ON THE MAP INDUCED ON HOCHSCHILD HOMOLOGY OF MATRIX FACTORIZATION CATEGORIES BY THE INCLUSION OF A DIVISOR

Abstract Given a smooth variety X over <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:math> $\mathbb {C}$ double struck upper C , a smooth divisor <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>i</mml:mi> <mml:mo>:</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">↪</mml:mo> <mml:mi>X</mml:mi> </mml:math> $i:Y\hookrightarrow X$ i colon upper Y right arrow with hook upper X and a global function f on X which vanishes on Y and on its critical locus, we compute the map induced on Hochschild homology by the pushforward functor <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mi>i</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msub> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>b</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false" accent="false">→</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> $i_{\ast }:D^b(Y)\to D^{abs}(MF(X,f))$ i Subscript asterisk Baseline colon upper D Superscript b Baseline left parenthesis upper Y right parenthesis right arrow upper D Superscript a b s Baseline left parenthesis upper M upper F left parenthesis upper X comma f right parenthesis right parenthesis in terms of the Hochschild-Kostant-Rosenberg isomorphisms.