First, reduce to the case that (1) \(x\) is a closed point in \(X\). By the definition of smooth morphisms, it is then enough to consider the case of a closed point \(x\in \operatorname{Spec}k[X_\bullet ](f_\bullet )\) where the Jacobian matrix has full rank. By Theorem 2.17 (2) and (4) it is enough to show that the images of the \(f_i\) in \(\mathfrak m_x/\mathfrak m_x^2\) are linearly independent. This is clear (cf. Example 2.4) if \(\kappa (x) = k\), and the general case can be reduced to this one, using that in the base change \(X\otimes _k\kappa (x)\) there exists a point \(\overline{x}\) with residue class field \(\kappa (x)\) projecting to \(x\in X\) and that we have an inclusion \(\mathfrak m_x/\mathfrak m_x^2 \to \mathfrak m_{\overline{x}}/\mathfrak m_{\overline{x}}^2\) of \(\kappa (x)\)-vector spaces.

Write \(d = \dim {\mathscr O}_{X, x}\). After renumbering the \(g_i\), if necessary, we may assume that the first \(n-d\) columns of \(J_{g_\bullet }(x)\) are linearly independent. We then have

and \(x\) is smooth over \(k\) as a point of \(Y\). By the previous lemma, \(\dim {\mathscr O}_{Y, x} = d\). It follows that \({\mathscr O}_{X,x} = {\mathscr O}_{Y,x}\), and together with Lemma 2.12 we obtain the claim.

For \(\ge \) choose some affine open neighborhood of \(x\), apply Noether normalization, and use that the properties finite and injective of a ring homomorphism are preserved under the base change \(-\otimes _kK\).

For \(\le \), use that the morphism \(X_K\to X\) (being obtained by base change from \(\operatorname{Spec}(K)\to \operatorname{Spec}(k)\)) is flat, and that flat ring homomorphisms satisfy a going down theorem. The key fact for the going down property is that for every flat local ring homomorphism \(A\to B\) between local rings, the map \(\operatorname{Spec}(B)\to \operatorname{Spec}(A)\) is surjective. Cf.  [ GW1 ] Lemma 14.9 or  [ M2 ] Theorem 7.3, Theorem 9.5. (In  [ GW1 ] , the proof of \(\le \) is given using the more difficult Proposition 5.44/Theorem 14.38 there, which is required in the book anyway; but at this point the above, related but simpler method works.)

The implications (i) \(\Rightarrow \) (ii) \(\Rightarrow \) (iii) and (iv) \(\Rightarrow \) (v) are easy.

Furthermore (iii) \(\Rightarrow \) (iv) and the regularity of \({\mathscr O}_{X,x}\) for a smooth point \(x\) follow from Lemma 2.18.

Next we show that the regularity of \({\mathscr O}_{X,x}\) implies that \(x\) is a smooth point if \(\kappa (x) = k\). Write \(d = \dim {\mathscr O}_{X, x} = \dim _k T_xX\). We embed an affine open neighborhood \(U\) into affine space, say as an open subscheme of \(V(g_1,\dots , g_s)\subseteq \mathbb {A}^n_k\). We are then in the situation of Lemma 2.19, and the lemma shows that \(x\) is a smooth point. This also shows (v) \(\Rightarrow \) (iii).

It remains to prove that (iii) \(\Rightarrow \) (i). It is enough to consider the case where \(x\) is a closed point of \(V(g_1,\dots , g_s)\subset \mathbb {A}^n_k\) for some polynomials \(g_i\). By Lemma 2.19, it is enough to show that \(\operatorname{rk}J_{g_\bullet }(x) = n-\dim {\mathscr O}_{X,x}\). But the rank of the Jacobian matrix does not change when we replace \(x\) by \(\overline{x}\) (and consider the polynomials \(g_i\) in \(K[X_\bullet ]\)), and \(\dim {\mathscr O}_{X,x} = \dim {\mathscr O}_{X_K, \overline{x}}\). Since \(\overline{x}\) is a regular point of \(X_K\) by (iii), which we now assume to hold, we are done.

Assume that \(\Omega _{B/A} = 0\), and consider \(I\subset C\) and a commutative diagram as above. We need to show that there is at most one ring homomorphism \(B\to C\) making the diagram commutative. Assume that \(\varphi _1, \varphi _2\colon B\to C\) have this property. The \(C\)-module structure on \(I\) factors through a \(C/I\)-module structure since \(I^2=0\), so that we can view \(I\) as a \(B\)-module via the map \(B\to C/I\). Then the difference \(\varphi _1 - \varphi _2\) is an \(A\)-derivation \(B\to I\), and is hence zero by our assumption.

For the converse it is enough that every \(A\)-derivation \(B\to M\) vanishes. Let \(C = D_B(M)\) and \(I = M\). Then \(I^2 = 0\), and the assumption that \(B\) is formally unramified over \(A\) implies \(\operatorname{Der}_A(B, M)=0\).

To check the exactness in Part (1), it is enough to check that the sequence gives rise to an exact sequence whenever we apply the functor \(\operatorname{Hom}_C(-, M)\) for \(M\) a \(C\)-module. Note that \(\operatorname{Hom}_C(\Omega _{B/A}\otimes _BC, M) = \operatorname{Hom}_B(\Omega _{B/A}, M)\) (where on the right we view \(M\) as a \(B\)-module via \(g\)). See  [ ALG2 ] Satz 3.14.

Thus the first part follows, once we check that

is exact (as a sequence of \(A\)-modules or just abelian groups) for any \(C\)-module \(M\). But this is obvious.

Part (2). Now assume that \(g\) is formally smooth. Let us construct a \(C\)-module homomorphism \(\Omega _{C/A}\to \Omega _{B/A}\otimes _BC\) as follows. Constructing a homomorphism like this amounts to constructing an \(A\)-derivation \(C\to \Omega _{B/A}\otimes _BC=:M\). Similarly as above, we consider \(C\times M\) as a ring (with \(M^2=0\)). Let \(B\to C\times M\) be given by \(b\mapsto (g(b), db\otimes 1)\). One checks that this is a ring homomorphism. Since \(g\) is formally smooth, for this \(B\)-algebra structure we find a homomorphism \(C\to C\times M\) of \(B\)-algebras. Composing it with the projection to \(M\) we obtain an \(A\)-derivation \(C\to M\). One checks that the composition \(\Omega _{B/A}\otimes _BC\to \Omega _{C/A}\to \Omega _{B/A}\otimes _BC\) is the identity, and this finishes the proof.

See also  [ GW2 ] Proposition 18.18 (1) for a slightly different proof of the second part (which is more along the lines of our proof of the first part).

Part (3). This can be proved by similar arguments as for Parts (1) and (2). We omit the proof for the time being (see  [ GW2 ] Proposition 18.18 (2)).

All assertions in Part (1) follow from Theorem 2.33 and Lemma 2.28.

To prove Part (2), consider the short exact sequence

The assumption that \(g\circ f\) is formally smooth implies that \(p\) admits a section \(s\). Then \(s\circ p_{|{\mathfrak b}/{\mathfrak b}^2} = 0\), and \(p\circ (\operatorname{id}- s\circ p) = 0\). We obtain \(D := \operatorname{id}- s\circ p\colon B/{\mathfrak b}^2\to {\mathfrak b}/{\mathfrak b}^2\). This is an element of \(\operatorname{Der}_A(B/{\mathfrak b}^2, {\mathfrak b}/{\mathfrak b}^2) = \operatorname{Hom}_B(\Omega _{B/A}, {\mathfrak b}/{\mathfrak b}^2)\) and one checks that it gives rise to a retraction of the map \({\mathfrak b}/{\mathfrak b}^2 \to \Omega _{B/A}\otimes _BC\) in the sequence in Part (2).

First note that \(\operatorname{Hom}^g(Y[\varepsilon ], X)\) can be identified with \(\operatorname{Der}_S({\mathscr O}_X, g_*{\mathscr O}_Y)\). In fact, this can be checked on an affine open cover, and in the affine case we have seen this in Construction 2.31. Now we conclude by the following chain of isomorphisms:

Write \(X = \mathbb {P}^n_R\). We have the “universal” surjection \({\mathscr O}_X^{n+1} \to {\mathscr O}_X(1)\) and denote by \({\mathscr K}\) its kernel. We want to show that \({\mathscr K}(-1):={\mathscr K}\otimes _{{\mathscr O}_X}{\mathscr O}_X(-1) \cong \Omega _{X/R}\). All the \({\mathscr O}_X\)-modules involved here are locally free of finite rank, so it is enough to prove that \({\mathscr T}_{X/R}\cong \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits ({\mathscr K}, {\mathscr O}(1)) = \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits ({\mathscr K}(-1), {\mathscr O}) = {\mathscr K}(-1)^\vee \).

Let \(U=\operatorname{Spec}(A)\subseteq X\) be open affine and denote by \(g\colon U\to X\) the inclusion. The morphism \(g\) corresponds to a surjection \(A^{n+1}\to L\) onto a locally free \(A\)-module \(L\) of rank \(1\) whose kernel we denote by \(K\). Note that \(K = {\mathscr K}_{|U}\). We now use the notation of Proposition 2.45.

Claim. There is a natural identification \(\operatorname{Hom}_A(K, A^{n+1}/K)\xrightarrow {\cong }\operatorname{Hom}^g(U[\varepsilon ], X)\).

Proof of claim. An element of \(\operatorname{Hom}^g(U[\varepsilon ], X) \subseteq X(U(\varepsilon ))\) is given by a surjection \((A[\varepsilon ]/(\varepsilon ^2))^{n+1} \to L'\), where \(L'\) is locally free over \(A[\varepsilon ]/(\varepsilon ^2)\) of rank \(1\), or equivalently its kernel \(K'\subset (A[\varepsilon ]/(\varepsilon ^2))^{n+1}\), such that \(K'\otimes _{A[\varepsilon ]/(\varepsilon ^2)} A = K\).

Now take an \(A\)-module homomorphism \(\alpha \colon K\to A^{n+1}/K\). We define \(K'\) as the \(A[\varepsilon ]/(\varepsilon ^2)\)-module generated by the image of the map \(K\to (A[\varepsilon ]/(\varepsilon ^2))^{n+1}\), \(x\to x+\varepsilon \alpha (x)\). (We define \(\varepsilon \alpha (x)\) by choosing a lift of \(\alpha (x)\in A^{n+1}/K\) in \(A^{n+1}\). The resulting \(K'\) is independent of the choice of lift.)

Note that \(A[\varepsilon ]^{n+1}/K'\) is locally free over \(A[\varepsilon ]\). To check this, we may localize and thus assume that \(K\) and \(A^{n+1}/K\) are free \(A\)-modules. Now choosing lifts of bases of \(K\) and \(A^{n+1}/K\) to \(A[\varepsilon ]^{n+1}\) gives us a family of \(n+1\) vectors. Write them as the columns of a matrix \(M\) over \(A[\varepsilon ]\). By construction, \(\det (M)\) maps to a unit in \(A\) and hence is a unit in \(A[\varepsilon ]\). Thus the lifts form a basis of \(A[\varepsilon ]^{n+1}\) and in particular \(A[\varepsilon ]^{n+1}/K'\) (and \(K'\)) are free.

This defines the desired bijection.

With the claim and Proposition 2.45 we can identify \(\Gamma (U, {\mathscr T}_{X/R})\) with

This identification is compatible with restrictions to smaller subsets and therefore defines the isomorphism of \({\mathscr O}_X\)-modules we are looking for.

Since the assertion is local on \(X\), we may assume that \(Y = \operatorname{Spec}R\) and \(X = \operatorname{Spec}R[T_1, \dots , T_n]/(f_1, \dots , f_{n-d})\) with the \(df_i(x)\in \Omega _{\mathbb {A}^n_R/R}(x)\) linearly independent, as in Definition 2.49. We write \(\mathfrak a = (f_1, \dots , f_{n-d})\) and \(A = R[T_1,\dots , T_n]/\mathfrak a\). We have the exact sequence (Theorem 2.34)

Renumber the \(T_i\) (if necessary) so that the images of \(dT_1,\dots , dT_d, df_1,\dots , df_{n-d}\) are a basis of the fiber \((\Omega _{R[T_\bullet ]/R}\otimes _{R[T_\bullet ]}A)(x)\) over \(x\). By the lemma of Nakayama, these elements give us also a basis of the stalk, and hence even a basis on an open neighborhood \(U\) of \(x\). The image of \({\mathfrak a}/{\mathfrak a}^2\) is exactly the submodule generated by the \(df_i\), so this implies that \(\Omega _{A/R}\) is free over such a neighborhood.

If \(\Omega _{X/k}\) is locally free of rank \(\dim X\), then \(X\) is regular and hence, since \(k\) is algebraically closed, also smooth over \(k\) (Theorem 2.21). Conversely, the smoothness of \(f\) implies that \(\Omega _{X/k}\) is locally free by Proposition 2.50. Again using Theorem 2.21, we also obtain that \(X\) is regular, and it follows that \(f\) must be smooth of relative dimension \(\dim X\).

As in the proof of Proposition 2.50, it is enough to consider the local situation, and we again use the notation set up in the beginning of the proof of that proposition.

Consider a ring \(C\), an ideal \(I\) of \(C\) with \(I^2 = 0\) and a commutative diagram

We need to show that there exists a homomorphism \(\varphi \colon A\to C\) making the diagram commutative. We start by choosing arbitarily an \(R\)-algebra homomorphism \(\psi \colon R[T_1,\dots , T_n]\to C\) such that the diagram

is commutative. Then \(\psi ({\mathfrak a})\subseteq I\) (but of course there is no reason to expect that \(\psi \) will factor through \(A\); we will now change it appropriately to achieve that). In Remark 2.51 we have seen that the sequence

is split exact, at least after replacing \(A\) by a suitable localization. Since the proposition makes only a local statement the localization is harmless and we suppress it from the notation. The restriction of \(\psi \) to \({\mathfrak a}\) induces a map \({\mathfrak a}/{\mathfrak a}^2 \to I/I^2=I\), as we have already noted, and since the sequence is split, we can extend that map to a map \(\xi \colon \Omega _{R[T_\bullet ]/R}\otimes _{R[T_\bullet ]}A \to I\). We define \(D\) as the composition

an \(R\)-derivation with the property that \(\psi _{|{\mathfrak a}} = D_{|{\mathfrak a}}\). Setting \(\varphi = \psi -D\), we obtain a map that maps \({\mathfrak a}\) to \(0\) and (since \(D\) is a derivation) is a ring homomorphism. Thus \(\varphi \) factors through a homomorphism \(\varphi \colon A\to C\). This makes the above diagram commutative, so we are done.

Let \(f\) be formally smooth and locally of finite presentation. To show that \(f\) is smooth, we may work locally on \(X\) and \(Y\) and therefore pass to an affine situation, i.e., assume that \(f\) is given by a ring homomorphism \(R\to R[T_\bullet ]\to A\) with \(R[T_\bullet ]\to A\) surjective with kernel \({\mathfrak a}\). Then Theorem 2.34 shows that the sequence

is split exact. Choosing a basis of \({\mathfrak a}/{\mathfrak a}^2\) and lifting its elements to polynomials \(f_1,\dots , f_{n-d}\in {\mathfrak a}\), we see that the conditions of Definition 2.49 are satisfied and the morphism \(\operatorname{Spec}A\to \operatorname{Spec}R\) is smooth.

For the converse, note that the previous proposition shows already that a smooth morphism is at least “locally formally smooth”. We only give some very sketchy indications on how to get a global version. See  [ Bo ] Ch. 8.5 for more details. See also  [ GW2 ] Section (18.10) for a slightly different approach.

Consider a diagram

where, as usual, \(I\subseteq C\) is an ideal with \(I^2=0\). We have seen that there exists an open cover \(X=\bigcup _i U_i\) such that each \(U_i\) is formally smooth over \(Y\). In particular after restricting \(a_0\) to \(U_i\) and the inverse image of \(U_i\) in \(\operatorname{Spec}C/I\), we can find the desired diagonal morphism that extends \(a_0\) to \(\operatorname{Spec}C\). In other words, we find an open cover \((V_i)_i\) of \(\operatorname{Spec}C\) (which topologically is \(=\operatorname{Spec}C/I\)) and morphisms \(\varphi _i\colon V_i\to U_i\subseteq X\) making the above diagram commutative. The idea is to replace the \(\varphi _i\) by \(\varphi '_i\) such that \(\varphi '_i\) and \(\varphi '_j\) coincide on \(V_i\cap V_j\). By gluing one obtains the desired map \(\operatorname{Spec}C\to X\).

Here, we want to set \(\varphi '_i = \varphi _i - D_i\) for some derivation \(D_i\) (cf. Construction 2.31 where we have seen this principle). Writing this out one sees that there exists a family \((D_i)_i\) with the desired properties if and only if a certain class (depending on the \(\varphi _i\)) in \(\check{H}^1(\operatorname{Spec}C/I, \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_{\operatorname{Spec}C/I}}(a_0^*\Omega _{X/Y}, \tilde{I})\) vanishes. But this cohomology group vanishes entirely since \(\operatorname{Spec}C/I\) is affine and \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_{\operatorname{Spec}C/I}}(a_0^*\Omega _{X/Y}, \tilde{I})\), \(\Omega _{X/Y}\) being of finite presentation by our assumptions, is quasi-coherent.