Integral transforms and Drinfeld centers in derived algebraic geometry

2010/04/01 by Dani Ben‐Zvi, John Francis, David Nadler · 34 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Hochschild homology #Mathematics #Functor #Pure mathematics #Base change #Tensor product #Cohomology #Derived algebraic geometry #Coherent sheaf #Stack (abstract data type) #Derived category #Product (mathematics) #Algebra over a field #Geometry #Mathematical analysis #Computer science

paper · pdf · doi:10.1090/s0894-0347-10-00669-7

published in Journal of the American Mathematical Society 23(4), 909-966 (American Mathematical Society)

openalex publication_date 2010/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

We study the interaction between geometric operations on stacks and algebraic operations on their categories of sheaves. We work in the general setting of derived algebraic geometry: our basic objects are derived stacks <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and their <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:annotation encoding="application/x-tex">∞</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -categories <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Q normal upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Q</mml:mi> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm QC(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of quasi-coherent sheaves. (When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a familiar scheme or stack, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Q normal upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Q</mml:mi> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm QC(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an enriched version of the usual quasi-coherent derived category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript q c Baseline left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">D_qc(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .) We show that for a broad class of derived stacks, called perfect stacks, algebraic and geometric operations on their categories of sheaves are compatible. We identify the category of sheaves on a fiber product with the tensor product of the categories of sheaves on the factors. We also identify the category of sheaves on a fiber product with functors between the categories of sheaves on the factors (thus realizing functors as integral transforms, generalizing a theorem of Toën for ordinary schemes). As a first application, for a perfect stack <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , consider <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Q normal upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Q</mml:mi> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm QC(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with its usual monoidal tensor product. Then our main results imply the equivalence of the Drinfeld center (or Hochschild cohomology category) of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Q normal upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Q</mml:mi> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm QC(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the trace (or Hochschild homology category) of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Q normal upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Q</mml:mi> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm QC(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the category of sheaves on the loop space of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . More generally, we show that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper E Subscript n"> <mml:semantics>

Citations

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