I am trying to read through the Cech Cohomology section in these notes
I would be really grateful if someone could explain to me what $f_*$ means. Sorry in advance if this is just something standard.
Extract:
Proposition 2. $\mathrm{H}^{0}(\check{\mathrm{c}}(\mathcal{U}, \mathcal{F}))=\mathcal{F}(X)$
Proof. Exercise: this is just a reformulation of the sheaf property. There is a sheaf-theoretic version of the Cech complex. For any open $V \subset X$ denote by $f: V \rightarrow X$ the inclusion. Then define $$ \mathcal{C}^{p}(\mathcal{U}, \mathcal{F})=\prod_{v \in I^{p+1}} f_{*} \mathcal{F}_{\mid U_{v}} $$ These form a complex $\mathcal{C}(\mathcal{U}, \mathcal{F})$ of sheaves on $X$ with the property that $\Gamma(X, \mathcal{C})=\tilde{\mathbb{C}}$. Note that there is a natural map $\mathcal{F} \rightarrow \mathcal{C}^{0}(\mathcal{U}, \mathcal{F})$ given by sending a section $s$ to its restrictions on all the components.
If $f : X \to Y$ is a continuous map and $\mathscr{F}$ is a sheaf on $X$, then $f_* \mathscr{F}$ is the sheaf on $Y$ defined by the following formula: $$f_* \mathscr{F} (V) = \mathscr{F} (f^{-1} V)$$ This is called the direct image sheaf. You should verify that the formula indeed defines a sheaf.