## 23.9 Local complete intersection maps

Let $A \to B$ be a local homomorphism of Noetherian complete local rings. A consequence of the Cohen structure theorem is that we can find a commutative diagram

\[ \xymatrix{ S \ar[r] & B \\ & A \ar[lu] \ar[u] } \]

of Noetherian complete local rings with $S \to B$ surjective, $A \to S$ flat, and $S/\mathfrak m_ A S$ a regular local ring. This follows from More on Algebra, Lemma 15.39.3. Let us (temporarily) say $A \to S \to B$ is a *good factorization* of $A \to B$ if $S$ is a Noetherian local ring, $A \to S \to B$ are local ring maps, $S \to B$ surjective, $A \to S$ flat, and $S/\mathfrak m_ AS$ regular. Let us say that $A \to B$ is a *complete intersection homomorphism* if there exists some good factorization $A \to S \to B$ such that the kernel of $S \to B$ is generated by a regular sequence. The following lemma shows this notion is independent of the choice of the diagram.

Lemma 23.9.1. Let $A \to B$ be a local homomorphism of Noetherian complete local rings. The following are equivalent

for some good factorization $A \to S \to B$ the kernel of $S \to B$ is generated by a regular sequence, and

for every good factorization $A \to S \to B$ the kernel of $S \to B$ is generated by a regular sequence.

**Proof.**
Let $A \to S \to B$ be a good factorization. As $B$ is complete we obtain a factorization $A \to S^\wedge \to B$ where $S^\wedge $ is the completion of $S$. Note that this is also a good factorization: The ring map $S \to S^\wedge $ is flat (Algebra, Lemma 10.97.2), hence $A \to S^\wedge $ is flat. The ring $S^\wedge /\mathfrak m_ A S^\wedge = (S/\mathfrak m_ A S)^\wedge $ is regular since $S/\mathfrak m_ A S$ is regular (More on Algebra, Lemma 15.43.4). Let $f_1, \ldots , f_ r$ be a minimal sequence of generators of $\mathop{\mathrm{Ker}}(S \to B)$. We will use without further mention that an ideal in a Noetherian local ring is generated by a regular sequence if and only if any minimal set of generators is a regular sequence. Observe that $f_1, \ldots , f_ r$ is a regular sequence in $S$ if and only if $f_1, \ldots , f_ r$ is a regular sequence in the completion $S^\wedge $ by Algebra, Lemma 10.68.5. Moreover, we have

\[ S^\wedge /(f_1, \ldots , f_ r)R^\wedge = (S/(f_1, \ldots , f_ n))^\wedge = B^\wedge = B \]

because $B$ is $\mathfrak m_ B$-adically complete (first equality by Algebra, Lemma 10.97.1). Thus the kernel of $S \to B$ is generated by a regular sequence if and only if the kernel of $S^\wedge \to B$ is generated by a regular sequence. Hence it suffices to consider good factorizations where $S$ is complete.

Assume we have two factorizations $A \to S \to B$ and $A \to S' \to B$ with $S$ and $S'$ complete. By More on Algebra, Lemma 15.39.4 the ring $S \times _ B S'$ is a Noetherian complete local ring. Hence, using More on Algebra, Lemma 15.39.3 we can choose a good factorization $A \to S'' \to S \times _ B S'$ with $S''$ complete. Thus it suffices to show: If $A \to S' \to S \to B$ are comparable good factorizations, then $\mathop{\mathrm{Ker}}(S \to B)$ is generated by a regular sequence if and only if $\mathop{\mathrm{Ker}}(S' \to B)$ is generated by a regular sequence.

Let $A \to S' \to S \to B$ be comparable good factorizations. First, since $S'/\mathfrak m_ R S' \to S/\mathfrak m_ R S$ is a surjection of regular local rings, the kernel is generated by a regular sequence $\overline{x}_1, \ldots , \overline{x}_ c \in \mathfrak m_{S'}/\mathfrak m_ R S'$ which can be extended to a regular system of parameters for the regular local ring $S'/\mathfrak m_ R S'$, see (Algebra, Lemma 10.106.4). Set $I = \mathop{\mathrm{Ker}}(S' \to S)$. By flatness of $S$ over $R$ we have

\[ I/\mathfrak m_ R I = \mathop{\mathrm{Ker}}(S'/\mathfrak m_ R S' \to S/\mathfrak m_ R S) = (\overline{x}_1, \ldots , \overline{x}_ c). \]

Choose lifts $x_1, \ldots , x_ c \in I$. These lifts form a regular sequence generating $I$ as $S'$ is flat over $R$, see Algebra, Lemma 10.99.3.

We conclude that if also $\mathop{\mathrm{Ker}}(S \to B)$ is generated by a regular sequence, then so is $\mathop{\mathrm{Ker}}(S' \to B)$, see More on Algebra, Lemmas 15.30.13 and 15.30.7.

Conversely, assume that $J = \mathop{\mathrm{Ker}}(S' \to B)$ is generated by a regular sequence. Because the generators $x_1, \ldots , x_ c$ of $I$ map to linearly independent elements of $\mathfrak m_{S'}/\mathfrak m_{S'}^2$ we see that $I/\mathfrak m_{S'}I \to J/\mathfrak m_{S'}J$ is injective. Hence there exists a minimal system of generators $x_1, \ldots , x_ c, y_1, \ldots , y_ d$ for $J$. Then $x_1, \ldots , x_ c, y_1, \ldots , y_ d$ is a regular sequence and it follows that the images of $y_1, \ldots , y_ d$ in $S$ form a regular sequence generating $\mathop{\mathrm{Ker}}(S \to B)$. This finishes the proof of the lemma.
$\square$

In the following proposition observe that the condition on vanishing of Tor's applies in particular if $B$ has finite tor dimension over $A$ and thus in particular if $B$ is flat over $A$.

Proposition 23.9.2. Let $A \to B$ be a local homomorphism of Noetherian local rings. Then the following are equivalent

$B$ is a complete intersection and $\text{Tor}^ A_ p(B, A/\mathfrak m_ A)$ is nonzero for only finitely many $p$,

$A$ is a complete intersection and $A^\wedge \to B^\wedge $ is a complete intersection homomorphism in the sense defined above.

**Proof.**
Let $F_\bullet \to A/\mathfrak m_ A$ be a resolution by finite free $A$-modules. Observe that $\text{Tor}^ A_ p(B, A/\mathfrak m_ A)$ is the $p$th homology of the complex $F_\bullet \otimes _ A B$. Let $F_\bullet ^\wedge = F_\bullet \otimes _ A A^\wedge $ be the completion. Then $F_\bullet ^\wedge $ is a resolution of $A^\wedge /\mathfrak m_{A^\wedge }$ by finite free $A^\wedge $-modules (as $A \to A^\wedge $ is flat and completion on finite modules is exact, see Algebra, Lemmas 10.97.1 and 10.97.2). It follows that

\[ F_\bullet ^\wedge \otimes _{A^\wedge } B^\wedge = F_\bullet \otimes _ A B \otimes _ B B^\wedge \]

By flatness of $B \to B^\wedge $ we conclude that

\[ \text{Tor}^{A^\wedge }_ p(B^\wedge , A^\wedge /\mathfrak m_{A^\wedge }) = \text{Tor}^ A_ p(B, A/\mathfrak m_ A) \otimes _ B B^\wedge \]

In this way we see that the condition in (1) on the local ring map $A \to B$ is equivalent to the same condition for the local ring map $A^\wedge \to B^\wedge $. Thus we may assume $A$ and $B$ are complete local Noetherian rings (since the other conditions are formulated in terms of the completions in any case).

Assume $A$ and $B$ are complete local Noetherian rings. Choose a diagram

\[ \xymatrix{ S \ar[r] & B \\ R \ar[u] \ar[r] & A \ar[u] } \]

as in More on Algebra, Lemma 15.39.3. Let $I = \mathop{\mathrm{Ker}}(R \to A)$ and $J = \mathop{\mathrm{Ker}}(S \to B)$. The proposition now follows from Lemma 23.7.6.
$\square$

To finish of this section we compare the notion defined above with the notion introduced in More on Algebra, Section 23.8.

Lemma 23.9.4. Consider a commutative diagram

\[ \xymatrix{ S \ar[r] & B \\ & A \ar[lu] \ar[u] } \]

of Noetherian local rings with $S \to B$ surjective, $A \to S$ flat, and $S/\mathfrak m_ A S$ a regular local ring. The following are equivalent

$\mathop{\mathrm{Ker}}(S \to B)$ is generated by a regular sequence, and

$A^\wedge \to B^\wedge $ is a complete intersection homomorphism as defined above.

**Proof.**
Omitted. Hint: the proof is identical to the argument given in the first paragraph of the proof of Lemma 23.9.1.
$\square$

Lemma 23.9.5. Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map. The following are equivalent

$A \to B$ is a local complete intersection in the sense of More on Algebra, Definition 15.33.2,

for every prime $\mathfrak q \subset B$ and with $\mathfrak p = A \cap \mathfrak q$ the ring map $(A_\mathfrak p)^\wedge \to (B_\mathfrak q)^\wedge $ is a complete intersection homomorphism in the sense defined above.

**Proof.**
Choose a surjection $R = A[x_1, \ldots , x_ n] \to B$. Observe that $A \to R$ is flat with regular fibres. Let $I$ be the kernel of $R \to B$. Assume (2). Then we see that $I$ is locally generated by a regular sequence by Lemma 23.9.4 and Algebra, Lemma 10.68.6. In other words, (1) holds. Conversely, assume (1). Then after localizing on $R$ and $B$ we can assume that $I$ is generated by a Koszul regular sequence. By More on Algebra, Lemma 15.30.7 we find that $I$ is locally generated by a regular sequence. Hence (2) hold by Lemma 23.9.4. Some details omitted.
$\square$

Lemma 23.9.6. Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map such that the image of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ contains all closed points of $\mathop{\mathrm{Spec}}(A)$. Then the following are equivalent

$B$ is a complete intersection and $A \to B$ has finite tor dimension,

$A$ is a complete intersection and $A \to B$ is a local complete intersection in the sense of More on Algebra, Definition 15.33.2.

**Proof.**
This is a reformulation of Proposition 23.9.2 via Lemma 23.9.5. We omit the details.
$\square$

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