I am reading the book GTM 81 `Lectures on Riemann Surfaces' written by Otto Forster. I am confused about the definition of cochains and cocycles, when I read Section $\textbf{14.8}$ on page 115:
`Let $(X,\tau)$ be a topology space, $Y\in\tau(X)$, and $\mathscr{F}$ be a sheaf of abelian group on $X$. For every open covering $\mathcal{U}=(U_i)_{i\in I}$ of $X$, $\mathcal{U}\cap Y:=(U_i\cap Y)_{i\in I}$ is an open covering of $Y$. The natural restriction mapping $\phi:Z^1(\mathcal{U},\mathscr{F})\rightarrow Z^1(\mathcal{U}\cap Y,\mathscr{F})$ induces a homomorphism $H^1(\mathcal{U},\mathscr{F})\rightarrow H^1(\mathcal{U}\cap Y,\mathscr{F})$...' $\qquad\qquad(0)$
$\textbf{My question is how to define $\phi:Z^1(\mathcal{U},\mathscr{F})\rightarrow Z^1(\mathcal{U}\cap Y,\mathscr{F})$.}$
For each $f\in Z^1(\mathcal{U},\mathscr{F})$, I want to define $\phi(f)$ by \begin{equation}[\phi(f)]_{ij}:=f_{ij}|_{U_i\cap U_j\cap Y}.\qquad(1)\end{equation}
However, for fixed $i,j$, if $Y=U_i\cap U_j$, then $U_i\cap Y=Y=U_j\cap Y$ and by (1) we know that $$[\phi(f)]_{ij}\neq[\phi(f)]_{ji}\qquad\qquad (2)$$ can be hold sometimes: For example, $U_i,U_j,U_k\in\tau(\mathbb{R})$, $X=U_i\cup U_j\cup U_k$, $\mathcal{U}=\{U_i,U_j,U_k\}$, $\mathscr{F}$ be the sheaf of smooth functions over $X$, and $f\in B^1(\mathcal{U},\mathscr{F})$. Let $g\in C^0(\mathcal{U},\mathscr{F})$ with $g_i\neq g_j$ on $Y$, and $f:=\delta g$. Then $f_{ij}=g_i-g_j\neq g_j-g_i=f_{ji}$ on $Y$.
Since the definition of cochains in GTM81 is not clear enough for me (on page 96): $$C^1(\mathcal{U},\mathscr{F}):=\prod_{(i_0,...,i_q)\in I^{q+1}}\mathscr{F}(U_{i_0}\cap\cdots\cap U_{i_q}),$$ I find another definition of cochains from `'Analytic Function Theory of Several Variables' written by Junjiro Noguchi (on page 74):
`Denote $$N_q(\mathcal{U}):=\{\sigma=(U_0,...,U_q):U_i\in\mathcal{U}\}.$$ A map $$f:\in \sigma\in N_q(\mathcal{U})\rightarrow f(\sigma)\in\Gamma(|\sigma|,\mathscr{F})$$ satisfying the alternating property $$f(U_0,...,U_i,....,U_j,...,U_q)=-f(U_0,...,U_j,....,U_i,...,U_q)$$ is called an $q$-cochain.'
In the sense of Noguchi's definition, if $U_i=U_j=:U$ then for each $f\in C^0(\mathcal{U},\mathscr{F})$, we have $$f_{ij}=f(U_i,U_j)=f(U,U)=f(U_j,U_i)=f_{ji}.\qquad (In\ fact\ by\ alternative\ property,\ f_{ij}=0.)$$ It is a contradiction to (2).
I would like to know：
a). Is the definition (1) wrong? What is the corrected one?
b). Do the cochains $f_{ij}$ depend on the index $i\in I$, not $U_i$, in GTM81?
c). Is there a book for cochains with a detailed definition?
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(Not sure) I think may be two book define $H^1(\mathcal{U},\mathscr{F})\rightarrow H^1(\mathcal{U}\cap Y,\mathscr{F})$ by different ways. I just see Lemma 3.4.7 from Noguchi's book:
Assume that $\mathcal{V}$ is a refinement of $\mathcal{U}$ by two maps $\mu:\mathcal{V}\rightarrow\mathcal{U}$ and $v:\mathcal{V}\rightarrow\mathcal{U}$. Then $$\mu^*=v^*:H^q(\mathcal{U},\mathscr{F})\rightarrow H^q(\mathcal{V},\mathscr{F}).$$ Here $\mu :\mathcal{V}\rightarrow\mathcal{U}$ is a map satisfies $V\subset\mu (V)$, for each $V\in\mathcal{V}$ and $$\mu^* :f\in C^q(\mathcal{U},\mathscr{F})\rightarrow C^q(\mathcal{V},\mathscr{F})$$ is defined by $$\mu^*(f)(V_0,...,V_q)=f(\mu(V_0,...,V_q))|_{V_0\cap...\cap V_q}.$$
In Noguchi's sense, $C^1(\mathcal{U},\mathscr{F})\rightarrow C^1(\mathcal{U}\cap Y,\mathscr{F})$ is depended on the choice of the map $\mu$ of refinements. However, $H^1(\mathcal{U},\mathscr{F})\rightarrow H^1(\mathcal{U}\cap Y,\mathscr{F})$ is not.
I do not sure what is Forster's method.
I think maybe the word `natural' in $(0)$ confused me. The restriction map $\phi$ may be induced by one of the refining maps $\mu:\mathcal{U}\cap Y\rightarrow\mathcal{U}$. Then the methods of the two books coincide.