Parahoric Hecke Ext-algebras in characteristic 𝑝

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a nonarchimedean local field of residual characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the group of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -points of a connected reductive group over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For an open compact subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper U"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">U</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {U}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a unital commutative ring <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> , we let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper X Subscript script upper U"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">X</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">U</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbf {X}_{\mathscr {U}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the space of compactly supported <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> -valued functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash script upper U"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">U</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">G/\mathscr {U}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Building on work of Ollivier–Schneider, we investigate the graded <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"> <mml:semantics> <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:annotation encoding="application/x-tex">\mathrm {Ext}</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 E Subscript script upper U Superscript asterisk Baseline colon-equal normal upper E normal x normal t Subscript upper G Superscript asterisk Baseline left-parenthesis bold upper X Subscript script upper U Baseline comma bold upper X Subscript script upper U Baseline right-parenthesis Superscript normal o normal p"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">U</mml:mi> </mml:mrow> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msubsup> <mml:mo>≔</mml:mo> <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>G</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">X</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX