Definition (Space of differentials). The space of differentials ${\mathrm{\Omega }}_C$ is the $K(C)$-vector space generated by $df$ for $f\in K(C)$, subject to the relations

• (i) $d(f+g)=df+dg$
• (ii) $d(fg)=fdg+gdf$
• (iii) $da=0 \forall a\in K$