Mayer vietoris sequence in de rham cohomology

Given: An open cover $M=U\cup V$ and countable choice.

Short exact mayer vietoris sequence of de rham complexes: Under countable choice, $0\to {\mathrm{\Omega }}^{\bullet }(M)\stackrel{r}{\to }{\mathrm{\Omega }}^{\bullet }(U)\oplus {\mathrm{\Omega }}^{\bullet }(V)\stackrel{s}{\to }{\mathrm{\Omega }}^{\bullet }(U\cap V)\to 0$ is short exact as a sequence of real cochain complexes.

De rham cohomology: The real de Rham cohomology is $H_{\mathrm{d}\mathrm{R}}^k(M)=Z^k(M)\mathrm{/}{B}^k(M)$, with $Z^k,B^k$ as in def-closed-and-exact-differential-forms. This is def-cohomology-object-of-a-cochain-complex in real vector spaces. Only a closed form $\omega$ represents a class $[\omega ]$. For closed forms $\omega ,\omega \mathrm{\prime }$, equality $[\omega ]=[\omega \mathrm{\prime }]$ means precisely $\omega \mathrm{\prime }-\omega =d\eta$ for some $(k-1)$-form $\eta$. Addition and real scalar multiplication are induced by those of forms. All groups on the empty manifold are zero.

The long exact sequence in cohomology: Let $0\to A^{\bullet }\to B^{\bullet }\to C^{\bullet }\to 0$ be a short exact sequence of cochain complexes in an abelian category. Then there is a natural exact sequence $\cdots \to H^n(A)\to H^n(B)\to H^n(C)\stackrel{{\mathrm{\partial }}^n}{\to }{H}^{n+1}(A)\to H^{n+1}(B)\to H^{n+1}(C)\to \cdots \mathrm{.}$