A relative version of Bass’ theorem about finite-dimensional algebras

As a special case of Bass’ theory of perfect rings, one obtains the assertion that, over a finite-dimensional associative algebra over a field, all flat modules are projective. In this paper we prove the following relative version of this result. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R long right-arrow upper A"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo stretchy="false"> ⟶ </mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">R\longrightarrow A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a homomorphism of associative rings such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finitely generated projective right <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -module. Then every flat left <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -module is a direct summand of an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -module filtered by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -modules <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A circled-times Subscript upper R Baseline upper F"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:msub> <mml:mo> ⊗ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>R</mml:mi> </mml:mrow> </mml:msub> <mml:mi>F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A\otimes _{R}F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> induced from flat left <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -modules <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In other words, a left <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -module is cotorsion if and only if its underlying left <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -module is cotorsion. The proof is based on the cotorsion periodicity theorem.