Let \(\varphi \colon A \to B\) be a ring map, and write \(B^{\otimes n} = B \otimes_A \cdots \otimes_A B\) with \(n\) factors.
(Amitsur complex)
For an \(A\)-module \(M\), the Amitsur complex of \(M\) along \(\varphi\) is \[0 \to M \xrightarrow{d^0} M \otimes_A B \xrightarrow{d^1} M \otimes_A B^{\otimes 2} \xrightarrow{d^2} M \otimes_A B^{\otimes 3} \to \cdots,\] where \(d^0(m) = m \otimes 1\) and, for \(n \geq 1\), \(d^n = \sum_{i=0}^{n} (-1)^i e_i\) with \[e_i(m \otimes b_1 \otimes \cdots \otimes b_n) = m \otimes b_1 \otimes \cdots \otimes b_i \otimes 1 \otimes b_{i+1} \otimes \cdots \otimes b_n .\]
(Exactness of the Amitsur complex)
If \(\varphi\) is faithfully flat, the Amitsur complex of every \(A\)-module \(M\) is exact. In particular the sequence \(0 \to M \to M \otimes_A B \rightrightarrows M \otimes_A B \otimes_A B\), with the two maps \(m \otimes b \mapsto m \otimes b \otimes 1\) and \(m \otimes b \mapsto m \otimes 1 \otimes b\), is an equalizer.
(Descent data)
A descent datum for \(A \to B\) is a \(B\)-module \(N\) with an isomorphism of \(B \otimes_A B\)-modules \(\theta \colon B \otimes_A N \to N \otimes_A B\) satisfying the cocycle condition \(\theta_{13} = \theta_{23} \circ \theta_{12}\) as maps \(B \otimes_A B \otimes_A N \to N \otimes_A B \otimes_A B\), where \(\theta_{ij}\) is \(\theta\) acting on the factors in positions \(i\) and \(j\). For an \(A\)-module \(M\), the module \(M \otimes_A B\) has the canonical descent datum \(b \otimes (m \otimes b') \mapsto (m \otimes b) \otimes b'\).
(Faithfully flat descent)
If \(A \to B\) is faithfully flat, then \(M \mapsto (M \otimes_A B, \text{canonical } \theta)\) is an equivalence from \(A\)-modules to \(B\)-modules with descent data. A quasi-inverse sends \((N, \theta)\) to \(\{ n \in N : \theta(1 \otimes n) = n \otimes 1 \}\).
Replacing modules by quasicoherent sheaves and \(\Spec B \to \Spec A\) by a faithfully flat quasicompact morphism \(S' \to S\) gives descent of quasicoherent sheaves; applied to \(\Hom\) it shows that morphisms of schemes descend, which is the statement that \(h_X\) is an fppf sheaf in D-ETFPPF.
Dependencies
D-DEFFFLATFaithfully flat modules, and descent of exactnessuses