Endomorphism Algebras Over Commutative Rings and Torsion in Self Tensor Products

Abstract

Abstract Let R be a commutative Noetherian local ring. We study tensor products involving a finitely generated R -module M through the natural action of its endomorphism ring. In particular, we study torsion properties of self tensor products in the case where $${\text {End}}_R(M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>End</mml:mtext> <mml:mi>R</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> has an $$R^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebra structure, and prove that if M is indecomposable, then $$M \otimes _{{\text {End}}_R(M)} M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:msub> <mml:mo>⊗</mml:mo> <mml:mrow> <mml:msub> <mml:mtext>End</mml:mtext> <mml:mi>R</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:msub> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> must always have torsion in this case under mild hypotheses.