Gorenstein derived functors

<p>Over any associative ring <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> it is standard to derive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper H normal o normal m Subscript upper R Baseline left-parenthesis minus comma minus right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">H</mml:mi> <mml:mi mathvariant="normal">o</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mi>R</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−</mml:mo> <mml:mo>,</mml:mo> <mml:mo>−</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {Hom}_R(-,-)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> using projective resolutions in the first variable, or injective resolutions in the second variable, and doing this, one obtains <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper E normal x normal t Subscript upper R Superscript n Baseline left-parenthesis minus comma minus right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">E</mml:mi> <mml:mi mathvariant="normal">x</mml:mi> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−</mml:mo> <mml:mo>,</mml:mo> <mml:mo>−</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {Ext}_R^n(-,-)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in both cases. We examine the situation where projective and injective modules are replaced by Gorenstein projective and Gorenstein injective ones, respectively. Furthermore, we derive the tensor product <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus circled-times minus"> <mml:semantics> <mml:mrow> <mml:mo>−</mml:mo> <mml:msub> <mml:mo>⊗</mml:mo> <mml:mi>R</mml:mi> </mml:msub> <mml:mo>−</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">-\otimes _R-</mml:annotation> </mml:semantics> </mml:math> </inline-formula> using Gorenstein flat modules.</p>