In “Sur la cohomologie de certains espaces de modules de fibrés vectoriels”, Beauville calculates the Chern class of the diagonal $\Delta$ of the moduli space $M$ of certain stable bundles on a curve $C$. For that, he uses that $\Delta$ can be identified with the degeneracy locus of a map $u: K^0\to K^1$ of vector bundles on $M \times M$ that is of expected codimension. He then uses Porteus’ formula to conclude that $$ [\Delta] = c_m(K^1 - K^0) = c_m(-\pi_! H) $$ holds. Here, $m$ is the dimension of the moduli space considered and H is a vector bundle on $C \times M \times M$ such that for the projection $\pi: C \times M \times M \to M \times M$ the relation $$\mathbf{R}\pi_{\ast}H \simeq \left[ 0\to K^0 \to K^1 \to 0\right]$$ holds. This is not the form of the Porteus formula I’m familiar with - it describes a different Chern class than e.g. in Eisenbud-Harris’ “3264”-book. Also, I’m not super sure how to interpret the Chern class of an element of the Grothendieck group in this context. Is he omitting the Chern character? And if so, how does this relate to the “usual” Porteus formula (I don’t see why these two expression should be the same, but maybe I’m just missing some obvious relations between Chern characters)?
Sorry if this a very basic question or if I’m unaware of something simple!
