On the number of generators of an algebra over a commutative ring
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Abstract
A theorem of O. Forster says that if <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> is a noetherian ring of Krull dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then every projective <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 of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be generated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d plus n"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">d+n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> elements. S. Chase and R. Swan subsequently showed that this bound is sharp: there exist examples that cannot be generated by fewer than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d plus n"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">d+n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> elements. We view rank- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> projective <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 as <italic> <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> -forms </italic> of the non-unital <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> -algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">R^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where the product of any two elements is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The first two authors generalized Forster’s theorem to forms of other algebras (not necessarily commutative, associative or unital); A. Shukla and the third author then showed that this generalized Forster bound is optimal for finite étale algebras. In this paper, we prove new upper and lower bounds on the number of generators of an <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> -form of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an infinite field and <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> is a finitely generated <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -ring of Krull dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In particular, we show that, contrary to expectations, for most types of algeb
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