Proof that direct image of quasi-coherent module is quasi-coherent

Question

I was looking at the following proof (source) but I'm having trouble understanding why we have that $\mathcal{F}(f^{-1}V)$ is equal to the kernel of the map. Can someone please explain this to me?

Answer 1

Recall that if we have a topological space $X$ equipped with a sheaf of abelian groups $\mathcal{F}$, then for any open set $U\subset X$ with an open covering $U=\bigcup_{i\in I} U_i$, then we have an exact sequence $$0\to \mathcal{F}(U)\to \prod_{i\in I} \mathcal{F}(U_i) \to \prod_{i,j \in I, i\neq j} \mathcal{F}(U_i\cap U_j) $$ where the first map is the restriction of sections, and the second map sends a section $s$ over $U_i$ to $res_{U_i,U_i\cap U_j}(s)$ if $i<j$ and $-res_{U_i,U_i\cap U_j}(s)$ if $j<i$ (you might complain about the ordering here, but all choices of an order produce the same kernel, and there's always at least one choice of an ordering).

This gives us the first part of the sequence of maps. We write $f^{-1} (V) = \bigcup_i (f^{-1}(V)\cap U_i)$, and so we have an exact sequence $$0\to \mathcal{F}(f^{-1} (V))\to \bigoplus_i \mathcal{F}(f^{-1}(V)\cap U_i) \to \bigoplus_{i\neq j} \mathcal{F}(f^{-1}(V)\cap U_i\cap U_j) \qquad (1)$$ where we have used the fact that the open covering is finite and finite direct products are the same as finite direct sums to change from products to sums.

Next, for each intersection $U_i\cap U_j$, we have covered it by finitely many open affines $U_{ijk}$ and therefore $f^{-1}(V)\cap U_i\cap U_j$ is covered by $f^{-1}(V)\cap U_{ijk}$. Applying the first step of the main exact sequence to each $V\cap U_i\cap U_j$ with the open cover $f^{-1}(V)\cap U_{ijk}$, we get an injective map from each $\mathcal{F}(f^{-1}(V)\cap U_i\cap U_j)$ to $\bigoplus_k\mathcal{F}(f^{-1}(V)\cap U_{ijk})$ and thus a map $$\bigoplus_{i\neq j}\mathcal{F}(f^{-1}(V)\cap U_i\cap U_j)\to\bigoplus_{i,j,k} \mathcal{F}(f^{-1}(V)\cap U_{ijk}) \qquad (2)$$

where we have again used finiteness of the cover to replace products with sums. We can compose with our exact sequence (1) to get $$0\to \mathcal{F}(f^{-1} (V))\to \bigoplus_i \mathcal{F}(f^{-1}(V)\cap U_i) \to \bigoplus_{i,j,k} \mathcal{F}(f^{-1}(V)\cap U_{ijk})$$ without changing exactness since (2) was injective. The above sequence is compatible with restrictions, so it patches to the desired exact sequence of sheaves at the end of the image you quote.