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Localized Chern class
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<p>In algebraic geometry, a localized Chern class is a variant of a <a href="/facts/Chern_class/Tw2WD5hM">Chern class</a>, that is defined for a chain complex of vector bundles as opposed to a single vector bundle. It was originally introduced in Fulton's <i>intersection theory</i>, as an algebraic counterpart of the similar construction in <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>. The notion is used in particular in the <a href="/facts/Riemann%E2%80%93Roch-type_theorem/J6nEhlcE">Riemann–Roch-type theorem</a>. </p><p>S. Bloch later generalized the notion in the context of <a href="/facts/Arithmetic_scheme/JwpLlcMw">arithmetic schemes</a> (schemes over a Dedekind domain) for the purpose of giving #Bloch's conductor formula that computes the non-constancy of Euler characteristic of a <a href="/facts/Degeneration_(algebraic_geometry)/B0NJbDLR">degenerating family</a> of algebraic varieties (in the mixed characteristic case). </p>