Given a variety $X$, a coherent sheaf $F$ which may not have finite locally free resolution, and a curve（integral） $C$ in $X$. How can we define intersection $F.C$ properly?(I mean 'properly' by having good behavior on short exact sequence and have projection formula)

Here is my attempt:

One can define $F.C$ as degree of $F|_C$ on $C$, suggest by Riemann-Roch, namely $\chi(F|C)-rank(F|C)\chi(\mathcal{O}_C)$. However, in this setting, projection formula only hold for locally free sheaf。

we take locally free resolution of $F$, and then pull them back（in fact we get $Li^\ast(F)$, where $i:C\rightarrow X$ is the inclusion） to $C$ and define intersection of $Li^\ast(F)$ with $C$. Where we have problem that $Li^\ast(F)$ may be unbounded.

So both attempt fail.