸ J(eAe) ≃ ē (A ⧸ J(A)) ē ≅ ∏ᵢ Dᵢ (IsIdempotentElem.cornerQuotientJacobsonEquiv), so eAe is basic;
• and a full idempotent makes A Morita equivalent to its corner (IsIdempotentElem.isMoritaEquivalent_corner).

In module-theoretic terms Ae is the direct sum of one indecomposable projective A-module for each simple module, and eAe is its endomorphism ring (up to taking the opposite ring): the classical basic algebra A^b of A. This is the normalisation step before presenting a finite-dimensional algebra over an algebraically closed field by a quiver with relations.

Main results

• TauCeti.IsSemiperfectRing.exists_isIdempotentElem_full_isBasic_corner: a semiperfect ring has a full idempotent whose corner ring is basic.
• TauCeti.IsSemiperfectRing.exists_isBasic_isMoritaEquivalent: a semiperfect algebra is Morita equivalent to a basic algebra.

References

• I. Assem, D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras I, CUP (2006), Section I.6.
• F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, 2nd ed., GTM 13, Springer (1992), §27 (basic rings).

theorem TauCeti.IsSemiperfectRing.exists_isIdempotentElem_full_isBasic_corner (A : Type u) [Ring A] [IsSemiperfectRing A] : ∃ (e : A) (he : IsIdempotentElem e), TwoSidedIdeal.span {e} = ⊤ ∧ IsBasic he.Corner

A semiperfect ring has a full idempotent with a basic corner. The idempotent lifts the matrix unit E₁₁ of every Wedderburn block of A ⧸ J(A); its corner eAe has radical quotient the product of the division rings of the blocks.

theorem TauCeti.IsSemiperfectRing.exists_isBasic_isMoritaEquivalent (R : Type u_1) [CommSemiring R] (A : Type u) [Ring A] [Algebra R A] [IsSemiperfectRing A] : ∃ (B : Type u) (x : Ring B) (x_1 : Algebra R B), IsBasic B ∧ IsMoritaEquivalent R A B

Morita reduction to a basic algebra. A semiperfect algebra, for instance a finite-dimensional algebra over a field (TauCeti.IsSemiperfectRing.of_finite), is Morita equivalent to a basic algebra: the corner ring of TauCeti.IsSemiperfectRing.exists_isIdempotentElem_full_isBasic_corner.