The purpose of the article under review is to describe an animation of the Dubson-Kashiwara formula [M. Kashiwara, Astérisque 130, 193–209 (1985; Zbl 0568.32017)]. Given any unital ring \(R\), a perfect complex \(P\) of \(R\)-modules defines a point \([P]\) in the \(K\)-theory spectrum \(K(R)\) associated to \(R\). The point \([P]\) is an animation of the Euler characteristic \(\xi(P)\) in the sense that its path component equals \(\xi(P)\in \pi_0 K(R)\). If \(R\) is commutative, \([P]\) maps to the determinant line det \(P\) in the Picard groupoid of \(R\).
Suppose now that \(F\) is a perfect complex of sheaves of \(R\)-modules on a compact real analytic manifold \(X\), with derived global sections \(R\Gamma(X,F)\). The author provides two animations of \(\xi(R\Gamma(X,F))\). The first animation is essentially given as follows. Let \(Y\subset X\) be a closed subset and \(\nu\) a continuous 1-form on \(X\smallsetminus Y\) taking values in the complement of the micro-support \(SS(F)\) of \(F\). Restricting \(F\) and \(\nu\) to any neighborhood of \(Y\) in \(X\) determines \( \varepsilon_{\nu Y}(F)\in K(R) \) having naturally the same path component as \([R\Gamma(X,F)]\). If \(R\) is commutative, this identification yields the so-called \(\varepsilon\)-factorization isomorphism of associated determinant lines.
The second animation is given by a natural cycle \(\varepsilon(F)_{SS(F)}\) with coefficients in \(K(R)\) and support on \(SS(F)\) whose intersection with the zero section of the cotangent bundle yields \([R\Gamma(X,F)]\) up to natural homotopy. If again \(R\) is commutative, the Dubson-Kashiwara formula for det \(R\Gamma(X,F)\) follows.