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I am trying to understand the mayer-vietoris sequence for DeRham Cohomology.

Assume $M=U \cup V$ and we want to show that the sequence $$ 0 \to \Omega(M) \to \Omega(U) \oplus \Omega(V) \to \Omega(U \cap V) \to 0$$ is a exact.

Here I am bit unsure why the preimage is defined like that. I am trying to show the surjectivity of the map $$\Omega(U) \oplus \Omega(V) \to \Omega(U \cap V)$$.

Assume $\theta \in \Omega(U \cap V)$ then Bott-Tu says the element $(-\rho_V \theta, \rho_U \theta)$ is preimage of $\theta$ but I want to understand why $(-\rho_U \theta, \rho_V \theta)$ wouldn't work. Here $\{\rho_U, \rho_V\}$ is a partition of unity subordinate to $\{U,V\}$. I understood the significance of the $-$ sign there but why it is defined like that?

permutation_matrix
  • 1,356
  • probably we defined it that way because in $(-\rho_U \theta, \rho_V \theta)$ the problem is that the multiplication isn't defined as we do not know anything about $\theta$ in $U \ V$ but the other way round we can write safely because the compact support of $\rho_V$ lies inside $V$ so $\rho_V$ is $0$ on $U \V$ and so we extended it smoothly and other conditions are also being satisfied. @MarianoSuárez-Álvarez – permutation_matrix Nov 24 '22 at 07:36
  • This is what you meant? – permutation_matrix Nov 24 '22 at 07:36
  • You should also fix your typos. You have $\Omega(M)$ in place of $\Omega(V)$. Make sure you understand that "preimage" is certainly not unique. You're trying to find some element of $\Omega(U)\oplus\Omega(V)$ that maps to $\theta$. – Ted Shifrin Nov 29 '22 at 17:43
  • Yes, its V instead of M. Alright. Preimage need not be unique – permutation_matrix Nov 29 '22 at 17:45

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