Faithfully flat descent of modules and algebras is effective

First consider a cover $g:T\to S$ with a section $s:S\to T$ and a module $N$ on $T$ with the given transport. Put $M=s^{\ast }N$. Pull back $\theta$ along $T\to T{\times }_{S}T$, $t\mapsto (s(g(t)),t)$. This gives an isomorphism $g^{\ast }M\to N$. Its compatibility with the original datum follows by pulling the cocycle identity back along $(s(g(t_1)),t_1,t_2)$ on $T{\times }_{S}T$. Pulling the cocycle back to the diagonal shows that diagonal transport is an invertible idempotent and hence the identity. Pulling back along $(t,s(g(t)),t)$ then shows that reverse transport is the inverse. These identities prove both effectiveness and that a map between descended objects is determined uniquely by its pullback: its descent-compatible upstairs map is recovered by pulling back along the section. This argument applies to affine modules, and to algebras when the transports preserve multiplication and unit.

Define $M$ to be the equalizer inside $N$ displayed in the Statement, considered as an $A$-module. Flat tensor product preserves this equalizer. For any flat base extension $A\to A^{\mathrm{\prime }}$, the invariant module for the base-changed datum is therefore $A^{\mathrm{\prime }}{\otimes }_{A}M$, since it is the kernel of the base-changed difference of the two transport maps. Take $A^{\mathrm{\prime }}=B$. The new covering ring is $B\to B{\otimes }_{A}B$ and has a retraction given by multiplication; equivalently its affine covering has a section. By step 1.1 the base-changed datum is effective. For an effective datum coming from a module $Q$, the invariant equalizer is $Q$: for a split cover, apply the section to an invariant element to recover its unique downstairs element. Thus the base change of $B{\otimes }_{A}M\to N$ is an isomorphism. Faithful flatness in [F1] proves that the original map is an isomorphism.

For completeness, the invariant equalizer of the canonical datum on $B{\otimes }_{A}Q$ is $Q$ also before this split base change. Indeed $Q\to B{\otimes }_{A}Q\rightrightarrows B{\otimes }_{A}B{\otimes }_{A}Q$ becomes a split exact equalizer after tensoring with $B$: multiplication supplies the section and step 1.1 applies. Its kernel and cokernel are detected by [F1]. Consequently a descent-compatible map $N\to N^{\mathrm{\prime }}$ takes invariants to invariants, and the isomorphisms of step 2.1 identify it uniquely with the base change of that restricted $A$-linear map. This proves full faithfulness as well as effectiveness.

If $N$ is an algebra, its invariant subset is closed under sums, products, scalar multiplication and the unit because $\theta$ is an algebra isomorphism. It is therefore an $A$-algebra. The map of step 2.1 is an algebra homomorphism and an isomorphism of modules, hence an algebra isomorphism. Restriction to invariants preserves algebra maps and step 3.1 proves their full faithfulness. This proves both equivalences and the displayed construction.