Faithfully Flat Descent

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

One of the most common constructions in algebraic geometry is gluing local objects into a single object, the most familiar example being gluing sections of a sheaf along an open cover. This essentially uses only a special kind of base change given by localization, but many situations we actually want to deal with are more general than this. For example, \(\Spec \mathbb{L}\rightarrow \Spec \mathbb{K}\) corresponding to a field extension \(\mathbb{L}/\mathbb{K}\) cannot be viewed as an open embedding since the residue fields differ.

The idea for resolving this is to treat not only open embeddings but also faithfully flat and quasi-compact morphisms as valid coverings. We first examine the algebraic reasons why these can be treated as valid coverings, and then carry this over to geometry.

Faithfully Flat Morphisms

As the name suggests, faithful flatness is a condition that adds a kind of surjectivity to flatness.

Definition 1 A ring homomorphism \(\phi: A \rightarrow B\) is faithfully flat if \(B\) is a flat \(A\)-module and, at the same time, for every \(A\)-module \(M\), \(M\otimes_A B=0\) implies \(M=0\).

In other words, the additional faithfulness attached to flatness is that \(-\otimes_A B\) does not send a nonzero module to \(0\). This has the following equivalent conditions.

Proposition 2 For a flat ring homomorphism \(\phi: A \rightarrow B\), the following are equivalent.

1. \(\phi\) is faithfully flat.
2. A sequence \(M' \rightarrow M \rightarrow M''\) of \(A\)-modules is exact if and only if the sequence \(M'\otimes_A B \rightarrow M\otimes_A B \rightarrow M''\otimes_A B\) of \(B\)-modules is exact.
3. The scheme morphism \(\varphi: \Spec B \rightarrow \Spec A\) corresponding to \(\phi\) is surjective.

Proof

We first show that the first and second conditions are equivalent. To this end, assume the first condition. Then since \(B\) is flat by assumption, an exact sequence remains exact after applying \(-\otimes_A B\), so one direction of the second condition is trivial. The key is the converse; to show it, suppose

\[M'\otimes_AB \overset{f\otimes_AB}{\longrightarrow} M\otimes_AB \overset{g\otimes_AB}{\longrightarrow} M''\otimes_AB\]

is an exact sequence. Then first,

\[0=(g\otimes B)\circ(f\otimes B)=(g\circ f)\otimes B,\]

so the image of

\[\im(g\circ f)\otimes_A B \rightarrow M''\otimes_A B\]

coming from the inclusion is also \(0\). Viewing this as coming from \(\im(g\circ f)\hookrightarrow M''\), since \(B\) is flat the above morphism is also injective, and hence by faithfulness \(\im(g\circ f)=0\) and \(\im f\subseteq \ker g\) holds. Now, to check exactness, it suffices to show that \(H=\ker g/\im f\) is \(0\), which is trivial since \(B\) is flat. Then by the same argument as before, \(H=0\) by faithfulness, and therefore the original sequence is exact.

Conversely, assume the second condition and let us show the first. If \(M\otimes_A B=0\), then the sequence \(0 \rightarrow M \rightarrow 0\) is exact after applying \(-\otimes_A B\), so by assumption \(0 \rightarrow M \rightarrow 0\) is exact. That is, \(M=0\) and \(\phi\) is faithfully flat.

We now show the equivalence of the first and third conditions. First assume the first condition and choose an arbitrary \(\mathfrak{p}\in \Spec A\). Then \(\mathfrak{p}\) belongs to the image of \(\varphi\) if and only if its fiber \(\Spec(B\otimes_A \kappa(\mathfrak{p}))\) is nonempty, that is, \(B\otimes_A \kappa(\mathfrak{p})\neq 0\). But by faithful flatness, if \(\kappa(\mathfrak{p})\neq 0\) then its base change \(B\otimes_A \kappa(\mathfrak{p})\) is also nonzero, so this holds.

Finally, assume the third condition and let us show the first. For this, it suffices to show that \(M\otimes_A B\neq 0\) for every \(A\)-module \(M\) with \(M\neq 0\). Choosing \(0\neq x\in M\), we have \(Ax\cong A/{\ann(x)}\), which is a submodule of \(M\). Now choose a maximal ideal \(\mathfrak{m}\) with \(\ann(x)\subseteq \mathfrak{m}\). Then by assumption this belongs to the image of \(\varphi\), so \(\kappa(\mathfrak{m})\otimes_A B\neq 0\). Meanwhile, applying \(-\otimes_A B\) to the surjection \(A/{\ann(x)}\twoheadrightarrow A/\mathfrak{m}=\kappa(\mathfrak{m})\) yields a surjection

\[(A/{\ann(x)})\otimes_A B\twoheadrightarrow \kappa(\mathfrak{m})\otimes_A B,\]

so \((A/{\ann(x)})\otimes_A B\neq 0\). Also, since \(B\) is flat, applying \(-\otimes_A B\) to the inclusion \(A/{\ann(x)}\cong Ax\hookrightarrow M\) yields an inclusion

\[(A/{\ann(x)})\otimes_A B\hookrightarrow M\otimes_A B.\]

Therefore \(M\otimes_A B\neq 0\), and \(B\) is a faithful \(A\)-module.

What is particularly noteworthy in this proposition is the second condition, which shows that exactness may be checked after base change. The third condition shows that this algebraically defined property corresponds exactly to the surjectivity of the morphism \(\Spec B \rightarrow \Spec A\), and therefore a faithfully flat ring homomorphism can be thought of as an affine faithfully flat morphism in the sense of §Flat Morphisms, ⁋Definition 1.

A special example is \(A \rightarrow \prod_i A_{f_i}\), obtained when elements \(f_1,\ldots, f_n\) of \(A\) generate all of \(A\): since each \(A_{f_i}\) is flat, their product is also flat, and since \(\Spec \prod A_{f_i}=\coprod D(f_i)\) covers \(\Spec A\), it is surjective. This is exactly a Zariski cover of an affine scheme, showing that the above notion can cover the everyday gluing of sheaves. Moreover, in the case of a field extension \(\mathbb{L}/\mathbb{K}\), which could not be explained by open embeddings above, \(\mathbb{L}\) is (of course) free as a \(\mathbb{K}\)-vector space, and the geometric morphism between them is a surjective morphism sending a point to a point, so it is faithfully flat.

The Amitsur Complex

Given a ring homomorphism \(\phi: A \rightarrow B\), we are given two morphisms

\[d^0, d^1: B \rightrightarrows B\otimes_A B,\qquad d^0(b)=b\otimes 1,\quad d^1(b)=1\otimes b.\]

These are two homomorphisms reflecting the two ways of inserting \(B\) into \(B\otimes_AB\), and by considering their difference \(d=d^1-d^0\) we obtain a morphism \(d: B\rightarrow B\otimes_AB\). Moreover, since an element coming from \(A\) belongs to the kernel of \(d\) by the properties of the tensor product, we can consider the sequence

\[0\rightarrow A \overset{\phi}{\longrightarrow}B\overset{d}{\longrightarrow}B\otimes_AB.\]

More generally, write \(C^n=B^{\otimes (n+1)}\) for the \((n+1)\)-th tensor power of \(B\), and consider the morphisms from \(C^n\) to \(C^{n+1}\)

\[\delta_i: C^n \rightarrow C^{n+1};\qquad \delta_i(b_0\otimes \cdots \otimes b_n)=b_0\otimes \cdots \otimes b_{i-1}\otimes 1\otimes b_i\otimes \cdots \otimes b_n.\]

That is, this is the morphism inserting \(1\) in the \(i\)-th slot, and their alternating sum

\[\partial^n=\sum_{i=0}^{n+1}(-1)^i\delta_i: C^n \rightarrow C^{n+1}\]

is a morphism from \(C^n\) to \(C^{n+1}\). A small computation confirms that this forms a complex, and the

\[0 \rightarrow A \overset{\phi}{\longrightarrow} B \overset{\partial^0}{\longrightarrow} B\otimes_A B \overset{\partial^1}{\longrightarrow} B\otimes_A B\otimes_A B \rightarrow \cdots\]

obtained by prepending \(\phi\) as above is called the Amitsur complex of \(\phi\).

The key observation is that if \(\phi\) is faithfully flat, then this complex becomes exact, so that \(C^\bullet\) is a resolution of \(A\). However, what we will actually use in this post is not the entire resolution but only the first two morphisms, so we claim only the following.

Lemma 3 For a ring homomorphism \(\phi: A \rightarrow B\), the sequence of \(B\)-modules

\[0 \rightarrow B \overset{\phi\otimes B}{\longrightarrow} B\otimes_A B \overset{d\otimes B}{\longrightarrow} B\otimes_A B\otimes_A B\]

obtained by applying \(-\otimes_A B\) to the above sequence is split exact. In particular, if \(\phi\) is faithfully flat, then the sequence

\[0 \rightarrow A \overset{\phi}{\longrightarrow} B \overset{d}{\longrightarrow} B\otimes_A B\]

is exact.

Proof

By Proposition 2, to show that this sequence is exact, it suffices to show that the sequence obtained after applying \(-\otimes_A B\) is exact. Consider the sequence

\[0 \rightarrow B \overset{\phi\otimes B}{\longrightarrow} B\otimes_A B \overset{d\otimes B}{\longrightarrow} B\otimes_A B\otimes_A B\]

obtained by applying \(-\otimes_A B\). Our claim is that this is a split exact sequence.

To check this, consider the two maps

\[s: B\otimes_AB \rightarrow B;\quad b\otimes b'\mapsto bb',\qquad t: B\otimes_AB\otimes_AB\rightarrow B\otimes_AB;\quad b\otimes b'\otimes b''\mapsto b\otimes b'b''.\]

If we give each of \(B\otimes_AB\) and \(B\otimes_AB\otimes_AB\) the \(B\)-module structure acting on the rightmost \(B\), then these two maps become \(B\)-linear maps, and it is immediate that

\[s\circ(\phi\otimes B)=\id_B\]

holds. That is, \(\phi\otimes B\) is injective. Moreover, if \(b\otimes b'\in\ker(d\otimes B)\), then by its definition

\[0=(d\otimes B)(b\otimes b')=1\otimes b\otimes b'-b\otimes 1\otimes b',\]

so \(1\otimes b\otimes b'=b\otimes 1\otimes b'\), and applying \(t\) to this gives

\[b\otimes b'=t(b\otimes 1\otimes b')=t(1\otimes b\otimes b')=1\otimes bb'=(\phi\otimes B)(s(b\otimes b')),\]

so \(\ker(d\otimes B)\subseteq \im(\phi\otimes B)\). The opposite inclusion is trivial from \(d\circ \phi=0\), so we can confirm that the base-changed sequence is exact, and since \(\phi\) is faithfully flat, this means the original sequence is exact.

To complete the claim of split exactness, looking back at the \(t\) side, for an arbitrary \(b\otimes b'\) we have \(t((d\otimes B)(b\otimes b'))=1\otimes bb'-b\otimes b'\), so

\[(\phi\otimes B)\circ s-t\circ (d\otimes B)=\id_{B\otimes_AB},\]

and together with \(s\circ (\phi\otimes B)=\id_B\), this confirms that \((s,-t)\) is a contracting homotopy of this sequence.

Looking at the Zariski open cover of an affine scheme seen earlier, it becomes clear why this is the engine of gluing. Suppose \(A=(f_1,\ldots, f_n)\), and let \(B=\prod_i A_{f_i}\). Then

\[B\otimes_AB=\left(\prod_i A_{f_i}\right)\otimes_A \left(\prod_j A_{f_j}\right),\]

and since the products here are finite, we can combine them and think of

\[B\otimes_AB \cong\prod_{i,j} A_{f_i}\otimes A_{f_j}\cong\prod_{i,j} A_{f_if_j}.\]

([Commutative Algebra] §Properties of Localization, ⁋Lemma 1) Then under this identification, \(d^0\) puts an element of \(B\) into the front \(i\) component, and \(d^1\) puts an element of \(B\) into the back \(j\) component.

Geometrically, since \(D(f_i)\cap D(f_j)=D(f_if_j)\), we can think of \(B\otimes_AB\) as the ring of functions defined on \(D(f_i)\cap D(f_j)\), and in this case \(d^0\) and \(d^1\) become the restrictions

\[d^0\bigl((s_i)_i\bigr)=\bigl(s_i\vert_{D(f_if_j)}\bigr)_{i,j},\qquad d^1\bigl((s_i)_i\bigr)=\bigl(s_j\vert_{D(f_if_j)}\bigr)_{i,j},\]

respectively. That is, the difference of the two morphisms depends on which of \(s_i\) and \(s_j\) one looks at on the overlap \(D(f_if_j)\), and an element \((s_i)_i\) of \(B\) belonging to the kernel of \(d\) means that \(s_i\) and \(s_j\) agree on that overlap for all \(i,j\). In other words, it gives the gluing condition for the \(f_i\)’s defined on each \(D(f_i)\). Moreover, the injectivity of \(\phi\) is precisely the claim that an element of \(A\) vanishing on every \(D(f_i)\) is \(0\), giving the identity condition of a sheaf, and therefore Lemma 3 is nothing but the sheaf condition of \(\mathcal{O}_{\Spec A}\) for the open cover \(\{D(f_i)\}_i\). More generally, the remaining terms of the Amitsur complex form the Čech complex of this cover.

Descent Data

Lemma 3 tells us exactly how \(A\) is recovered from the data of \(B\); the slogan is that if we collect the elements of \(B\) on which the two ways of base change agree, the result is exactly \(A\).

Descent is this principle lifted to modules. Suppose a \(B\)-module \(N\) is given, and consider the process of obtaining a \(B\otimes_AB\)-module structure by applying \(-\otimes_AB\) or \(B\otimes_A-\) to it. As in the case of rings, these two \(B\otimes_AB\)-modules \(N\otimes_AB\) and \(B\otimes_AN\) are two structures that differ depending on whether \(N\) goes into the first factor or the second factor, and our goal is to compare these two and consider their equalizer. The problem is that, unlike the situation for rings, \(N\otimes_AB\) and \(B\otimes_AN\) are genuinely different¹ objects. Therefore, to compare them and compute the equalizer, an additional input is needed, namely an identification between \(N\otimes_AB\) and \(B\otimes_AN\), and this is exactly a descent datum.

To deal with this, let us fix notation. We define the morphisms

\[p_1: B\rightarrow B\otimes_AB;\quad b\mapsto b\otimes 1, \qquad p_2: B\rightarrow B\otimes_AB;\quad b\mapsto 1\otimes b,\]

and similarly define

\[p_{12}, p_{13}, p_{23}: B\otimes_A B \rightarrow B\otimes_A B\otimes_A B\]

as the morphisms sending into the two factors, among the three factors, specified by the index. Then for a \(B\)-module \(N\), we know that \(p_1^\ast N=N\otimes_A B\) and \(p_2^\ast N=B\otimes_A N\).

Definition 4 A descent datum for a ring homomorphism \(\phi: A \rightarrow B\) is a pair \((N, \Phi_N)\) of a \(B\)-module \(N\) and a \(B\otimes_A B\)-module isomorphism

\[\Phi_N: p_1^\ast N=N\otimes_A B \overset{\sim}{\longrightarrow} B\otimes_A N=p_2^\ast N\]

satisfying the cocycle condition

\[p_{13}^\ast \Phi_N=p_{23}^\ast \Phi_N\circ p_{12}^\ast \Phi_N\]

over \(B\otimes_A B\otimes_A B\). A morphism between two descent data \((N, \Phi_N)\) and \((N', \Phi_{N'})\) is a \(B\)-module homomorphism \(g: N \rightarrow N'\) satisfying \(\Phi_{N'}\circ(g\otimes B)=(B\otimes g)\circ \Phi_N\). We write \(\Desc(B/A)\) for the category they form.

Here, the cocycle condition says that the gluing is well-defined without contradiction over triple intersections, and it can be expressed by the diagram

where each morphism, for example \(p_{12}^\ast \Phi_N: p_1^\ast N\rightarrow p_2^\ast N\), is given by the formula

\[N\otimes_A B\otimes_A B \rightarrow B\otimes_A N\otimes_A B;\qquad n\otimes b\otimes b'\mapsto \Phi_N(n\otimes b)\otimes b'.\]

To see the Amitsur complex in the case of rings, we first observed that the kernel of \(d=d^1-d^0\) already contains \(A\). What corresponds to this in the module situation is the case where a \(B\)-module \(N\) comes from the base change \(M\otimes_A B\) of some \(A\)-module \(M\); in this case, the identification demanded as an additional input above is given automatically. In this case, the two base changes are

\[p_1^\ast N=M\otimes_A B\otimes_A B,\qquad p_2^\ast N=B\otimes_A M\otimes_A B,\]

which, as in the general case examined above, are different \(B\otimes_AB\)-modules. However, in both of these two cases, the \(B\otimes_AB\)-module structure is given in such a way that the first factor of \(B\otimes_AB\) acts on the left \(B\)-factor and the second factor acts on the right \(B\)-factor, and using the commutativity of \(A\) to move the \(M\)-factor of \(p_2^\ast N\) to the front gives a \(B\otimes_AB\)-module isomorphism, through which we can compare \(p_1^\ast N\) and \(p_2^\ast N\).

Example 5 Given an \(A\)-module \(M\), setting \(N=M\otimes_A B\), the \(B\otimes_A B\)-module isomorphism

\[\sigma_M: M\otimes_A B\otimes_A B \overset{\sim}{\longrightarrow} B\otimes_A M\otimes_A B;\qquad m\otimes x\otimes y\mapsto x\otimes m\otimes y\]

between the two base changes \(p_1^\ast N=M\otimes_A B\otimes_A B\) and \(p_2^\ast N=B\otimes_A M\otimes_A B\) we saw above defines a descent datum \((M\otimes_A B, \sigma_M)\). We call this the canonical descent datum attached to \(M\).

More generally, an \(A\)-module homomorphism \(M \rightarrow M'\) base-changes to a morphism between the canonical descent data, so the assignment \(M\mapsto (M\otimes_A B, \sigma_M)\) defines a functor

\[\rMod{A} \rightarrow \Desc(B/A)\]

.

Faithfully Flat Descent

We are now ready to state the central claim of this post. It is essentially nothing more than a rewriting of the principle we verified in Lemma 3.

Theorem 6 (Grothendieck) If a ring homomorphism \(\phi: A \rightarrow B\) is faithfully flat, then the functor

\[\rMod{A} \rightarrow \Desc(B/A);\qquad M\mapsto (M\otimes_A B, \sigma_M)\]

from Example 5 is a categorical equivalence. Its inverse functor is given, for a descent datum \((N, \Phi_N)\), by

\[N^\Phi=\{n\in N\mid \Phi_N(n\otimes 1)=1\otimes n\}\]

.

Proof

First, suppose an \(A\)-module \(M\) is given, and consider the canonical descent datum \((M\otimes_A B, \sigma_M)\) it defines. Applying the inverse functor above to it, we obtain

\[(M\otimes_AB)^\sigma=\{x\in M\otimes_A B\mid \sigma_M(x\otimes 1)=1\otimes x\}\]

, and since \(\sigma_M\) merely moves the \(M\)-factor, moving the \(M\)-factor on both sides back to the front and reading the condition inside \(M\otimes_A B\otimes_A B\), the condition becomes \(x\otimes 1=1\otimes x\). Hence what we must show is the exactness of the sequence

\[0 \rightarrow M \rightarrow M\otimes_A B \rightarrow M\otimes_A B\otimes_A B\]

, and this is obtained by repeating the proof of Lemma 3 verbatim with coefficient \(M\) attached.

The essentially substantive part is the opposite direction. Namely, given a descent datum \((N, \Phi_N)\), setting \(M=N^\Phi\), we must show that the \(B\)-module morphism

\[u: M\otimes_A B \rightarrow N;\qquad m\otimes b\mapsto bm\]

is an isomorphism from the descent datum \((M\otimes_AB, \sigma_M)\) to \((N, \Phi_N)\).

We now construct the inverse \(v: N\rightarrow M\otimes_AB\) of \(u\). The idea is that, in order to take an element of \(N\) and land it in \(M\otimes_AB\), one has no choice but to send \(n\) to an element of the same sort as \(n\otimes 1\), and keeping this in mind, a little computation tells us that we must define it using the descent datum as \(n\mapsto \Phi_N^{-1}(1\otimes n)\). For convenience, writing \(\Psi=\Phi_N^{-1}\), our claim is that the image of this assignment \(n\mapsto \Psi(1\otimes n)\) lands in \(M\otimes_AB\).

To verify this, write \(\Psi(1\otimes n)=\sum_j n_j\otimes c_j\). Then by the cocycle condition, \(p_{13}^\ast \Psi=p_{12}^\ast \Psi\circ p_{23}^\ast \Psi\), and evaluating both sides at the element \(1\otimes 1\otimes n\in B\otimes_A B\otimes_A N\) gives

\[p_{13}^\ast \Psi(1\otimes 1\otimes n)=\sum_j n_j\otimes 1\otimes c_j,\qquad (p_{12}^\ast \Psi\circ p_{23}^\ast \Psi)(1\otimes 1\otimes n)=\sum_j \Psi(1\otimes n_j)\otimes c_j\]

. Now identifying these two and applying \(p_{12}^\ast \Phi_N\) to both sides, we obtain the identity

\[\sum_j \Phi_N(n_j\otimes 1)\otimes c_j=\sum_j (1\otimes n_j)\otimes c_j\]

. Therefore, if we define \(d_N: N \rightarrow B\otimes_A N\) by \(d_N(n)=\Phi_N(n\otimes 1)-1\otimes n\), then \((d_N\otimes B)\bigl(\sum_j n_j\otimes c_j\bigr)=0\) holds. Our claim is then that \(\ker(d_N\otimes B)=M\otimes_AB\), and hence that the above assignment lands in \(M\otimes_AB\). Now by the definition of \(M\), the sequence \(0 \rightarrow M \rightarrow N \overset{d_N}{\longrightarrow} B\otimes_A N\) is exact, and since \(B\) is flat, applying \(-\otimes_AB\) to it yields

\[0 \rightarrow M\otimes_A B \rightarrow N\otimes_A B \overset{d_N\otimes B}{\longrightarrow} B\otimes_A N\otimes_A B\]

, which is also exact. That is, inside \(N\otimes_A B\), we have \(M\otimes_A B=\ker(d_N\otimes B)\).

Let us now show that the assignment \(v: n\mapsto \Psi(1\otimes n)\) thus constructed is indeed the inverse of \(u\). First,

\[v(u(m\otimes b))=\Psi(1\otimes bm)=(1\otimes b)\Psi(1\otimes m)=(1\otimes b)(m\otimes 1)=m\otimes b\]

is immediate. Conversely, for \(u(v(n))=n\): since \(v(n)\) is an element of \(M\otimes_A B\), we may choose \(m_k\in M\) and write \(v(n)=\sum_k m_k\otimes b_k\); applying \(\Phi_N\) to this gives \(\Phi_N(v(n))=\Phi_N(\Psi(1\otimes n))=1\otimes n\), and hence

\[1\otimes n=\sum_k (1\otimes b_k)\Phi_N(m_k\otimes 1)=\sum_k (1\otimes b_k)(1\otimes m_k)=1\otimes \sum_k b_km_k\]

. But the injectivity of \(n\mapsto n\otimes 1\) is given by Lemma 3 (in its \(M\)-coefficient version), and the same holds for \(n\mapsto 1\otimes n: N \rightarrow B\otimes_A N\), which differs only in the order of factors, so \(u(v(n))=\sum_k b_km_k=n\).

Finally, that \(u\) is indeed an isomorphism of descent data can be checked by applying the two composites to \(m\otimes b\otimes b'\), and naturality can likewise be shown by a short computation.

As a direct consequence of this theorem, one may check various properties of an \(A\)-module \(M\) not on \(M\) itself but on \(M\otimes_A B\) lifted up to \(B\). For instance, if \(M\otimes_A B\) is a finitely generated \(B\)-module then \(M\) is also finitely generated; if \(M\otimes_A B\) is finitely presented then \(M\) is finitely presented; and if \(M\otimes_A B\) is flat then so is \(M\). This is because each of these properties can be expressed in terms of exact sequences, and Proposition 2 reflects that exactness back down to \(A\).

Proposition 7 Let \(\phi: A \rightarrow B\) be a faithfully flat ring homomorphism and let \(M\) be an \(A\)-module. Then \(M\) is finitely generated (resp. finitely presented, flat, locally free of finite rank) if and only if \(M\otimes_A B\) is finitely generated (resp. finitely presented, flat, locally free of finite rank) as a \(B\)-module.

Proof

The direction that if \(M\) has the property then so does \(M\otimes_A B\) is trivial since each property is preserved under base change; the heart of this proposition lies in the converse directions.

First, suppose \(M\otimes_AB\) is generated by \(y_1,\ldots, y_n\). Then each \(y_i\) can be written as a sum of finitely many \(m_{ij}\otimes b_{ij}\), so collecting all the \(m_{ij}\) we can define a finitely generated submodule \(M_0\subseteq M\) of \(M\). Then \(M_0\otimes_AB\rightarrow M\otimes_AB\) is surjective, so \((M/M_0)\otimes_A B=0\), and by faithfulness \(M/M_0=0\). That is, \(M=M_0\) is finitely generated.

Now consider the case of finite presentation. Under this hypothesis, we have already obtained above that \(M\) is finitely generated, so it suffices to show that the kernel \(K\) of \(A^n \twoheadrightarrow M\) is finitely generated. For this, base-changing the exact sequence

\[0 \rightarrow K \rightarrow A^n \rightarrow M \rightarrow 0\]

to \(B\) yields

\[0 \rightarrow K\otimes_A B \rightarrow B^n \rightarrow M\otimes_A B \rightarrow 0\]

, which is exact, and since \(M\otimes_A B\) is finitely presented, \(K\otimes_A B\) is finitely generated (the discussion following [Commutative Algebra] §Flatness, ⁋Corollary 6). Therefore, applying the finitely generated result above to \(K\), \(K\) is also finitely generated and \(M\) is finitely presented.

For flatness, to show that \(M\) is flat, it suffices to show that for every injective \(A\)-module morphism \(M' \hookrightarrow M''\), the map \(M'\otimes_A M \rightarrow M''\otimes_A M\) is injective. Again applying \(-\otimes_AB\), we obtain the morphism \(M'\otimes_A M\otimes_A B \rightarrow M''\otimes_A M\otimes_A B\), which can be regarded as obtained by tensoring the injective homomorphism \(M'\otimes_A B \rightarrow M''\otimes_A B\) with the flat \(B\)-module \(M\otimes_A B\), so it is again injective.

Finally, being locally free of finite rank is equivalent to being finitely presented and flat ([Commutative Algebra] §Flatness, ⁋Corollary 6), so there is nothing more to prove.

Descent of Quasi-coherent Sheaves

We now have all the tools for gluing. What remains is merely to attach appropriate names to them. Namely, since we have seen that gluing works well even when the notion of open embedding is extended to faithfully flat morphisms, we may as well rewrite the notion of an open set afresh using these faithfully flat morphisms.

Definition 8 A Grothendieck topology on a category \(\mathcal{C}\) having fiber products is an assignment, to each object \(U\), of a collection of families \(\{f_i: U_i \rightarrow U\}_{i\in I}\) of morphisms with codomain \(U\), whose elements are called coverings of \(U\). These satisfy the following three conditions.

1. If \(f: V \rightarrow U\) is an isomorphism, then \(\{f: V \rightarrow U\}\) is a covering.
2. If \(\{f_i: U_i \rightarrow U\}\) is a covering and \(g: V \rightarrow U\) is an arbitrary morphism, then the family \(\{U_i\times_U V \rightarrow V\}_{i\in I}\) given by base change is also a covering.
3. If \(\{f_i: U_i \rightarrow U\}\) is a covering and, for each \(i\), \(\{g_{ij}: U_{ij} \rightarrow U_i\}_{j\in J_i}\) is a covering, then the family \(\{f_i\circ g_{ij}: U_{ij} \rightarrow U\}_{i, j}\) given by composition is also a covering.

In particular, since \(\Sch\) has fiber products (§Fiber Products, ⁋Theorem 8), we can apply this definition. Reading an open cover \(\{U_i\}\) of a topological space as the family of inclusions \(\{U_i\hookrightarrow U\}\), the three conditions above hold, and in that case \(U_i\times_U V\) is the intersection \(U_i\cap V\). That is, the three conditions only require that a space covers itself, that a restriction of a covering is again a covering, and that a covering of a covering is a covering. The topology we will use takes as coverings the morphisms that are faithfully flat and quasi-compact; shortening its name fidèlement plat quasi-compact, it is called the fpqc topology.

Definition 9 A family of morphisms \(\{\psi_i: U_i \rightarrow X\}_{i\in I}\) over a scheme \(X\) is an fpqc cover if each \(\psi_i\) is flat, \(\coprod_i U_i \rightarrow X\) is surjective, and the quasi-compact condition holds: every affine open \(V\subseteq X\) is covered by the images of finitely many affine opens \(W_{ij}\) of the \(U_i\). The Grothendieck topology on \(\Sch\) defined by these coverings is called the fpqc topology.

In the fpqc topology, the simplest covering of a single affine scheme \(\Spec A\) is \(\{\Spec B \rightarrow \Spec A\}\), consisting of a single faithfully flat ring homomorphism \(A \rightarrow B\).

The reason we bothered to lift Lemma 3 to modules is, of course, to deal with quasi-coherent sheaves (§Quasi-coherent Sheaves, ⁋Definition 8).

Theorem 10 For any scheme \(X\) and any quasi-coherent sheaf \(\mathcal{F}\) on \(X\), the presheaf

\[T\mapsto \Gamma(T, \psi^\ast \mathcal{F})\qquad (\psi: T \rightarrow X)\]

is a sheaf for the fpqc topology. That is, for every fpqc cover \(\{T_i \rightarrow T\}\), the sequence

\[\Gamma(T, \psi^\ast\mathcal{F}) \rightarrow \prod_i \Gamma(T_i, \psi_i^\ast\mathcal{F}) \rightrightarrows \prod_{i,j}\Gamma(T_i\times_T T_j, \psi_{ij}^\ast\mathcal{F})\]

is exact.

Proof

The problem is local, and thanks to the quasi-compact condition it reduces to finite coverings, so it suffices to treat the case where \(T=\Spec A\) is affine and the covering is a single faithfully flat morphism \(\{\Spec B \rightarrow \Spec A\}\). In this case, choosing an \(A\)-module \(M\) with \(\mathcal{F}=\widetilde M\), the pullback is given by base change (§Quasi-coherent Sheaves, ⁋Proposition 15), so the above sequence becomes

\[M \rightarrow M\otimes_A B \rightrightarrows M\otimes_A B\otimes_A B\]

. The claim is then the exactness of the sequence

\[0 \rightarrow M \rightarrow M\otimes_A B \rightarrow M\otimes_A B\otimes_A B\]

obtained by generalizing Lemma 3 with \(M\) as coefficient, which was already shown in the proof of Theorem 6 (Grothendieck). Since the fact that the equalizer of the two morphisms \(d^0, d^1\) is \(M\) is precisely the sheaf condition above, we obtain the conclusion.

Theorem 10 allows us to compute global sections of a quasi-coherent sheaf over a faithfully flat covering. From this we obtain descent for quasi-coherent sheaves themselves.

Theorem 11 Let the family \(\{\psi_i: U_i \rightarrow X\}\) be an fpqc cover. Then giving a quasi-coherent sheaf on \(X\) is equivalent to giving the data of quasi-coherent sheaves \(\mathcal{F}_i\) on each \(U_i\), together with isomorphisms \(\Phi_{ij}: \pr_2^\ast \mathcal{F}_j\cong \pr_1^\ast \mathcal{F}_i\) over \(U_i\times_X U_j\) satisfying the cocycle condition.

Proof

Since the problem is local, it suffices to consider the case where \(X=\Spec A\) and the covering is a single faithfully flat morphism \(\Spec B \rightarrow \Spec A\). In this situation, \(U_i\times_X U_j\) is \(\Spec(B\otimes_A B)\), and the given data are exactly a \(B\)-module \(N=\Gamma(\Spec B, \mathcal{F}_1)\) together with a \(B\otimes_A B\)-module isomorphism \(\Phi_N\) forming a cocycle pair, that is, a descent datum in the sense of Definition 4. These data correspond precisely to an object of \(\Desc(B/A)\), so by Theorem 6 (Grothendieck) they come from a unique \(A\)-module \(M\), i.e. from a unique quasi-coherent sheaf \(\widetilde M\), and this correspondence also preserves morphisms.

For a general fpqc cover, use the quasi-compact condition to pick a finite subcover, form its disjoint union into a single affine faithfully flat morphism, apply the affine case above, and then glue the results over the affine opens of \(X\). The consistency of the gluing is guaranteed by the sheaf property of Theorem 10.

Again, the essential fact in Theorem 11 is that when a descent datum \((\mathcal{F}_i, \Phi)\) of the above form is given, one can actually glue these together into a single sheaf \(\mathcal{F}\).

Descent of Morphisms

We now turn to the problem of gluing objects one step more geometric than quasi-coherent sheaves. Our first goal is to glue schemes: suppose an fpqc cover \(\{\psi_i:U_i\rightarrow S\}\) is given, that for each \(i\) a \(U_i\)-scheme structure \(V_i\rightarrow U_i\) is given, and that over each overlap \(U_i\times_SU_j\) an identification of these via isomorphisms satisfying the cocycle condition is already given. Our goal is to find an \(S\)-scheme \(V\rightarrow S\) gluing these \(U_i\)-schemes together, where the condition that it extends the \(V_i\) is given by the isomorphisms

\[V\times_SU_i\cong V_i\]

. In general such a construction is not always possible, and the most basic condition making it possible is that the \(V_i\rightarrow U_i\) be affine.

Theorem 12 Suppose we are given an fpqc cover \(\{\psi_i:U_i \rightarrow S\}\), affine morphisms \(V_i \rightarrow U_i\) defined over each of them, and cocycle isomorphism data identifying them over the intersections \(U_i\times_S U_j\). Then there exist an affine morphism \(V\rightarrow S\) over \(S\) and isomorphisms \(V\times_SU_i\cong V_i\) compatible with the given cocycle isomorphisms, and such a \(V\) is unique up to unique isomorphism.

Proof

Our strategy is to use the gluing of quasi-coherent sheaves that we already have; to this end, we regard the affine morphisms \(\varphi_i: V_i\rightarrow U_i\) as quasi-coherent \(\mathcal{O}_{U_i}\)-algebras

\[\mathcal{A}_i=(\varphi_i)_\ast\mathcal{O}_{V_i}\]

(§Quasi-coherent Sheaves). That is, we think of \(V_i\) as the relative spec \(\rSpec_{U_i}(\mathcal{A}_i)\); it then suffices to glue these quasi-coherent algebras into a single quasi-coherent algebra and turn it back into an affine morphism.

Now the cocycle isomorphisms between the \(V_i\) translate in this language into cocycle isomorphisms between pullbacks of the \(\mathcal{A}_i\), so applying Theorem 11 yields a quasi-coherent sheaf \(\mathcal{A}\) on \(S\) together with isomorphisms

\[\psi_i^\ast\mathcal{A}\cong\mathcal{A}_i.\]

We must now endow it with an algebra structure. As we checked in the proof of §Quasi-coherent Sheaves, pullback is compatible with tensor products and \(\psi_i^\ast\mathcal{O}_S\cong\mathcal{O}_{U_i}\), so the multiplication and unit of each \(\mathcal{A}_i\)

\[\mu_i:\mathcal{A}_i\otimes\mathcal{A}_i\rightarrow\mathcal{A}_i,\qquad \eta_i:\mathcal{O}_{U_i}\rightarrow\mathcal{A}_i\]

can be regarded as morphisms between the pullbacks of \(\mathcal{A}\otimes\mathcal{A}\) and \(\mathcal{A}\), and between those of \(\mathcal{O}_S\) and \(\mathcal{A}\), respectively. Since these are compatible with the given algebra isomorphisms, the correspondence for morphisms in Theorem 11 gives unique morphisms

\[\mu:\mathcal{A}\otimes\mathcal{A}\rightarrow\mathcal{A},\qquad \eta:\mathcal{O}_S\rightarrow\mathcal{A}\]

whose pullbacks agree with \(\mu_i\) and \(\eta_i\), respectively. Associativity and the unit law hold after pulling back to the \(U_i\), and sheaf morphisms that agree over an fpqc cover also agree over \(S\); hence \(\mathcal{A}\) is a quasi-coherent \(\mathcal{O}_S\)-algebra. Therefore, setting

\[V=\rSpec_S(\mathcal{A}),\]

this is an affine scheme over \(S\). Since the relative spectrum is compatible with base change by §Quasi-coherent Sheaves, we have

\[V\times_SU_i\cong\rSpec_{U_i}(\psi_i^\ast\mathcal{A})\cong\rSpec_{U_i}(\mathcal{A}_i)\cong V_i,\]

and these isomorphisms recover the cocycle data given at the start. Moreover, \(\mathcal{A}\) and its algebra structure are unique up to unique isomorphism by Theorem 11, and an affine morphism is recovered from its quasi-coherent algebra, so \(V\) is unique in the same sense.

More generally, a quasi-compact, quasi-separated scheme morphism \(\varphi:V\rightarrow U\) is called quasi-affine if the canonical morphism \(V\rightarrow\rSpec_U(\varphi_\ast\mathcal{O}_V)\) is a quasi-compact open immersion. The conclusion of Theorem 12 holds in this case as well. A generalization in another direction is the case of quasi-projective morphisms; being quasi-projective alone is not enough, and one also needs an ample line bundle together with a compatible descent datum on it. Roughly, the proof descends the section algebra of the ample line bundle to form a relative Proj; the original scheme then appears as an open subscheme inside it, so one glues these together.

Meanwhile, the essential property of faithfully flat base change is not only that it is an exact functor, but also that the exactness checked there can be transferred back to the original; Proposition 7 used this to descend flatness and finiteness conditions of modules. Applying the same argument affine-locally, one can also verify properties of an already given scheme morphism \(\psi:X\rightarrow Y\) over a cover. For this, take an fpqc cover \(\{Y_i\rightarrow Y\}\) of \(Y\); then \(\psi\) defines morphisms

\[\psi_i:X\times_YY_i\rightarrow Y_i.\]

Promoting Proposition 7 scheme-theoretically then yields the flatness and finiteness parts of the following proposition, while surjectivity and affineness can be handled in the same way as Theorem 12.

Proposition 13 Let \(\psi: X\rightarrow Y\) be a scheme morphism and \(\{Y_i \rightarrow Y\}\) an fpqc cover of \(Y\). Then \(\psi\) has one of the following properties if and only if each base change \(\psi_i: X\times_Y Y_i \rightarrow Y_i\) has that property.

Flat, faithfully flat, affine, locally of finite type, locally of finite presentation, surjective.

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References

[Vak] R. Vakil, The rising sea: Foundations of algebraic geometry. Available online.
[FGA] B. Fantechi, L. Göttsche, L. Illusie, S. Kleiman, N. Nitsure, A. Vistoli, Fundamental algebraic geometry: Grothendieck’s FGA explained. Mathematical Surveys and Monographs. American Mathematical Society, 2005.

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1. In the case of rings, both of these objects were \(B\otimes_AB\). ↩