I am reading Hartshorne. I can basically understand gluing sheaves like this. But, I still get confused about gluing sheaves of modules on schemes.
Suppose I have a scheme $X$ obtained by gluing $\mathrm{Spec}(A_i)$ for a family of commutative rings $A_i$. Specifically, by Exercise 2.12 in Hartshorne, we may assume there are ring isomorphisms $$ g_{ij}: A_j\to A_i $$ such that $g_{ik}=g_{ij}\circ g_{jk}$, and use this data to interpret $X$. (More precisely, we should think $\mathrm{Spec}(A_ij) \subseteq \mathrm{Spec}(A_i)$, but let's abuse the notations. Hope this does not cause serious issues).
According to Proposition 5.1 in Hartshorne, a local sheaf $\mathscr F_i$ of modules on $\mathrm{Spec}(A_i)$ is equivalent to an $A_i$-module $M_i$ by setting $\mathscr F_i=\tilde M_i$.
Now, we aim to glue these local sheaves. Let's get back to Exercise 1.22 for gluing sheaves. What is the condition for gluing these sheaves $\mathscr F_i=\tilde M_i$ on $\mathrm{Spec}(A_i)$?
By Proposition 5.2, I was wondering if we should consider homomorphisms of modules $$ G_{ij}: M_i\to A_i\otimes_{A_j}M_j \quad \text{or}\quad A_j\otimes_{A_i} M_i\to M_j $$ where for the tensor product, we need to view $A_j$ as an $A_i$-module via $g_{ij}:A_i\to A_j$. However, if so, the cocycle condition in Exercise 1.22 makes me confused; we cannot simply set $G_{ij}\circ G_{jk}=G_{ik}$ since $M_i$ are modules over different $A_i$'s.
Question: Is there a purely algebraic way to understand the cocycle condition for gluing sheaves of modules? Here "purely algebraic" means that we only use rings, modules, morphisms among them, etc.
These should be maps of modules over the same sheaf of rings. There is nothing special about the case of schemes here so let's work in the general case that $(X, \mathcal{O}_X)$ is a ringed space.
Let $\{U_i\}$ be an open cover of $X$ and $\mathscr{F}_i$ a collection of $\mathcal{O}_{U_i}$-modules. To glue these we need to give isomorphisms of $\mathcal{O}_{U_i \cap U_j}$ modules $\psi_{ij}: \mathscr{F}_i|_{U_i \cap U_j} \to \mathscr{F}_j|_{U_i \cap U_j}$ which needs to satisfy the cocycle condition. Let $i, j, k$ be indices and denote $U_{ijk} = U_i \cap U_j \cap U_j$. Then this condition requires that the composition $$\mathscr{F}_i|_{U_{ijk}} \stackrel{\psi_{ij}|_{U_{ijk}}}\longrightarrow \mathscr{F}_j|_{U_{ijk}} \stackrel{\psi_{jk}|_{U_{ijk}}}\longrightarrow \mathscr{F}_k|_{U_{ijk}} $$ agrees with $\psi_{ik}|_{U_{ijk}}$. We see here that these are all modules over the same sheaf of rings (namely, $\mathcal{O}_{U_{ijk}}$), so nothing funny is happening.
EDIT: Let's look at the situation that $U_i = U_{ii} = \operatorname{Spec} A_i$ are affine schemes, $U_{ij} \subset \operatorname{Spec} A_i$ are open subschemes together with isomorphisms $g_{ij}: U_{ij} \to U_{ji}$ which satisfy the requirements of exercise II.2.12 to glue these $U_i$ to a scheme $X$, together with open embeddings $j_i: U_i \hookrightarrow X$.
Now, suppose are given $\mathscr{F}_i$ on each $U_i$. We may identify $U_i$ with $j_i(U_i)$ and similarly $\mathscr{F}_i$ with ${j_i}_* \mathscr{F}_i$. Under these identifications, the gluing data for a quasicohernet sheaf of modules is the same as above; we should give sheaves of modules on each $U_i$ with isomorphisms on the overlaps satisfying the cocylce condition.
We can untangle some of these indentifications to perhaps see what's going on better. Without these identifications, $\mathscr{F}_i|_{U_{ij}}$ and $\mathscr{F}_j|_{U_{ji}}$ are not sheaves on the samed ringed space so as you said, we would need to specify isomorphisms of $\mathcal{O}_{U_{ji}}$-modules $\psi_{ji}: {g_{ij}}_*(\mathscr{F}_i|_{U_{ij}}) \to \mathscr{F}_j|_{U_{ji}}$ satisfying the cocycle condition.
This is a bit strange because we would also want $\psi_{ij}$ to be the inverse of $\psi_{ji}$ somehow. We can can do this by defining/insisting that $\psi_{ij} = {g_{ji}}_*(\psi_{ji}^{-1})$.
As for the cocycle condition, we can insist that for $i, j, k$, on $U_{ki} \cap U_{kj} \subset \operatorname{Spec} A_k$ the composition $${g_{ik}}_*(\mathscr{F}_i|_{U_{ij} \cap U_{ik}}) \stackrel{{g_{jk}}_*\psi_{ji}}\longrightarrow {g_{jk}}_*(\mathscr{F}_j|_{U_{ji} \cap U_{jk}}) \stackrel{\psi_{kj}}\longrightarrow \mathscr{F}_k|_{U_{ki} \cap U_{kj}}$$ agrees with $\psi_{ki}|_{U_{ki} \cap U_{kj}}$ as maps of $\mathcal{O}_{U_{ki} \cap U_{kj}}$ modules. Note that if you push forward by $j_k$ you obtain the same cocycle condition described above. (also, to suppress notation I ommitted some of the restrictions on the maps in the above sequence)