Éléments de Géométrie Algébrique — English

§22. Differential criteria for formal smoothness and regularity

The principal results of this section are:

a) The criteria (22.2.2) and (22.5.8), which give necessary and sufficient conditions for a complete Noetherian local ring $A$ containing a field $k$ to be formally smooth over $k$. The criterion (22.5.8) is stated without involving any differential notion, and completes $(0_{IV},\mathrm{19.6.6})$; the statement (22.2.2), of a somewhat more technical nature, allows one to give a classification, for $k$ fixed, of the local rings $A$ considered (22.2.5): they are determined by their dimension $n$ and their residue field $K$, subject only to the condition $n\ge r{g}_K({{\rm Y}}_{K/k})$.

b) The theorem (22.3.3), and the corollary (22.7.6) of Nagata's Jacobian criterion (22.7.3), which give conditions allowing one to assert that a localization of a complete Noetherian local ring $A$ containing a field $k$ is geometrically regular over $A$; they will play an important role in the study of the "fine" properties of local rings carried out in Chap. IV, §7. For the moment we possess no proof of (22.7.6) independent of Nagata's Jacobian criterion.

c) Zariski's Jacobian criterion (22.6.7) and its variants, which are easy consequences of $(0_{IV},\mathrm{20.7.8})$.

22.1. Lifting of formal smoothness

Theorem (22.1.1).

Let $s:\Lambda \to A$, $u:A\to B$ be two ring homomorphisms, $\mathfrak{J}$ an ideal of $B$, $C$ the quotient ring $B/\mathfrak{J}$. Suppose that $C$ is a $\Lambda$-algebra formally smooth for the discrete topology.

(i) Suppose the following conditions are satisfied:

α) $\mathfrak{J}/{\mathfrak{J}}^2$ is a projective $C$-module.

β) The canonical homomorphism $S_C^{\bullet }(\mathfrak{J}/{\mathfrak{J}}^2)\to g{r}_{\mathfrak{J}}^{\bullet }(B)$ $(0_{IV},\mathrm{19.5.2})$ is bijective.

γ) The characteristic homomorphism $\chi :{{\rm Y}}_{C/A/\Lambda }\to \mathfrak{J}/{\mathfrak{J}}^2$ $(0_{IV},\mathrm{20.6.20})$ is injective.

δ) The $C$-module ${\Omega }_{B/\Lambda }^1{\otimes }_{B}C$ is projective.

Then $B$, equipped with the $\mathfrak{J}$-preadic topology, is a $\Lambda$-algebra formally smooth.

(ii) Conversely, suppose that $B$ is a $\Lambda$-algebra formally smooth for the $\mathfrak{J}$-preadic topology, and that $A$ is a $\Lambda$-algebra formally smooth for the discrete topology. Then the conditions α), β), γ), δ) of (i) are satisfied.

By virtue of the exact sequence $(0_{IV},\mathrm{20.6.22.1})$ (which is applicable, since $B/{\mathfrak{J}}^2$ is a $\Lambda$-trivial extension of $C$ by $\mathfrak{J}/{\mathfrak{J}}^2$, $C$ being supposed to be a $\Lambda$-algebra formally smooth $(0_{IV},\mathrm{19.4.4})$), the condition γ) is equivalent to ${{\rm Y}}_{B/A/\Lambda }^C=0$, or equivalently to the following:

γ′) The homomorphism $u_{B/A/\Lambda }\otimes 1_C:{\Omega }_{A/\Lambda }^1{\otimes }_{A}C\to {\Omega }_{B/\Lambda }^1{\otimes }_{B}C$ is injective.

(i) The conditions α) and β) entail that $B$ is a $\Lambda$-algebra formally smooth for the $\mathfrak{J}$-preadic topology $(0_{IV},\mathrm{19.5.4})$. It therefore amounts to the same to say that $B$ is (for the $\mathfrak{J}$-preadic topology) a $\Lambda$-algebra formally smooth, or a $\Lambda$-algebra formally smooth relative to $A$. To see that $B$ is a $\Lambda$-algebra formally smooth, it suffices, by virtue of $(0_{IV},\mathrm{20.7.2})$, to prove that the continuous homomorphism of topological $B$-modules $u_{B/A/\Lambda }:{\Omega }_{A/\Lambda }^1{\otimes }_{A}B\to {\Omega }_{B/\Lambda }^1$ is formally left-invertible. Now, since $B$ is a $\Lambda$-algebra formally smooth, the topological $B$-module ${\Omega }_{B/\Lambda }^1$ is formally projective $(0_{IV},\mathrm{20.4.9})$ and its topology is deduced from that of $B$ $(0_{IV},\mathrm{20.4.5})$. By virtue of $(0_{IV},\mathrm{19.1.9})$, it therefore suffices, for $u_{B/A/\Lambda }$ to be formally left-invertible, that $u_{B/A/\Lambda }\otimes 1_C$ be left-invertible. But by virtue of γ′), this last map is injective, so one has the exact sequence $(0_{IV},\mathrm{20.6.14.7})$

  0 → Ω^1_{A/Λ} ⊗_A C → Ω^1_{B/Λ} ⊗_B C → Ω^1_{B/A} ⊗_B C → 0.

Finally, by virtue of δ), the $C$-module ${\Omega }_{B/\Lambda }^1{\otimes }_{B}C$ is projective, so the preceding exact sequence is split, which completes the proof of (i).

(ii) If $B$ is a $\Lambda$-algebra formally smooth for the $\mathfrak{J}$-preadic topology, and $A$ is a $\Lambda$-algebra formally smooth for the discrete topology, then $B$ is a $A$-algebra formally smooth for the $\mathfrak{J}$-preadic topology $(0_{IV},\mathrm{19.3.5},(ii))$; the conditions α) and β) therefore result from $(0_{IV},\mathrm{19.5.6})$. Moreover, $(0_{IV},\mathrm{20.7.2})$ shows that $u_{B/A/\Lambda }$ is formally left-invertible, hence $u_{B/A/\Lambda }\otimes 1_C$ is left-invertible $(0_{IV},\mathrm{19.1.7})$, and a fortiori injective.

Finally, ${\Omega }_{B/\Lambda }^1$ is a formally projective $B$-module $(0_{IV},\mathrm{20.4.9})$, and consequently ${\Omega }_{B/\Lambda }^1{\otimes }_{B}C$ is a projective $C$-module $(0_{IV},\mathrm{19.2.4})$.

Corollary (22.1.2).

Let $s:\Lambda \to A$, $u:A\to B$ be two ring homomorphisms, $\mathfrak{m}$ a maximal ideal of $B$, $K=B/\mathfrak{m}$ the quotient field. Suppose that the $\Lambda$-algebras $A$ and $K$ are formally smooth for the discrete topology. Then the following conditions are equivalent:

a) $B$ is a $\Lambda$-algebra formally smooth for the $\mathfrak{m}$-preadic topology.

b) The canonical homomorphism

$$(\mathrm{22.1.2.1})S_K^{\bullet }(\mathfrak{m}/{\mathfrak{m}}^2)\to g{r}_{\mathfrak{m}}^{\bullet }(B)$$

is bijective, and the characteristic homomorphism

$$(\mathrm{22.1.2.2})\chi :{{\rm Y}}_{K/A/\Lambda }\to \mathfrak{m}/{\mathfrak{m}}^2$$

is injective.

This follows immediately from (22.1.1), the conditions α) and δ) being here trivially satisfied, since $K$ is a field.

Remark (22.1.3).

Suppose the hypotheses of (22.1.2) are satisfied and moreover suppose that $\mathfrak{m}/{\mathfrak{m}}^2$ is a $K$-vector space of finite rank. Then, for $B$ to be a $\Lambda$-algebra formally smooth for the $\mathfrak{m}$-preadic topology, it suffices that the following conditions be satisfied:

(22.1.3.1) Given a polynomial algebra $E=K[T_1,\cdots ,T_n]$, the maximal ideal $\mathfrak{n}$ of $E$ generated by the $T_i$ ($1\le i\le n$), and an ideal $\mathfrak{b}\supset {\mathfrak{n}}^k$ of $E$, every $\Lambda$-homomorphism $v:B\to E/\mathfrak{b}$ which, by passage to the quotients, gives the identity $K\to K$, factors as $B\stackrel{w}{\to }E/{\mathfrak{n}}^k\to E/\mathfrak{b}$, where $w$ is a $\Lambda$-homomorphism.

(22.1.3.2) If $F$ is the trivial extension $D_K(K)$ ("algebra of dual numbers over $K$"), $\rho :\Lambda \to F$ a ring homomorphism such that the composite $\Lambda \stackrel{\rho }{\to }F\to K$ is the canonical homomorphism, then every $\Lambda$-homomorphism $v:B\to K$ which, by passage to the quotients, gives the identity $K\to K$, factors as $B\stackrel{w}{\to }F\to K$, where $w$ is a $\Lambda$-homomorphism (for the $\Lambda$-algebra structure on $F$ defined by $\rho$).

Indeed, we have seen $(0_{IV},\mathrm{19.5.5})$ that condition (22.1.3.1) entails that the canonical homomorphism (22.1.2.1) is bijective. By virtue of (22.1.2), it then suffices to see that the canonical homomorphism $u_{B/A/\Lambda }\otimes 1_K:{\Omega }_{A/\Lambda }^1{\otimes }_{A}K\to {\Omega }_{B/\Lambda }^1{\otimes }_{B}K$ is injective, or equivalently (since these are $K$-vector spaces) that for every $K$-vector space $L$, the canonical homomorphism ${\mathrm{Der}}_{\Lambda }(B,L)\to {\mathrm{Der}}_{\Lambda }(A,L)$ is surjective (by virtue of $(0_{IV},\mathrm{20.4.8})$); it is clear that for this it is necessary and sufficient that the canonical homomorphism ${\mathrm{Der}}_{\Lambda }(B,K)\to {\mathrm{Der}}_{\Lambda }(A,K)$ be surjective, whence our assertion, by virtue of $(0_{IV},\mathrm{20.1.5})$.

One notes that the two conditions (22.1.3.1) and (22.1.3.2) are entailed by the following:

(22.1.3.3) For every local Artinian $\Lambda$-algebra $E$ with residue field equal to $K$, and every nilpotent ideal $\mathfrak{r}$ of $E$, every $\Lambda$-homomorphism $B\to E/\mathfrak{r}$ which, by passage to the quotients, gives the identity $K\to K$, factors as $B\stackrel{u}{\to }E\to E/\mathfrak{r}$, where $u$ is a $\Lambda$-homomorphism.

This result generalizes as follows.

Proposition (22.1.4).

Let $\varphi :A\to B$ be a local homomorphism of Noetherian local rings, $k$ (resp. $K$) the residue field of $A$ (resp. $B$). For $B$ to be a $A$-algebra formally smooth (for the preadic topologies), it is necessary and sufficient that for every local Artinian ring $C$ with residue field equal to $K$, every local homomorphism $A\to C$ (making $C$ a topological $A$-algebra), and every nilpotent ideal $\mathfrak{J}$ of $C$, every local $A$-homomorphism $B\to C/\mathfrak{J}$ such that the homomorphism $K\to K$ obtained by passage to the quotients is the identity, factors as $B\stackrel{f}{\to }C\to C/\mathfrak{J}$, where $f$ is a local $A$-homomorphism.

The condition being necessary by definition $(0_{IV},\mathrm{19.3.1})$, we show that it is sufficient. Since the $A$-homomorphisms of $B$ into a discrete local Â-algebra arise by extension from local $A$-homomorphisms of $B$ into this algebra, one may reduce to the case where $A$ and $B$ are complete, by virtue of $(0_{IV},\mathrm{19.3.6})$. When $A$ is a field $k$, the proposition follows from (22.1.3.3), where one takes the prime subfield of $k$ for $\Lambda$, and from $(0_{IV},\mathrm{19.6.1})$. In the general case, set $B_0=B{\otimes }_{A}k=B/\mathfrak{m}B$ ($\mathfrak{m}$ maximal ideal of $A$), which is complete. If C_0 is a local Artinian $k$-algebra with residue field equal to $K$, ${\mathfrak{J}}_0$ a nilpotent ideal of C_0, every local $k$-homomorphism $g_0:B_0\to C_0/{\mathfrak{J}}_0$ giving the identity on $K$ by passage to the quotients, gives by composition a local $A$-homomorphism

  g : B → B_0 → C_0/𝔍_0

which by hypothesis therefore factors as $B\to C_0\to C_0/{\mathfrak{J}}_0$, where $f$ is a local $A$-homomorphism; but since C_0 is a $k$-algebra, $f$ factors as $B\to B_0\stackrel{f_0}{\to }{C}_0$, where $f_0$ is a local $k$-homomorphism. We see therefore that the hypotheses of the statement are still satisfied when one substitutes B_0 and $k$ for $B$ and $A$ respectively; hence B_0 is a $k$-algebra formally smooth according to what we have just seen. Applying $(0_{IV},\mathrm{19.7.2})$ and $(0_{IV},\mathrm{19.7.1})$, we see that there exists a complete Noetherian local ring $B^{\prime }$, a local homomorphism $A\to B^{\prime }$ making $B^{\prime }$ a flat $A$-module and a $A$-algebra formally smooth, and an $A$-isomorphism $u_0:B_0\stackrel{\sim }{\to }{B}^{\prime }{\otimes }_{A}k=B_0^{\prime }$. Let $v_0$ be the inverse isomorphism of $u_0$; we propose to show that there exists a local $A$-homomorphism $v:B\to B^{\prime }$ such that $v_0$ arises from $v$ by passage to the quotients. For this, denote by $\mathfrak{n}$ and ${\mathfrak{n}}^{\prime }$ the respective maximal ideals of $B$ and $B^{\prime }$, and, for each $j$, let $w_j:B/({\mathfrak{n}}^{j+1}+\mathfrak{m}B)\to B^{\prime }/({\mathfrak{n}}^{\prime j+1}+\mathfrak{m}{B}^{\prime })$ be the homomorphism deduced from $v_0$ by passage to the quotients; it will suffice to form for each $j$ a local $A$-homomorphism $v_j:B/{\mathfrak{n}}^{j+1}\to B^{\prime }/{\mathfrak{n}}^{\prime j+1}$ such that the diagrams

  B/𝔫^{j+1} ──v_j──▸ B′/𝔫′^{j+1}              B/𝔫^{j+1} ────v_j────▸ B′/𝔫′^{j+1}
       │                  │                          │                       │
       ▼                  ▼                          ▼                       ▼
  B/𝔫^j ──v_{j-1}──▸ B′/𝔫′^j         B/(𝔫^{j+1} + 𝔪B) ─w_j─▸ B′/(𝔫′^{j+1} + 𝔪B′)

be commutative; $B$ and $B^{\prime }$ being complete, the homomorphism $v=\lim v_j$ will answer the question. Now, the recursive formation of $v_j$ results from the hypothesis on $B$ and from the

same reasoning as in $(0_{IV},\mathrm{19.3.10.1})$, using the lemma $(0_{IV},\mathrm{19.3.10.2})$ and the fact that ${\mathfrak{n}}^{\prime j}+({\mathfrak{n}}^{\prime j+1}+\mathfrak{m}{B}^{\prime })={\mathfrak{n}}^{\prime j}+\mathfrak{m}{B}^{\prime }$. It then follows from $(0_{IV},\mathrm{19.7.1.4})$ that $v$ is an $A$-isomorphism, for $B^{\prime }$ is a flat $A$-module, and $B$ is complete for the $\mathfrak{m}$-preadic topology, the ideals ${\mathfrak{m}}^{j}B$ being closed in $B$ for the $\mathfrak{n}$-adic topology $(0_I,\mathrm{7.3.5})$. Q.E.D.

22.2. Differential characterization of local algebras formally smooth over a field

(22.2.1) Let $k$ be a field, $P$ its prime subfield, $A$ a $k$-algebra which is a local ring, $\mathfrak{m}$ its maximal ideal, $K=A/\mathfrak{m}$ its residue field. Since $K$ is separable over $P$, $K$ is a $P$-algebra formally smooth for the discrete topology $(0_{IV},\mathrm{19.6.1})$, and consequently the $P$-extension $A/{\mathfrak{m}}^2$ of $K$ by $\mathfrak{m}/{\mathfrak{m}}^2$ is $P$-trivial, and the characteristic homomorphism

$$(\mathrm{22.2.1.1}){\chi }_{A/k}:{{\rm Y}}_{K/k}\to \mathfrak{m}/{\mathfrak{m}}^2$$

is therefore defined $(0_{IV},\mathrm{20.6.23})$.

Theorem (22.2.2).

Under the conditions of (22.2.1), for $A$ to be a $k$-algebra formally smooth for the $\mathfrak{m}$-preadic topology, it is necessary and sufficient that the following conditions be satisfied:

(i) The canonical homomorphism $S_K^{\bullet }(\mathfrak{m}/{\mathfrak{m}}^2)\to g{r}_{\mathfrak{m}}^{\bullet }(A)$ is bijective.

(ii) The characteristic homomorphism ${\chi }_{A/k}:{{\rm Y}}_{K/k}\to \mathfrak{m}/{\mathfrak{m}}^2$ is injective.

This is a particular case of (22.1.2), where one replaces $\Lambda$ by $P$, $A$ by $k$, and $B$ by $A$; $K$ and $k$, being separable over $P$, are indeed $P$-algebras formally smooth $(0_{IV},\mathrm{19.6.1})$.

Remark (22.2.3).

Condition (ii) of (22.2.2) may be replaced by any of the following:

a) The canonical homomorphism ${\Omega }_k^1{\otimes }_{k}K\to {\Omega }_A^1{\otimes }_{A}K$ is injective.

b) For every $n$, the canonical homomorphism ${\Omega }_k^1{\otimes }_k(A/{\mathfrak{m}}^{n+1})\to {\Omega }_{A/k}^1{\otimes }_A(A/{\mathfrak{m}}^{n+1})$ is left-invertible.

c) The canonical homomorphism ${\Omega }_k^1{\hat{\otimes }}_{k}A\to {\hat{\Omega }}_A^1$ is left-invertible.

d) $A$ is a $k$-algebra formally smooth (for the $\mathfrak{m}$-preadic topology) relative to the prime field.

Indeed, we have already remarked in (22.1.1) that the injectivity of ${\chi }_{A/k}$ is equivalent to a), and that the conjunction of a) and condition (i) of (22.2.2) is equivalent to that of d) and (i). Finally, we have also seen in (22.1.1) that b) and d) are equivalent, and the equivalence of b) and c) results from $(0_{IV},\mathrm{19.1.8})$.

(22.2.4) The principal application of (22.2.2) concerns local Noetherian and complete $k$-algebras (case where $\mathfrak{m}/{\mathfrak{m}}^2$ is a $K$-vector space of finite rank). More precisely, let $k$ be a field, $K$ an extension of $k$, $V$ a $K$-vector space of

finite rank; we consider the triples $(A,\alpha ,\beta )$, where $A$ is a complete Noetherian local $k$-algebra formally smooth with maximal ideal $\mathfrak{m}$, $\alpha :A/\mathfrak{m}\stackrel{\sim }{\to }K$ an isomorphism of $k$-algebras, $\beta :\mathfrak{m}/{\mathfrak{m}}^2\stackrel{\sim }{\to }V$ a di-isomorphism of vector spaces (for the isomorphism $\alpha$). Given two such triples $(A,\alpha ,\beta )$, $(A^{\prime },{\alpha }^{\prime },{\beta }^{\prime })$, an equivalence of the first onto the second is a $k$-isomorphism $u:A\stackrel{\sim }{\to }{A}^{\prime }$ such that (if ${\mathfrak{m}}^{\prime }$ is the maximal ideal of $A^{\prime }$) the diagrams

       gr^0(u)                         gr^1(u)
  A/𝔪 ────────▸ A′/𝔪′             𝔪/𝔪² ────────▸ 𝔪′/𝔪′²
    │              │                  │                │
   α│              │α′               β│                │β′
    ▼              ▼                  ▼                ▼
    K ───1_K────▸ K                   V ────1_V────▸ V

are commutative. It is immediate that the equivalence classes of triples $(A,\alpha ,\beta )$ form a set, which we denote $FL(K/k,V)$. We now remark that to every triple $(A,\alpha ,\beta )$ is associated the composite $K$-homomorphism of vector spaces ${{\rm Y}}_{K/k}\stackrel{{\chi }_{A/k}}{\to }\mathfrak{m}/{\mathfrak{m}}^2\stackrel{\beta }{\to }V$ (where ${\chi }_{A/k}$ is regarded as a di-homomorphism relative to ${\alpha }^{-1}$); the preceding definition proves (by $(0_{IV},\mathrm{20.6.22.2})$) that this $K$-homomorphism depends only on the equivalence class of $(A,\alpha ,\beta )$ in $FL(K/k,V)$; in other words one has defined a canonical map

  (22.2.4.1)   FL(K/k, V) → Hom_K(Υ_{K/k}, V).

Proposition (22.2.5).

Let $k$ be a field, $K$ an extension of $k$, $V$ a $K$-vector space of finite rank. The canonical map (22.2.4.1) is a bijection of $FL(K/k,V)$ onto the set of injective $K$-homomorphisms of ${{\rm Y}}_{K/k}$ into $V$.

(i) To show that (22.2.4.1) is injective, one must prove the following: let $A$, $A^{\prime }$ be two complete Noetherian local $k$-algebras formally smooth, $\mathfrak{m}$, ${\mathfrak{m}}^{\prime }$ their respective maximal ideals, $\alpha :A/\mathfrak{m}\stackrel{\sim }{\to }K$, ${\alpha }^{\prime }:A^{\prime }/{\mathfrak{m}}^{\prime }\stackrel{\sim }{\to }K$ two $k$-isomorphisms; suppose given two $k$-isomorphisms $v^0:A/\mathfrak{m}\stackrel{\sim }{\to }{A}^{\prime }/{\mathfrak{m}}^{\prime }$, $v^1:\mathfrak{m}/{\mathfrak{m}}^2\stackrel{\sim }{\to }{\mathfrak{m}}^{\prime }/{\mathfrak{m}}^{\prime 2}$ such that the diagrams

                    v^0                                       v^1
  (22.2.5.1)   A/𝔪 ────▸ A′/𝔪′                 𝔪/𝔪² ────▸ 𝔪′/𝔪′²
                ╲       ╱                          ↖             ↗
                 α     α′                        χ_A         χ_{A′}
                  ╲   ╱                              ╲         ╱
                    K                                  Υ_{K/k}

are commutative. It is required to prove that there exists a $k$-isomorphism $u:A\stackrel{\sim }{\to }{A}^{\prime }$ such that $g{r}^0(u)=v^0$ and $g{r}^1(u)=v^1$. Note first that since $K$ is a field, the canonical homomorphism $(0_{IV},\mathrm{20.6.7.1})$ is bijective $(0_{IV},\mathrm{20.6.11})$, hence there already exists a $k$-isomorphism $v:A/{\mathfrak{m}}^2\stackrel{\sim }{\to }{A}^{\prime }/{\mathfrak{m}}^{\prime 2}$ such that $g{r}^0(v)=v^0$ and $g{r}^1(v)=v^1$. Note next that since $A^{\prime }$ is metrisable and complete, and $A$ a $k$-algebra formally smooth, the composite homomorphism $A\to A/{\mathfrak{m}}^2\stackrel{v}{\to }{A}^{\prime }/{\mathfrak{m}}^{\prime 2}$ factors as $A\stackrel{u}{\to }{A}^{\prime }\to A^{\prime }/{\mathfrak{m}}^{\prime 2}$, where $u$ is

a $k$-homomorphism $(0_{IV},\mathrm{19.3.11})$, and we therefore already have $g{r}^0(u)=v^0$ and $g{r}^1(u)=v^1$. Now, one has the commutative diagram

  S_K^•(𝔪/𝔪²)  ────────▸  gr_𝔪^•(A)
       │                         │
       │ w                       │ gr^•(u)
       ▼                         ▼
  S_K^•(𝔪′/𝔪′²) ───────▸  gr_{𝔪′}^•(A′)

where the horizontal arrows are the canonical homomorphisms, and $w$ arises from $v^0$ and $v^1$; since $v^0$ and $v^1$ are bijective, the same holds for $w$, and since $A$ and $A^{\prime }$ are $k$-algebras formally smooth, one deduces from (22.2.2, (i)) that $g{r}^{\bullet }(u)$ is bijective. But since $A$ and $A^{\prime }$ are separated and complete, this entails that $u$ itself is bijective (Bourbaki, Alg. comm., chap. III, §2, n° 8, cor. 3 of th. 1).

(ii) It remains to show that the image of (22.2.4.1) is the set of injective homomorphisms of ${{\rm Y}}_{K/k}$ into $V$; we already know by (22.2.2, (ii)) that it is contained in this set; it will then suffice to prove the

Lemma (22.2.5.2).

For every $K$-homomorphism $h:{{\rm Y}}_{K/k}\to V$, there exists a $k$-algebra $A$ which is a complete regular Noetherian local ring, with maximal ideal $\mathfrak{m}$, residue field $K$, and a $K$-isomorphism $\beta :\mathfrak{m}/{\mathfrak{m}}^2\stackrel{\sim }{\to }V$ such that $h=\beta \circ {\chi }_{A/k}$.

Indeed, if this lemma is proved, the fact that $A$ is regular entails that condition (i) of (22.2.2) is satisfied $(0_{IV},\mathrm{17.1.1})$; if moreover $h$ is injective, (22.2.2) will show that $A$ is a $k$-algebra formally smooth whose class will have $h$ for image (on the other hand, if $h$ is not injective, the $k$-algebra $A$ whose existence (22.2.5.2) proves is not formally smooth).

To prove (22.2.5.2), let $B$ be the $K$-algebra $S_K^{\bullet }(V)$, $\mathfrak{n}$ the augmentation ideal of $B$, and $A=\hat{B}$ the completed $K$-algebra of $B$ for the $\mathfrak{n}$-preadic topology (so that $A$ is isomorphic to the algebra of formal series $K[[T_1,\cdots ,T_n]]$ if $n=r{g}_K(V)$). By virtue of $(0_{IV},\mathrm{20.6.11})$ there exists on $A/{\mathfrak{m}}^2$ (where $\mathfrak{m}=\mathfrak{n}A$) a structure of $k$-extension of $K$ by $V$ (distinct in general from the $k$-trivial structure defined by the canonical injection $k\to K\to A$) such that $h$ is the characteristic homomorphism of this extension. If $f:k\to A/{\mathfrak{m}}^2$ is the structural homomorphism, the fact that $k$ is separable over the prime field permits $(0_{IV},\mathrm{19.6.1})$ factoring $f$ as $k\stackrel{g}{\to }A\to A/{\mathfrak{m}}^2$, and the $k$-algebra $A$ so defined answers the question $(0_{IV},\mathrm{20.6.22.2})$.

Theorem (22.2.6).

Let $k$ be a field, $K$ an extension of $k$. For there to exist a Noetherian local $k$-algebra $A$, with maximal ideal $\mathfrak{m}$, such that $A/\mathfrak{m}=K$ and $A$ is formally smooth (for its $\mathfrak{m}$-preadic topology), it is necessary and sufficient that ${{\rm Y}}_{K/k}$ be a $K$-vector space of finite rank. The $k$-algebra $A$ is then determined (up to a $k$-isomorphism giving by passage to the quotients the identity $K\to K$) by its dimension, which is subject only to the condition $\ge r{g}_K({{\rm Y}}_{K/k})$.

Since $\mathfrak{m}/{\mathfrak{m}}^2$ is of finite rank over $K$, the necessity of the condition follows from (22.2.2, (ii)); conversely, (22.2.5) shows that the condition is sufficient and that one may take for $\mathfrak{m}/{\mathfrak{m}}^2$ a $K$-vector space of arbitrary rank $\ge r{g}_K({{\rm Y}}_{K/k})$.

In particular, the identity homomorphism of ${{\rm Y}}_{K/k}$ canonically associates to $K$ a complete Noetherian $k$-algebra, formally smooth, for which ${{\rm Y}}_{K/k}$ is isomorphic to $\mathfrak{m}/{\mathfrak{m}}^2$.

Remarks (22.2.7).

(i) Let $\Lambda$ be a Noetherian local ring with residue field $k$; it follows from $(0_{IV},\mathrm{19.7.1})$ and $(0_{IV},\mathrm{19.7.2})$ that the determination of the $\Lambda$-algebras $A$ which are complete Noetherian local rings and which are formally smooth is equivalent to the same problem where $\Lambda$ is replaced by $k$, and is thus in principle entirely resolved by (22.2.5), which reduces the question to the structure of the residue fields of the sought-for $k$-algebras, as extensions of $k$.

(ii) The fact that ${{\rm Y}}_{K/k}$ is of finite rank over $K$ does not entail that $K$ is "of finite radicial multiplicity" over $k$ (cf. $(0_{IV},\mathrm{19.6.6})$). For example, let $k_0$ be a perfect field of characteristic $p>0$, $k=k_0(T)$ the field of rational fractions in one indeterminate over $k_0$, and $K=k^{p^{-\infty }}$ the smallest perfect extension of $k$. One then has ${\Omega }_k^1={\Omega }_{k/k_0}^1=0$ $(0_{IV},\mathrm{21.4.4})$, hence by definition $(0_{IV},\mathrm{20.6.1.1})$ ${{\rm Y}}_{K/k}={\Omega }_K^1{\otimes }_{k}K$, and since $k^p=k_0(T^p)$, $T$ is a $p$-basis of $k$ over $k^p$, hence $(0_{IV},\mathrm{21.4.5})$, ${\Omega }_K^1$ is of rank 1 over $k$, and consequently ${{\rm Y}}_{K/k}$ of rank 1 over $K$. But for every $n$ the residue field of the local ring $K{\otimes }_k{k}^{p^{-n}}$ is isomorphic to $K$ and is therefore not separable over $k^{p^{-n}}$.

The theorem (22.2.2) generalizes as follows.

Proposition (22.2.8).

Under the conditions of (22.2.1), suppose that the characteristic $p$ of $k$ is > 0; let $(k_{\alpha })$ be a decreasing filtered family of subfields of $k$, such that ${\bigcap }_{\alpha }{k}_{\alpha }(k^p)=k^p$. Then condition (ii) of (22.2.2) can be replaced by any of the following:

a) There exists ${\alpha }_0$ such that the canonical homomorphism

  Ω^1_{k/k_α} ⊗_k K → Ω^1_{A/k_α} ⊗_A K

is injective for every $\alpha \ge {\alpha }_0$.

b) There exists ${\alpha }_0$ such that $A$ is a $k$-algebra formally smooth (for the $\mathfrak{m}$-preadic topology) relative to $k_{\alpha }$, for every $\alpha \ge {\alpha }_0$.

c) For every $\alpha$ and every integer $n\ge 0$, the canonical homomorphism

  Ω^1_{k/k_α} ⊗_k (A/𝔪^{n+1}) → Ω^1_{A/k_α} ⊗_A (A/𝔪^{n+1})

is left-invertible.

Indeed, if $A$ is a $k$-algebra formally smooth, it is so a fortiori relative to any subfield of $k$, which entails c) by virtue of $(0_{IV},\mathrm{20.7.2})$ and $(0_{IV},\mathrm{19.1.7})$; it is clear that c) implies b), by virtue of $(0_{IV},\mathrm{20.7.2})$, and that b) implies a). Finally, taking account of (22.2.3), it remains to prove that a) entails that the canonical homomorphism $u:{\Omega }_k^1{\otimes }_{k}K\to {\Omega }_A^1{\otimes }_{A}K$ is injective. Now, for every $\alpha$ one has a commutative diagram

  Ω^1_k ⊗_k K ──────u──────▸ Ω^1_A ⊗_A K
       │                            │
       │ v_α                        │
       ▼                            ▼
  Ω^1_{k/k_α} ⊗_k K ──u_α──▸ Ω^1_{A/k_α} ⊗_A K

and the hypothesis that $u_{\alpha }$ is injective for $\alpha \ge {\alpha }_0$ implies that $Ker(u)$ is contained in the intersection of the $Ker(v_{\alpha })$. But we have seen $(0_{IV},\mathrm{21.8.3})$ that this intersection is zero by virtue of the hypothesis ${\bigcap }_{\alpha }{k}_{\alpha }(k^p)=k^p$, which completes the proof.

Proposition (22.2.9).

Under the hypotheses of (22.2.1), the following conditions are equivalent:

a) $A/{\mathfrak{m}}^2$ is a $k$-trivial $k$-extension of $K$ by $\mathfrak{m}/{\mathfrak{m}}^2$.

b) ${\chi }_{A/k}=0$.

c) The canonical homomorphism (0_IV, 20.5.11.2)

  δ_{K/A/k} : 𝔪/𝔪² → Ω^1_A ⊗_A K

is injective (in other words, the sequence

  (22.2.9.1)   0 → 𝔪/𝔪² → Ω^1_A ⊗_A K → Ω^1_K → 0

is exact).

The equivalence of b) and c) results from the exact sequence $(0_{IV},\mathrm{20.6.22.1})$; the equivalence of c) and a) results from $(0_{IV},\mathrm{20.5.12})$, the sequence (22.2.9.1) being split if it is exact, since these are vector spaces.

One notes that if $K$ is separable over $k$, the equivalent conditions of (22.2.9) are satisfied, by virtue of the definition of ${\chi }_{A/k}$ (22.2.1) and of $(0_{IV},\mathrm{20.6.3})$.

Corollary (22.2.10).

Under the hypotheses of (22.2.1), consider a subfield $k_0$ of $k$. The following conditions are equivalent:

a) $A/{\mathfrak{m}}^2$ is a $k_0$-trivial $k$-extension of $K$ by $\mathfrak{m}/{\mathfrak{m}}^2$.

b) The composite homomorphism ${{\rm Y}}_{K/k_0}\to {{\rm Y}}_{K/k}\stackrel{{\chi }_{A/k}}{\to }\mathfrak{m}/{\mathfrak{m}}^2$ is zero.

c) The characteristic homomorphism ${\chi }_{A/k}:{{\rm Y}}_{K/k}\to \mathfrak{m}/{\mathfrak{m}}^2$ factors as ${{\rm Y}}_{K/k}\to {{\rm Y}}_{K/k/k_0}\stackrel{{\chi }^{\prime }}{\to }\mathfrak{m}/{\mathfrak{m}}^2$.

Moreover, when these conditions are fulfilled, the homomorphism ${\chi }^{\prime }$ is uniquely determined and coincides with the characteristic homomorphism of the $k_0$-trivial $k$-extension $A/{\mathfrak{m}}^2$ of $K$ by $\mathfrak{m}/{\mathfrak{m}}^2$.

Since the composite in b) is none other than ${\chi }_{A/k_0}$, the equivalence of a) and b) results from (22.2.9); on the other hand, b) and c) are equivalent by virtue of the exact sequence $(0_{IV},\mathrm{20.6.18.1})$

$${{\rm Y}}_{K/k_0}\to {{\rm Y}}_{K/k}\to {{\rm Y}}_{K/k/k_0}\to 0$$

and since the canonical homomorphism ${{\rm Y}}_{K/k}\to {{\rm Y}}_{K/k/k_0}$ is surjective, this implies the uniqueness of ${\chi }^{\prime }$; the fact that ${\chi }^{\prime }$ is the characteristic homomorphism of the $k_0$-extension $A/{\mathfrak{m}}^2$ results from $(0_{IV},\mathrm{20.6.22.2})$.

Corollary (22.2.11).

Under the hypotheses of (22.2.1), let $p$ be the characteristic exponent of $k$, and let $(k_{\alpha })$ be a decreasing filtered family of subfields of $k$ such that ${\bigcap }_{\alpha }{k}_{\alpha }(k^p)=k^p$. If ${{\rm Y}}_{K/k}$ is of finite rank over $K$, there exists an index $\alpha$ such that $A/{\mathfrak{m}}^2$ is a $k_{\alpha }$-trivial $k$-extension of $K$ by $\mathfrak{m}/{\mathfrak{m}}^2$.

We know $(0_{IV},\mathrm{21.8.3})$ that there exists then an index $\alpha$ such that the canonical homomorphism ${{\rm Y}}_{K/k}\to {{\rm Y}}_{K/k/k_{\alpha }}$ is bijective, hence condition b) of (22.2.10) (where $k_{\alpha }$ replaces $k_0$) is satisfied.

22.3. Application to the relations between certain local rings and their completions

Lemma (22.3.1).

Let A_0 be a Noetherian semi-local ring, $A$ a finite A_0-algebra (which is thus a Noetherian semi-local ring, cf. Bourbaki, Alg. comm., chap. IV, §2, n° 5, cor. 3 of prop. 9). If ${\hat{A}}_0$ and Â are the completions of A_0 and $A$ for their respective preadic topologies, then, when one equips A_0, $A$ and Â with the discrete topologies, Â is a $A$-algebra formally smooth relative to A_0 $(0_{IV},\mathrm{19.9.1})$.

We know indeed that $\hat{A}=A{\otimes }_{A_0}{\hat{A}}_0$ (Bourbaki, Alg. comm., chap. IV, §2, n° 5, cor. 3 of prop. 9 and chap. III, §3, n° 4, th. 1), so the proposition results from $(0_{IV},\mathrm{19.9.3})$.

Proposition (22.3.2).

Let $A$ be a regular Noetherian local ring, $K$ its field of fractions, $p$ the characteristic of $K$. Suppose one of the following hypotheses is satisfied:

(i) $p=0$.

(ii) $p>0$ and there exists a decreasing filtered family $(A_{\alpha })$ of Noetherian subrings of $A$, such that $A$ is a finite $A_{\alpha }$-algebra for every $\alpha$, that $A^{p^{h(\alpha )}}\subset A_{\alpha }$ for a suitable integer $h(\alpha )$, and that, if $K_{\alpha }$ denotes the field of fractions of $A_{\alpha }$, one has ${\bigcap }_{\alpha }{K}_{\alpha }(K^p)=K^p$.

Let then $B$ be an integral finite $A$-algebra containing $A$, $\mathfrak{n}$ a prime ideal of $B$, $C$ the local ring $B_{\mathfrak{n}}$, $\mathfrak{q}$ a prime ideal of the completion Ĉ such that $\mathfrak{q}\cap C=0$, so that the local ring ${\hat{C}}_{\mathfrak{q}}$ is an algebra over the field of fractions $L$ of $B$. Then ${\hat{C}}_{\mathfrak{q}}$ is an $L$-algebra formally smooth for its $\mathfrak{q}$-adic topology, and consequently a geometrically regular ring over $L$ $(0_{IV},\mathrm{19.6.5})$.

Let us distinguish two cases.

I) Suppose that $\mathfrak{n}$ contains the maximal ideal $\mathfrak{m}$ of $A$, and consequently $\mathfrak{n}\cap A=\mathfrak{m}$. Then $\mathfrak{n}$ is maximal (Bourbaki, Alg. comm., chap. V, §2, n° 1, prop. 1) and if $\mathfrak{r}$ is the radical of the semi-local ring $B$, Ĉ is the separated completion of $B$ for the $\mathfrak{n}$-preadic topology, and one of the components of the semi-local ring $\hat{B}$, the completion of $B$ for the $\mathfrak{r}$-preadic topology (Bourbaki, Alg. comm., chap. III, §3, n° 4, prop. 8); one may therefore write ${\hat{C}}_{\mathfrak{q}}={\hat{B}}_{{\mathfrak{q}}^{\prime }}$, where ${\mathfrak{q}}^{\prime }$ is the prime ideal of $\hat{B}$ inverse image of $\mathfrak{q}$. Note now that $B$ is a subring of $\hat{B}$ and the hypothesis $\mathfrak{q}\cap C=0$ entails ${\mathfrak{q}}^{\prime }\cap B=0$, hence ${\hat{B}}_{{\mathfrak{q}}^{\prime }}$ is also the localization of the ring $\hat{B}{\otimes }_{B}L$ at one of its prime ideals ${\mathfrak{q}}^{\prime \prime }$, of which ${\mathfrak{q}}^{\prime }$ is the inverse image in $\hat{B}$. Moreover one has $\hat{B}=\hat{A}{\otimes }_{A}B$ (Bourbaki, Alg. comm., chap. IV, §2, n° 5, cor. of prop. 9 and chap. III, §3, n° 4, th. 1), hence $\hat{B}{\otimes }_{B}L=\hat{A}{\otimes }_{A}L$; finally, if $\mathfrak{p}$ is the prime ideal of Â, inverse image of ${\mathfrak{q}}^{\prime \prime }$, the fact that $A\to B$ is injective and the commutativity of the diagram

  B ────▸ B̂
  ▲       ▲
  │       │
  A ────▸ Â

entail that $\mathfrak{p}\cap A=0$, hence $(\hat{A}{\otimes }_{A}L{)}_{{\mathfrak{q}}^{\prime \prime }}$ is also localized of the ring

  Â ⊗_A L = (Â ⊗_A K) ⊗_K L

at one of its prime ideals. To show that this local ring is an $L$-algebra formally smooth, it therefore suffices to prove that ${\hat{A}}_{\mathfrak{p}}{\otimes }_{A}K$ is a $K$-algebra formally smooth $(0_{IV},\mathrm{19.3.5},(iii)and(iv))$; moreover ${\hat{A}}_{\mathfrak{p}}{\otimes }_{A}K={\hat{A}}_{\mathfrak{p}}$ since $\mathfrak{p}\cap A=0$. We are going to apply the criteria (22.2.2) and (22.2.8). In the first place, since $A$ is regular, so is Â $(0_{IV},\mathrm{17.1.5})$, hence so is ${\hat{A}}_{\mathfrak{p}}$ $(0_{IV},\mathrm{17.3.2})$, and condition (i) of (22.2.2) is satisfied $(0_{IV},\mathrm{17.1.1})$. It therefore suffices (when $p>0$) to verify condition b) of (22.2.8). Now, (22.3.1) shows that, for every $\alpha$, Â is an $A$-algebra formally smooth relative to $A_{\alpha }$ (for the discrete topologies); consequently $\hat{A}{\otimes }_{A_{\alpha }}{K}_{\alpha }=\hat{A}{\otimes }_A(A{\otimes }_{A_{\alpha }}{K}_{\alpha })$ is a $(A{\otimes }_{A_{\alpha }}{K}_{\alpha })$-algebra formally smooth relative to $K_{\alpha }$, for the discrete topologies $(0_{IV},\mathrm{19.9.2},(iii))$. But since $A$ is a finite $A_{\alpha }$-algebra, one has $A{\otimes }_{A_{\alpha }}{K}_{\alpha }=K$; since ${\hat{A}}_{\mathfrak{p}}$ is a localization of $\hat{A}{\otimes }_{A}K$, one concludes $(0_{IV},\mathrm{19.9.2},(iv))$ that ${\hat{A}}_{\mathfrak{p}}$ is a $K$-algebra formally smooth relative to $K_{\alpha }$, for the discrete topology, and a fortiori for its $\mathfrak{p}$-adic topology ($(0_{IV},\mathrm{19.3.8})$ and $(0_{IV},\mathrm{19.9.5})$). But this is precisely condition b) of (22.2.8), hence the proposition is proved in this case for $p>0$. When $p=0$, the residue field of ${\hat{A}}_{\mathfrak{p}}$ is separable over $K$, and since ${\hat{A}}_{\mathfrak{p}}$ is a regular ring, it is a $K$-algebra formally smooth by virtue of $(0_{IV},\mathrm{19.6.4})$.

II) General case. Let $\mathfrak{s}$ be the prime ideal $\mathfrak{n}\cap A$ of $A$, and set $S=A-\mathfrak{s}$, so that $S^{-1}A=A_{\mathfrak{s}}$, and one has $C=(S^{-1}B{)}_{S^{-1}\mathfrak{n}}$, where this time $S^{-1}\mathfrak{n}$ contains the maximal ideal $\mathfrak{s}{A}_{\mathfrak{s}}$ of $A_{\mathfrak{s}}$. Since $A_{\mathfrak{s}}$ is regular $(0_{IV},\mathrm{17.3.2})$ and $S^{-1}B$ is an integral finite $A_{\mathfrak{s}}$-algebra containing $A_{\mathfrak{s}}$, all reduces to verifying condition (ii) for $A_{\mathfrak{s}}$ when $p>0$: now, set ${\mathfrak{s}}_{\alpha }=\mathfrak{s}\cap A_{\alpha }$ and consider the local ring $(A_{\alpha }{)}_{{\mathfrak{s}}_{\alpha }}$. For every $t\notin \mathfrak{s}$, we have by hypothesis, on setting $q=p^{h(\alpha )}$, $t^q\in A_{\alpha }$ and $t\notin \mathfrak{s}$, hence $t^q\notin {\mathfrak{s}}_{\alpha }$; if one sets $S_{\alpha }=A_{\alpha }-{\mathfrak{s}}_{\alpha }$, one sees that one has $A_{\mathfrak{s}}=S_{\alpha }^{-1}A$, and the hypothesis entails that $A_{\mathfrak{s}}$ is a finite $(A_{\alpha }{)}_{{\mathfrak{s}}_{\alpha }}$-algebra; moreover, $K_{\alpha }$ is also the field of fractions of $(A_{\alpha }{)}_{{\mathfrak{s}}_{\alpha }}$, which completes the proof.

Theorem (22.3.3).

Let $A$ be a complete Noetherian local ring, $\mathfrak{p}$ a prime ideal of $A$, $B$ the localized ring $A_{\mathfrak{p}}$, $\mathfrak{q}$ a prime ideal of the completion $\hat{B}$, $\mathfrak{r}=\mathfrak{q}\cap B$. Then the localized ring ${\hat{B}}_{\mathfrak{q}}$ of $\hat{B}$ is a $B_{\mathfrak{r}}$-algebra formally smooth for the preadic topologies.

The prime ideal $\mathfrak{r}$ of $B$ is of the form ${\mathfrak{n}}_{\mathfrak{p}}$, where $\mathfrak{n}\subset \mathfrak{p}$ is a prime ideal of $A$, and one has $B_{\mathfrak{r}}=A_{\mathfrak{n}}$; let $k$ be the residue field of $B_{\mathfrak{r}}$, which is also the field of fractions of the complete integral local ring $A^{\prime }=A/\mathfrak{n}$. Since $\hat{B}$ is a flat $B$-module $(0_I,\mathrm{7.3.5})$, ${\hat{B}}_{\mathfrak{q}}$ is a flat $B_{\mathfrak{r}}$-module $(0_I,\mathrm{6.3.2})$, and to prove that ${\hat{B}}_{\mathfrak{q}}$ is a $B_{\mathfrak{r}}$-algebra formally smooth, it suffices to prove that ${\hat{B}}_{\mathfrak{q}}{\otimes }_{B}k={\hat{B}}_{\mathfrak{q}}/\mathfrak{r}{\hat{B}}_{\mathfrak{q}}$ is a $k$-algebra formally smooth $(0_{IV},\mathrm{19.7.1})$. Now ${\hat{B}}_{\mathfrak{q}}/\mathfrak{r}{\hat{B}}_{\mathfrak{q}}=(\hat{B}/\mathfrak{r}\hat{B}{)}_{{\mathfrak{q}}^{\prime }}$, where ${\mathfrak{q}}^{\prime }$ is the canonical image of $\mathfrak{q}$ in $\hat{B}/\mathfrak{r}\hat{B}$; one has by definition ${\mathfrak{q}}^{\prime }\cap (B/\mathfrak{r})=0$ and $B/\mathfrak{r}=A_{{\mathfrak{p}}^{\prime }}$, where ${\mathfrak{p}}^{\prime }=\mathfrak{p}/\mathfrak{n}$; moreover $\hat{B}/\mathfrak{r}\hat{B}$ is the completion of $B/\mathfrak{r}$.

We are therefore reduced to proving the following corollary.

Corollary (22.3.4).

Let $A$ be a complete integral Noetherian local ring, $K$ its field of fractions, $\mathfrak{p}$ a prime ideal of $A$, $B$ the localized ring $A_{\mathfrak{p}}$. Then, for every prime ideal $\mathfrak{q}$

of the completion $\hat{B}$, such that $\mathfrak{q}\cap B=0$, the local ring ${\hat{B}}_{\mathfrak{q}}$ is a $K$-algebra formally smooth for its $\mathfrak{q}$-adic topology, hence a geometrically regular ring over $K$.

Indeed, it follows from $(0_{IV},\mathrm{19.8.9})$ that there exists a subring A_0 of $A$ which is a ring of formal series over a field of characteristic $p>0$ or over a Cohen ring (recall that Cohen rings have a field of fractions of characteristic 0), and is such that $A$ is a finite A_0-algebra. Now, one knows that the ring A_0 is regular $(0_{IV},\mathrm{17.3.8})$ and verifies one of the hypotheses (i), (ii) of (22.3.2), by virtue of $(0_{IV},\mathrm{21.8.8})$; it suffices therefore to apply (22.3.2), replacing $B$ by $A$ and $A$ by A_0.

22.4. Preliminary results on finite extensions of local rings whose maximal ideal has square zero

Proposition (22.4.1).

Let $A$ be a local ring whose maximal ideal $\mathfrak{m}$ has square zero, $K=A/\mathfrak{m}$ its residue field; denote by $V$ the ideal $\mathfrak{m}$ regarded as a $K$-vector space. Let $T_i$ ($1\le i\le r$) be indeterminates, and for each $i$, let $F_i$ be a unitary polynomial of $A[T_i]$ of degree $d_i$; denote by $A^{\prime }$ the quotient of the polynomial ring $A[T_1,\cdots ,T_r]$ by the ideal $\mathfrak{b}$ generated by the $r$ polynomials $F_i$. Suppose that $A^{\prime }$ is a local ring, and let ${\mathfrak{m}}^{\prime }$ be its maximal ideal, $K^{\prime }=A^{\prime }/{\mathfrak{m}}^{\prime }$ its residue field. Then:

(i) If one sets $d={\prod }_{i=1}^r{d}_i$, $A^{\prime }$ is a free $A$-module of rank $d$, and one has $[K^{\prime }:K]\le d$.

(ii) For $[K^{\prime }:K]=d$, it is necessary and sufficient that $A^{\prime }{\otimes }_{A}K=A^{\prime }/\mathfrak{m}{A}^{\prime }$ be a field (necessarily isomorphic to $K^{\prime }$), in other words, that ${\mathfrak{m}}^{\prime }=\mathfrak{m}{A}^{\prime }$. One has in this case ${\mathfrak{m}}^{\prime 2}=0$, and if $V^{\prime }$ denotes the ideal ${\mathfrak{m}}^{\prime }$ regarded as a $K^{\prime }$-vector space, the canonical homomorphism $V{\otimes }_K{K}^{\prime }\to V^{\prime }$ is bijective (and consequently $r{g}_{K^{\prime }}{V}^{\prime }=r{g}_{K}V$).

Without supposing $A^{\prime }$ local, it is evident, by Euclidean division and induction on $r$, that if $t_i$ is the class of $T_i$ mod. $\mathfrak{b}$, the monomials $M_{\nu }=t_1^{{\nu }_1}\cdots t_r^{{\nu }_r}$, where $0\le {\nu }_i\le d_i-1$, form a basis of the $A$-module $A^{\prime }$. If $A^{\prime }$ is supposed local, $K^{\prime }$ is a quotient of $A^{\prime }/\mathfrak{m}{A}^{\prime }=A^{\prime }{\otimes }_{A}K$, hence $[K^{\prime }:K]\le [A^{\prime }{\otimes }_{A}K:K]=d$, and there can be equality only if ${\mathfrak{m}}^{\prime }=\mathfrak{m}{A}^{\prime }$, which proves (i) and the first assertion of (ii). On the other hand, it is clear that when ${\mathfrak{m}}^{\prime }=\mathfrak{m}{A}^{\prime }$, one has ${\mathfrak{m}}^{\prime 2}=0$, and $V{\otimes }_K{K}^{\prime }=\mathfrak{m}{\otimes }_A{A}^{\prime }=\mathfrak{m}{A}^{\prime }=V^{\prime }$ since $A^{\prime }$ is a free $A$-module.

Remark (22.4.2).

When $A$ is Artinian (in other words $r{g}_K(V)$ finite), the hypothesis that $A^{\prime }$ is local always entails that $r{g}_{K^{\prime }}({\mathfrak{m}}^{\prime }/{\mathfrak{m}}^{\prime 2})\ge r{g}_K(V)$, as we shall see later (22.5.2). Note that these two ranks can be equal without one having ${\mathfrak{m}}^{\prime }=\mathfrak{m}{A}^{\prime }$.

Proposition (22.4.3).

The hypotheses on $A$ and the notations being those of (22.4.1), suppose that the $F_i$ are of the form

  (22.4.3.1)   F_i(T_i) = T_i^{d_i} + ∑_{1 ≤ k ≤ d_i} c_{ik} T_i^{d_i − k}

where $d_i\ge 2$ for every $i$, and $c_{ik}\in \mathfrak{m}$ ("Eisenstein polynomials"). Then:

(i) $A^{\prime }$ is a local ring, and its residue field $K^{\prime }=A^{\prime }/{\mathfrak{m}}^{\prime }$ is isomorphic to $K$.

(ii) The kernel of the canonical homomorphism $V\to V^{\prime }={\mathfrak{m}}^{\prime }/{\mathfrak{m}}^{\prime 2}$ is the ideal $\mathfrak{n}$ of $A$ generated by the ${\xi }_i=c_{i,d_i}$ ("constant terms" of the $F_i$), and the cokernel of this homomorphism has for basis over $K^{\prime }$ the images of the $T_i$; in particular, one has

  (22.4.3.2)   rg_{K′}(V′) = rg_K(V) + r − r′

where $r^{\prime }=r{g}_K(\mathfrak{n})$. When $r{g}_K(V)$ is finite (in other words when $A$ is Artinian), for $r{g}_{K^{\prime }}(V^{\prime })=r{g}_K(V)$, it is necessary and sufficient that the ${\xi }_i$ be linearly independent over $K$.

(i) It is clear that $A^{\prime }{\otimes }_{A}K=A^{\prime }/\mathfrak{m}{A}^{\prime }$ is isomorphic to $K[T_1,\cdots ,T_r]/{\mathfrak{b}}^{\prime }$, where ${\mathfrak{b}}^{\prime }$ is the ideal generated by the $r$ polynomials $T_i^{d_i}$; this is therefore a local ring of residue field $K$, whose maximal ideal is generated by the classes of the $T_i$ mod. ${\mathfrak{b}}^{\prime }$, whence (i).

(ii) Let $t_i$ ($1\le i\le r$) be the class of $T_i$ mod. $\mathfrak{b}$; it follows from (i) that the maximal ideal ${\mathfrak{m}}^{\prime }$ of $A^{\prime }$ is given by

  𝔪′ = 𝔪 A′ + ∑_{i=1}^r t_i A′

whence, taking account of ${\mathfrak{m}}^2=0$,

  𝔪′² = ∑_{i,j} t_i t_j A′ + ∑_i t_i 𝔪 A′.

One concludes that $A^{\prime }/{\mathfrak{m}}^{\prime 2}$ is isomorphic to the ring $A[T_1,\cdots ,T_r]/\mathfrak{c}$, where $\mathfrak{c}$ is the ideal generated by the elements

  F_i  (1 ≤ i ≤ r),   T_i T_j  (1 ≤ i ≤ r, 1 ≤ j ≤ r),   x T_i  (1 ≤ i ≤ r, x ∈ 𝔪).

Since $d_i\ge 2$ and the $T_i^{d_i}$ belong to $\mathfrak{c}$, the hypotheses made entail that $\mathfrak{c}$ is also generated by the elements

  ξ_i  (1 ≤ i ≤ r),   T_i T_j  (1 ≤ i ≤ r, 1 ≤ j ≤ r),   T_i x  (1 ≤ i ≤ r, x ∈ 𝔪).

Let ${\mathfrak{c}}_1={\sum }_{i,j}{T}_i{T}_{j}A[T_1,\cdots ,T_r]$, ${\mathfrak{c}}_2={\mathfrak{c}}_1+{\sum }_i\mathfrak{m}{T}_{i}A[T_1,\cdots ,T_r]$, so that $\mathfrak{c}={\mathfrak{c}}_2+{\sum }_i{\xi }_{i}A[T_1,\cdots ,T_r]$. It is clear that $A[T_1,\cdots ,T_r]/{\mathfrak{c}}_1$ is the direct sum of $A$ and the $A{t}_i^{(1)}$, where $t_i^{(1)}$ is the class of $T_i$ mod. ${\mathfrak{c}}_1$. One deduces that $A[T_1,\cdots ,T_r]/{\mathfrak{c}}_2$ is the direct sum of $A$ and the $K{t}_i^{(2)}$, where $t_i^{(2)}$ is the class of $T_i$ mod. ${\mathfrak{c}}_2$. Finally $A[T_1,\cdots ,T_r]/\mathfrak{c}$ is the direct sum of $A/\mathfrak{n}$ and the $K{t}_i^{\prime }$, where $t_i^{\prime }$ is the class of $T_i$ mod. $\mathfrak{c}$, and $\mathfrak{n}$ is the ideal of $A$ generated by the ${\xi }_i$; in the identification of $A^{\prime }/{\mathfrak{m}}^{\prime 2}$ with $A[T_1,\cdots ,T_r]/\mathfrak{c}$, the image of $V$ identifies with $\mathfrak{m}/\mathfrak{n}$, and what precedes therefore proves (ii).

Proposition (22.4.4).

The hypotheses on $A$ and the notations being those of (22.4.1), suppose that $K$ is of characteristic $p>0$ (hence $p\cdot 1\in \mathfrak{m}$ in $A$) and that the $F_i$ are of the form

  (22.4.4.1)   F_i(T_i) = T_i^p + p ∑_{k=1}^{p−1} c_{ik} T_i^{p−k} − f_i

with $c_{ik}\in A$ and $f_i\in A$ for every $i$ and every $k$ such that $1\le k\le p-1$; moreover, if $a_i$ is the class of $f_i$ mod. $\mathfrak{m}$, suppose that the family $(a_i{)}_{1\le i\le r}$ is $p$-free over $K^p$ ("absolutely $p$-free"). Then:

(i) $A^{\prime }$ is a local ring, of maximal ideal $\mathfrak{m}{A}^{\prime }$, whose residue field $K^{\prime }$ is isomorphic to the extension of $K$ obtained by adjunction of the $a_i^{1/p}$ ($1\le i\le r$).

(ii) The elements $d_A{f}_i\otimes 1_{K^{\prime }}$ form a basis of the kernel $N={{\rm Y}}_{A^{\prime }/A}^{K^{\prime }}$ of the canonical homomorphism ${\Omega }_A^1{\otimes }_A{K}^{\prime }\to {\Omega }_{A^{\prime }}^1{\otimes }_{A^{\prime }}{K}^{\prime }$.

(i) This time $A^{\prime }{\otimes }_{A}K=A^{\prime }/\mathfrak{m}{A}^{\prime }$ is isomorphic to $K[T_1,\cdots ,T_r]/{\mathfrak{b}}^{\prime }$, where ${\mathfrak{b}}^{\prime }$ is the ideal generated by the polynomials $T_i^p-a_i$; the hypothesis on the $a_i$ entails that this quotient ring is a field $(0_{IV},\mathrm{21.1.8})$, whence the assertions of (i).

Before proving (ii), we establish the

Lemma (22.4.4.2).

Let $\Lambda$ be a ring, $A$ a $\Lambda$-algebra, $\mathfrak{J}$ an ideal of $A$ such that the ring $C=A/\mathfrak{J}$ is of characteristic $p>0$ (which is equivalent to saying that $p\cdot 1\in \mathfrak{J}$ in $A$). Let $\pi$ be the canonical image of $p\cdot 1$ in $\mathfrak{J}/{\mathfrak{J}}^2$. If $C$ is a $(\Lambda /p\Lambda )$-algebra formally smooth for the discrete topologies, one has a canonical exact sequence

  (22.4.4.3)   0 → (𝔍/𝔍²)/C · π → Ω^1_{A/Λ} ⊗_A C → Ω^1_{C/Λ} → 0.

Set indeed $A^{\prime }=A/pA$, ${\mathfrak{J}}^{\prime }=\mathfrak{J}/pA$, so that $C=A^{\prime }/{\mathfrak{J}}^{\prime }$. Since $A^{\prime }=A{\otimes }_{\Lambda }(\Lambda /p\Lambda )$, one has ${\Omega }_{A/\Lambda }^1{\otimes }_A{A}^{\prime }={\Omega }_{A^{\prime }/{\Lambda }^{\prime }}^1$ up to a canonical isomorphism, ${\Lambda }^{\prime }$ denoting the ring $\Lambda /p\Lambda$ $(0_{IV},\mathrm{20.5.5})$. Applying then the exact sequence $(0_{IV},\mathrm{20.5.14.1})$ to the ${\Lambda }^{\prime }$-algebra $C=A^{\prime }/{\mathfrak{J}}^{\prime }$ formally smooth, one obtains the exact sequence

  0 → 𝔍′/𝔍′² → Ω^1_{A′/Λ′} ⊗_{A′} C → Ω^1_{C/Λ′} → 0

and since the homomorphism $\Lambda \to {\Lambda }^{\prime }$ is surjective, one has ${\Omega }_{C/{\Lambda }^{\prime }}^1={\Omega }_{C/\Lambda }^1$ up to a canonical isomorphism $(0_{IV},\mathrm{20.5.15})$; whence the exact sequence (22.4.4.3) by virtue of what precedes.

(ii) Since one has ${\mathfrak{m}}^2=0$, we shall again denote by $V^{\prime }$ the ideal ${\mathfrak{m}}^{\prime }$ regarded as a $K^{\prime }$-vector space, and we set

  (22.4.4.4)   V_0 = V/K · p = 𝔪/(𝔪² + p A),   V′_0 = V′/K′ · p = 𝔪′/(𝔪′² + p A′).

Applying the exact sequence (22.4.4.3) to the case $\Lambda =\mathbb{Z}$, to the $\mathbb{Z}$-algebra $A$ (resp. $A^{\prime }$) and to its ideal $\mathfrak{m}$ (resp. ${\mathfrak{m}}^{\prime }$), there arises, since $K$ (resp. $K^{\prime }$) is separable over the prime field $\mathbb{Z}/p\mathbb{Z}$, hence a $(\mathbb{Z}/p\mathbb{Z})$-algebra formally smooth $(0_{IV},\mathrm{19.6.1})$, the exact sequences

  (22.4.4.5)   0 → V_0 → Ω^1_A ⊗_A K → Ω^1_K → 0,    0 → V′_0 → Ω^1_{A′} ⊗_{A′} K′ → Ω^1_{K′} → 0.

Consider then the diagram (22.4.4.6) whose two middle columns are the second sequence (22.4.4.5) and the first tensored by $K^{\prime }$; since the first exact sequence (22.4.4.5) is formed of vector spaces over $K$, the two middle columns of (22.4.4.6) are exact; moreover, the diagram formed by these columns and the horizontal arrows connecting them is commutative, by virtue of the proof of (22.4.4.2) and of the commutativity of the diagram $(0_{IV},\mathrm{20.5.11.3})$. The last line of (22.4.4.6) is the exact sequence obtained from $(0_{IV},\mathrm{20.6.1.1})$ by replacing $A$, $B$, $C$ by $\mathbb{Z}$, $K$ and $K^{\prime }$; the middle line is obtained by tensorisation with $K^{\prime }$ of the exact sequence of $K$-modules deduced from $(0_{IV},\mathrm{20.6.1.1})$ by replacing $A$, $B$, $C$ by $\mathbb{Z}$, $A$, $K$ respectively. The diagram formed by these two lines and the vertical arrows connecting them is commutative by virtue of the commutativity of $(0_{IV},\mathrm{20.6.4.3})$. Note on the other hand that one is under the conditions of application of (22.4.1, (ii)), so the upper line of the diagram (22.4.4.6) is exact; the extreme columns of the diagram

                              0                       0
                              │                       │
                              ▼                       ▼
  (22.4.4.6)            V_0 ⊗_K K′ ───────────────▸ V′_0
                              │                       │
                              ▼                       ▼
  0 ──▸ N ──▸ Ω^1_A ⊗_A K′ ─────────────▸ Ω^1_{A′} ⊗_{A′} K′ ──▸ Ω^1_{A′/A} ⊗_{A′} K′ ──▸ 0
                              │                       │                        ║
                              ▼                       ▼                        ▼
  0 ──▸ Υ_{K′/K} ──▸ Ω^1_K ⊗_K K′ ────────────────▸ Ω^1_{K′} ───────────────▸ Ω^1_{K′/K} ──▸ 0
                              │                       │
                              ▼                       ▼
                              0                       0

are thus formed respectively of the kernels and cokernels of the middle horizontal arrows, which completes proving that the diagram is commutative. Moreover:

Lemma (22.4.4.7).

The homomorphisms

  N → Υ_{K′/K}    and    Ω^1_{A′/A} ⊗_{A′} K′ → Ω^1_{K′/K}

of the diagram (22.4.4.6) are bijective.

Indeed, the snake-diagram (Bourbaki, Alg. comm., chap. I, §1, n° 4, prop. 2) applied to the two middle columns of (22.4.4.6) gives an exact sequence

  0 → N → Υ_{K′/K} → 0 → Ω^1_{A′/A} ⊗_{A′} K′ → Ω^1_{K′/K} → 0.

Now, if $t_i$ is the canonical image of $T_i$ in $A^{\prime }$, the relation $F_i(t_i)=0$, together with the fact that in $A^{\prime }$ one has $p\cdot 1\in \mathfrak{m}{A}^{\prime }\subset {\mathfrak{m}}^{\prime }$, shows at once that $d_{A^{\prime }}{f}_i\in {\mathfrak{m}}^{\prime }{\Omega }_{A^{\prime }}^1$, hence $d_{A^{\prime }}{f}_i\otimes 1_{K^{\prime }}=0$; this proves that the $d_A{f}_i\otimes 1_{K^{\prime }}$ belong to $N$. Moreover, their images in ${{\rm Y}}_{K^{\prime }/K}$ are the $d_K{a}_i\otimes 1_{K^{\prime }}$, which form a basis of ${{\rm Y}}_{K^{\prime }/K}$ over $K^{\prime }$ $(0_{IV},\mathrm{21.4.7})$; taking account of (22.4.4.7), this completes the proof of (22.4.4).