# Regular local rings and complete intersections

*Written by Claude Opus 5.5 (Anthropic) on 22 October 2026. Self-checked by the writing AI. Public domain (CC0).*

For a noetherian local ring \(A\) with residue field \(k\), the cotangent complex \(L_{k/A}\) measures how far \(A\) is from being regular. Its first homology is always \(\mathfrak m/\mathfrak m^2\); its second homology vanishes exactly when \(A\) is regular; and its third homology vanishes exactly when \(A\) is a complete intersection, that is, when the completion of \(A\) is a regular local ring modulo an ideal generated by a regular sequence. A presentation of the completion by a regular local ring of the smallest possible dimension reads off the number of generators of \(\mathfrak m\) and the number of relations from \(D_1(k/A)\) and \(D_2(k/A)\). The lesson proves:

- \(A\) is regular if and only if \(D_2(k/A)=0\), if and only if \(L_{k/A}\cong(\mathfrak m/\mathfrak m^2)[1]\);
- \(L_{k/A}\) does not change under completion, and power series rings over fields and Cohen rings are regular;
- for a surjection from a regular local ring \(P\) onto \(A\) or its completion, with kernel \(I\), the shift isomorphisms \(D_q(k/A)\cong D_{q-1}(-/P,k)\) for \(q\geq3\), and an exact sequence relating \(D_2(k/A)\), \(I/\mathfrak m_PI\), \(\mathfrak m_P/\mathfrak m_P^2\) and \(\mathfrak m/\mathfrak m^2\);
- \(A\) is a complete intersection if and only if \(D_3(k/A)=0\), if and only if \(D_q(k/A)=0\) for every \(q\geq3\); the kernel of every presentation by a regular local ring is then generated by a regular sequence, and of none otherwise;
- presentations with \(I\subseteq\mathfrak m_P^2\) exist, and for them \(\dim_kD_2(k/A)\) is the minimal number of generators of \(I\).

**Prerequisites.** [*Weakly regular ideals of noetherian rings*](weakly-regular-ideals-of-noetherian-rings.md) (Proposition 3.4, Theorem 3.5); [*The fundamental spectral sequence*](the-fundamental-spectral-sequence.md) (Proposition 1.1 (3), Theorem 4.2, Exercise 5.1); [*Weak complete intersections*](weak-complete-intersections.md) (its conventions on the transitivity triangle and on \(\partial_L\)); [*The cotangent complex*](course:AG-HP/the-cotangent-complex#3-tor-independent-base-change) (Proposition 3.1, Theorem 4.1). From the AI-integrated Stacks project: regular local rings [[Stacks, Tag 00KU](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-definition-regular-local)], minimal generators of the maximal ideal of a regular local ring [[Tag 00NQ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-regular-ring-CM)], regularity modulo a regular sequence [[Tag 00NU](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-regular-mod-x)]; discrete valuation rings [[Tag 00PD](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-characterize-dvr)] and Cohen rings [[Tag 0327](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-definition-cohen-ring)]; power series rings over noetherian rings [[Tag 0306](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-Noetherian-power-series)]; the Cohen structure theorem [[Tag 032A](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-theorem-cohen-structure-theorem)]; completion of noetherian rings: noetherian [[Tag 0316](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-completion-Noetherian-Noetherian)], flat [[Tag 00MB](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-completion-flat)], complete with the same quotients [[Tag 031C](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-completion-complete)]; complete intersections [[Tag 09PZ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/dpa.html#dpa-lemma-ci-well-defined), [Tag 09Q3](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/dpa.html#dpa-definition-lci)]; Nakayama's lemma [[Tag 00DV](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-NAK)].

## Conventions

Rings are commutative, in the usual sense. \((A,\mathfrak m,k)\) denotes a noetherian local ring with maximal ideal \(\mathfrak m\) and residue field \(k=A/\mathfrak m\); \(\operatorname{edim}A=\dim_k\mathfrak m/\mathfrak m^2\) is its embedding dimension, the minimal number of generators of \(\mathfrak m\) [Tag 00DV]. A *regular local ring* is a noetherian local ring whose maximal ideal is generated by \(\dim A\) elements [Tag 00KU]. For a surjection \(A\to B\) and a \(B\)-module \(M\), \(D_q(B/A,M)=H_q(L_{B/A}\otimes_BM)\) as in the lesson on the fundamental spectral sequence, and \(D_q(B/A)=D_q(B/A,B)\). Since \(L_{k/A}\otimes_kk=L_{k/A}\), \(D_q(k/A,k)=D_q(k/A)=H_q(L_{k/A})\). The isomorphism \(\partial_L\colon H_1(L_{B/A})\to I/I^2\) for \(B=A/I\) is that of Proposition 1.1 (3) of the lesson on the fundamental spectral sequence; it is natural in the pair \((A,I)\). A homomorphism of local rings is *local* if it maps the maximal ideal into the maximal ideal; a surjective homomorphism of local rings is local.

## 1. Regular local rings

**Lemma 1.1.** \(H_0(L_{k/A})=0\), and \(\partial_L\colon H_1(L_{k/A})\to\mathfrak m/\mathfrak m^2\) is an isomorphism. So \(\dim_kD_1(k/A)=\operatorname{edim}A\).

**Proof.** Proposition 1.1 (3) of the lesson on the fundamental spectral sequence, for \(k=A/\mathfrak m\). \(\square\)

**Theorem 1.2.** For a noetherian local ring \((A,\mathfrak m,k)\) the following are equivalent:

1. \(A\) is regular;
2. \(\mathfrak m\) is generated by a regular sequence;
3. \(D_2(k/A)=0\);
4. \(D_q(k/A)=0\) for every \(q\geq2\);
5. \(\partial_L\) induces an isomorphism \(L_{k/A}\cong(\mathfrak m/\mathfrak m^2)[1]\) in \(D(k)\).

**Proof.** (1) implies (2): a minimal set of generators of the maximal ideal of a regular local ring is a regular sequence [Tag 00NQ].

(2) implies (1): if \(x_1,\dots,x_r\) is a regular sequence generating \(\mathfrak m\), then \(A/(x_1,\dots,x_r)=k\) is a regular local ring of dimension \(0\), so \(A\) is regular [Tag 00NU].

(2) and (3) are equivalent by Proposition 3.4 of the lesson on weakly regular ideals of noetherian rings, applied to \(I=\mathfrak m\): there \(D_2(k/A,\kappa)\) is \(D_2(k/A)\).

(2) implies (5) and (4): by the same proposition \(\mathfrak m\) is weakly regular, and Theorem 4.2 of the lesson on the fundamental spectral sequence gives (5), which implies (4). Finally (4) implies (3). \(\square\)

## 2. Completion and presentations

**Lemma 2.1.** Let \(\hat A\) be the \(\mathfrak m\)-adic completion of \(A\). Then \(\hat A\) is a noetherian local ring with maximal ideal \(\mathfrak m\hat A\) and residue field \(k\), and \(A\to\hat A\) is flat. The functoriality map \(L_{k/A}\to L_{k/\hat A}\) is an isomorphism; so \(D_q(k/A)\cong D_q(k/\hat A)\) for every \(q\), and \(\operatorname{edim}\hat A=\operatorname{edim}A\).

**Proof.** \(\hat A\) is noetherian [Tag 0316] and flat over \(A\) [Tag 00MB]. It is \(\mathfrak m\hat A\)-adically complete, and \(\hat A/\mathfrak m^n\hat A=A/\mathfrak m^n\) [Tag 031C]; in particular \(\hat A/\mathfrak m\hat A=k\). An element \(u\in\hat A\smallsetminus\mathfrak m\hat A\) maps to a unit of \(k\); if \(v\in\hat A\) lifts its inverse, then \(uv=1-w\) with \(w\in\mathfrak m\hat A\), and \(\sum_jw^j\) converges, so \(u\) is a unit. Hence \(\hat A\) is local with maximal ideal \(\mathfrak m\hat A\). As \(\hat A\) is flat over \(A\), \(\operatorname{Tor}^A_i(k,\hat A)=0\) for \(i>0\), and \(k\otimes_A\hat A=\hat A/\mathfrak m\hat A=k\). By Proposition 3.1 of *The cotangent complex*, the functoriality map \(L_{k/A}=L_{k/A}\otimes^{\mathbf L}_kk\to L_{k/\hat A}\) is an isomorphism. The last assertion follows from Lemma 1.1. \(\square\)

**Lemma 2.2.** Let \(\Lambda\) be a field or a Cohen ring, and \(n\geq0\). Then \(R=\Lambda[[x_1,\dots,x_n]]\) is a regular local ring, whose maximal ideal is generated by \(x_1,\dots,x_n\) and the maximal ideal of \(\Lambda\).

**Proof.** A field is a regular local ring of dimension \(0\). A Cohen ring is a complete discrete valuation ring [Tag 0327], hence a regular local ring of dimension \(1\) [Tag 00PD]. The ring \(R\) is noetherian [Tag 0306]. Let \(\mathfrak m_\Lambda\) be the maximal ideal of \(\Lambda\) and \(\mathfrak n=\mathfrak m_\Lambda R+(x_1,\dots,x_n)\). Elements of \(\mathfrak n\) are not units, as their images in \(\Lambda=R/(x_1,\dots,x_n)\) lie in \(\mathfrak m_\Lambda\). A series \(f\notin\mathfrak n\) has a constant term \(c\in\Lambda^\times\), and \(f=c(1-g)\) with \(g\in(x_1,\dots,x_n)\); since \(g^j\) has no terms of degree \(<j\), the series \(\sum_jg^j\) is defined and inverts \(1-g\). So \(R\) is local with maximal ideal \(\mathfrak n\). The quotient \(R/(x_1,\dots,x_{i-1})\) is \(\Lambda[[x_i,\dots,x_n]]\), on which multiplication by \(x_i\) is injective, and \(R/(x_1,\dots,x_n)=\Lambda\neq0\); so \(x_1,\dots,x_n\) is a regular sequence. As \(\Lambda\) is a regular local ring, so is \(R\) [Tag 00NU]. \(\square\)

**2.3. Complete intersections.** A noetherian complete local ring \(C\) is a *complete intersection* if there is a surjective homomorphism of local rings \(P\to C\), with \(P\) a regular local ring, whose kernel is generated by a regular sequence [Tag 09PZ]; a noetherian local ring is a complete intersection if its completion is one [Tag 09Q3]. Every noetherian complete local ring \(C\) has surjections from regular local rings: by the Cohen structure theorem [Tag 032A], \(C\cong\Lambda[[x_1,\dots,x_n]]/J\) with \(\Lambda\) a field or a Cohen ring, and \(\Lambda[[x_1,\dots,x_n]]\) is regular by Lemma 2.2. Theorem 2.6 shows that the condition does not depend on the surjection.

**Proposition 2.4.** Let \(C\) be a noetherian local ring with residue field \(k\), \(P\) a regular local ring with maximal ideal \(\mathfrak m_P\), and \(P\to C\) a surjective homomorphism with kernel \(I\subseteq\mathfrak m_P\). Then
\[
D_q(k/C)\cong D_{q-1}(C/P,k)\quad(q\geq3),
\]
and there is an exact sequence
\[
\begin{aligned}&0\to D_2(k/C)\to I/\mathfrak m_PI\\&\quad\to\mathfrak m_P/\mathfrak m_P^2\to\mathfrak m_C/\mathfrak m_C^2\to0,\end{aligned}
\]
in which \(I/\mathfrak m_PI\to\mathfrak m_P/\mathfrak m_P^2\) is induced by the inclusion \(I\subseteq\mathfrak m_P\) and \(\mathfrak m_P/\mathfrak m_P^2\to\mathfrak m_C/\mathfrak m_C^2\) by \(P\to C\).

**Proof.** The transitivity triangle of \(P\to C\to k\) (Theorem 4.1 of *The cotangent complex*) gives the long exact sequence
\[
\begin{aligned}&\cdots\to D_q(C/P,k)\to D_q(k/P)\\&\quad\to D_q(k/C)\to D_{q-1}(C/P,k)\\&\quad\to D_{q-1}(k/P)\to\cdots,\end{aligned}
\]
in which \(D_q(k/P)=0\) for \(q\geq2\), by Theorem 1.2. For \(q\geq3\) the outer terms around \(D_q(k/C)\to D_{q-1}(C/P,k)\) vanish, which gives the isomorphisms. In low degrees the sequence is
\[
\begin{aligned}&0\to D_2(k/C)\to D_1(C/P,k)\\&\quad\to D_1(k/P)\to D_1(k/C)\\&\quad\to D_0(C/P,k).\end{aligned}
\]
Since \(H_0(L_{C/P})=0\), \(D_0(C/P,k)=0\), and right exactness of \(\otimes_Ck\) gives \(D_1(C/P,k)=H_1(L_{C/P})\otimes_Ck\), which \(\partial_L\) identifies with \((I/I^2)\otimes_Ck=I/\mathfrak m_PI\), because \(I^2\subseteq\mathfrak m_PI\). By Lemma 1.1, \(D_1(k/P)\cong\mathfrak m_P/\mathfrak m_P^2\) and \(D_1(k/C)\cong\mathfrak m_C/\mathfrak m_C^2\). The maps are induced by the functoriality of the cotangent complex for the squares from \(P\to C\) to \(P\to k\) and from \(P\to k\) to \(C\to k\); by naturality of \(\partial_L\) in the pair, they are the maps induced by \(I\subseteq\mathfrak m_P\) and by \(P\to C\). \(\square\)

**Corollary 2.5.** Let \(P\) be a regular local ring and \(I\subseteq\mathfrak m_P\) an ideal, \(C=P/I\) with residue field \(k\). Then \(I\) is generated by a regular sequence if and only if \(D_3(k/C)=0\), and then \(D_q(k/C)=0\) for every \(q\geq3\).

**Proof.** By Proposition 2.4, \(D_q(k/C)\cong D_{q-1}(C/P,k)\) for \(q\geq3\). By Proposition 3.4 of the lesson on weakly regular ideals of noetherian rings, applied to \(I\subseteq\mathfrak m_P\), \(I\) is generated by a regular sequence if and only if \(D_2(C/P,k)=0\); and then \(I\) is weakly regular, so \(D_q(C/P,k)=0\) for every \(q\geq2\) (Theorem 3.5 there). \(\square\)

**Theorem 2.6.** For a noetherian local ring \((A,\mathfrak m,k)\) the following are equivalent:

1. \(A\) is a complete intersection;
2. \(D_q(k/A)=0\) for every \(q\geq3\);
3. \(D_3(k/A)=0\).

If they hold, the kernel of every surjective homomorphism from a regular local ring onto \(\hat A\) is generated by a regular sequence; if they fail, no such kernel is.

**Proof.** Let \(P\to\hat A\) be a surjection from a regular local ring, with kernel \(I\); it exists by 2.3, and \(I\) lies in the maximal ideal of \(P\). By Lemma 2.1, \(D_q(k/A)\cong D_q(k/\hat A)\) for every \(q\). By Corollary 2.5 for \(C=\hat A\), \(I\) is generated by a regular sequence if and only if \(D_3(k/A)=0\), and then \(D_q(k/A)=0\) for every \(q\geq3\). This holds for every such surjection. So (1), which asks it for one surjection, implies (2); (2) implies (3); and (3) gives it for every surjection, hence (1). \(\square\)

If \(A=P/I\) with \(P\) a regular local ring, Corollary 2.5 and Theorem 2.6 show directly, without completing, that \(A\) is a complete intersection if and only if \(I\) is generated by a regular sequence.

**Proposition 2.7.** Let \((A,\mathfrak m,k)\) be a noetherian local ring.

1. For every surjection \(P\to\hat A\) from a regular local ring with kernel \(I\),
\[
\begin{aligned}&\dim_kD_2(k/A)\\&\quad=\mu(I)-\operatorname{edim}P+\operatorname{edim}A,\end{aligned}
\]
where \(\mu(I)=\dim_kI/\mathfrak m_PI\) is the minimal number of generators of \(I\). The following are equivalent: \(I\subseteq\mathfrak m_P^2\); \(\dim P=\operatorname{edim}A\); \(D_2(k/A)\cong I/\mathfrak m_PI\).
2. There is a surjection \(P\to\hat A\) from a regular local ring with kernel contained in \(\mathfrak m_P^2\). For it, \(\dim P=\operatorname{edim}A\) and \(\dim_kD_2(k/A)=\mu(I)\).

**Proof.** (1) By Lemma 2.1, \(D_2(k/A)\cong D_2(k/\hat A)\) and \(\operatorname{edim}\hat A=\operatorname{edim}A\). The exact sequence of Proposition 2.4 for \(C=\hat A\) gives the dimension formula. Its second map \(I/\mathfrak m_PI\to\mathfrak m_P/\mathfrak m_P^2\) is zero exactly when \(I\subseteq\mathfrak m_P^2\), which by exactness holds exactly when \(D_2(k/A)\to I/\mathfrak m_PI\) is bijective, and exactly when \(\mathfrak m_P/\mathfrak m_P^2\to\mathfrak m/\mathfrak m^2\) is bijective, that is \(\operatorname{edim}P=\operatorname{edim}A\). For the regular local ring \(P\), a minimal set of generators of \(\mathfrak m_P\) has \(\dim P\) elements [Tag 00NQ], so \(\operatorname{edim}P=\dim P\).

(2) Start with any surjection \(P\to\hat A\) from a regular local ring (2.3). If its kernel \(I\) is not contained in \(\mathfrak m_P^2\), choose \(f\in I\smallsetminus\mathfrak m_P^2\). The class of \(f\) in \(\mathfrak m_P/\mathfrak m_P^2\) is nonzero, so \(f\) is part of a minimal set of generators of \(\mathfrak m_P\), and \(P/(f)\) is a regular local ring of dimension \(\dim P-1\) [Tag 00NQ]. The surjection factors through \(P/(f)\to\hat A\), with kernel \(I/(f)\). Repeating this, the dimension decreases each time, so after finitely many steps the kernel lies in the square of the maximal ideal. The rest follows from (1). \(\square\)

So \(\dim_kD_1(k/A)\) is the minimal number of generators of \(\mathfrak m\), and \(\dim_kD_2(k/A)\) the minimal number of relations among them, in a presentation of the completion by a regular local ring of the smallest possible dimension.

**Remark 2.8.** Quillen conjectured that a homomorphism of finite type \(A\to B\) of noetherian rings is a local complete intersection when \(B\) has finite flat dimension over \(A\) and \(L_{B/A}\) has finite projective dimension, and that a noetherian local ring with \(D_q(k/A)=0\) for all large \(q\) is a complete intersection. Avramov proved the first in a very general form, and the second follows from his theorem on rings of locally complete intersection type applied to \(A\to k\): L. Avramov, *Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology*, Ann. of Math. 150 (1999), Theorems 1.3 and 1.5, [arXiv:math/9909192](https://arxiv.org/abs/math/9909192). These results are not used in the course.

## 3. Exercises

**Exercise 3.1.** Let \(k\) be a field, \(P=k[x,y]_{(x,y)}\) and \(A=P/(xy)\). Show that \(P\) is regular, that \(A\) is a complete intersection but not regular, and that \(\dim_kD_1(k/A)=2\), \(\dim_kD_2(k/A)=1\) and \(D_q(k/A)=0\) for \(q\geq3\).

**Exercise 3.2.** With \(P\) as in Exercise 3.1, let \(A=P/(x^2,xy,y^2)\). Show that \(\dim_kD_2(k/A)=3\), that \(A\) is not a complete intersection, and that \(D_3(k/A)\neq0\).

**Exercise 3.3.** Let \((A,\mathfrak m,k)\) be a noetherian local ring and \(f\in\mathfrak m\) a nonzerodivisor. Show that \(D_q(k/A)\cong D_q(k/(A/fA))\) for \(q\geq3\), and conclude that \(A\) is a complete intersection if and only if \(A/fA\) is.

**Exercise 3.4.** Show that a noetherian local ring \(A\) with \(\operatorname{edim}A\leq1\) is a complete intersection.

## 4. Complete solutions

**Solution 3.1.** In the domain \(P\), \(x\) is a nonzerodivisor, \(P/xP\) is a localization of the domain \(k[y]\) in which \(y\) is nonzero, and \(P/(x,y)=k\neq0\); so \(x,y\) is a regular sequence generating the maximal ideal, and \(P\) is regular by Theorem 1.2. The element \(xy\neq0\) is a nonzerodivisor of \(P\), so \(I=(xy)\) is generated by a regular sequence, and \(A\) is a complete intersection by the remark after Theorem 2.6, with \(D_q(k/A)=0\) for \(q\geq3\). As \(I\subseteq\mathfrak m_P^2\), \(\mathfrak m_A/\mathfrak m_A^2=\mathfrak m_P/(\mathfrak m_P^2+I)=\mathfrak m_P/\mathfrak m_P^2\), which has dimension \(2\); so \(\dim_kD_1(k/A)=2\) by Lemma 1.1. In the exact sequence of Proposition 2.4 for \(C=A\), \(I/\mathfrak m_PI\) is spanned by the class of \(xy\), and its map to \(\mathfrak m_P/\mathfrak m_P^2\) is zero, as \(xy\in\mathfrak m_P^2\); so \(D_2(k/A)\cong I/\mathfrak m_PI\cong k\). As \(D_2(k/A)\neq0\), \(A\) is not regular (Theorem 1.2).

**Solution 3.2.** Here \(I=(x^2,xy,y^2)=\mathfrak m_P^2\), and \(x^2\), \(xy\), \(y^2\) form a minimal set of generators of \(I\): their classes span \(I/\mathfrak m_PI=\mathfrak m_P^2/\mathfrak m_P^3\), which has dimension \(3\). As \(I\subseteq\mathfrak m_P^2\), the exact sequence of Proposition 2.4 for \(C=A\) gives \(D_2(k/A)\cong I/\mathfrak m_PI\), of dimension \(3\). The sequence \(x^2,xy\) is not regular: \(x\cdot xy=y\cdot x^2\) lies in \(x^2P\), while \(x\) does not, since \(x=x^2a/s\) with \(s\notin\mathfrak m_P\) would give \(s=xa\in\mathfrak m_P\). By Proposition 3.4 of the lesson on weakly regular ideals of noetherian rings, if \(I\) were generated by a regular sequence, every minimal set of generators would be one; so \(I\) is not, \(A\) is not a complete intersection (remark after Theorem 2.6), and \(D_3(k/A)\neq0\) by Corollary 2.5.

**Solution 3.3.** Put \(B=A/fA\). The ideal \(fA\) is generated by the regular sequence \(f\) (\(B\neq0\), as \(f\in\mathfrak m\)), so it is weakly regular, \(L_{B/A}\cong(fA/f^2A)[1]\), and \(fA/f^2A\cong B\), \(f\) being a nonzerodivisor (Proposition 3.4 and Theorem 3.5 of the lesson on weakly regular ideals of noetherian rings). Hence \(D_q(B/A,k)\) is \(k\) for \(q=1\) and \(0\) otherwise. The transitivity triangle of \(A\to B\to k\) gives the exact sequences \(D_q(B/A,k)\to D_q(k/A)\to\) \(D_q(k/B)\to D_{q-1}(B/A,k)\), whose outer terms vanish for \(q\geq3\). So \(D_q(k/A)\cong D_q(k/B)\) for \(q\geq3\), and in particular \(D_3(k/A)=0\) if and only if \(D_3(k/B)=0\). Theorem 2.6 applies to \(A\) and to \(B\), whose residue field is \(k\).

**Solution 3.4.** By Proposition 2.7 (2), there is a surjection \(P\to\hat A\) from a regular local ring with \(\dim P=\operatorname{edim}A\leq1\). If \(\dim P=0\), \(P\) is a field and the kernel is \(0\), generated by the empty sequence. If \(\dim P=1\), \(P\) is a discrete valuation ring [Tag 00PD], so the kernel is \(0\) or generated by a nonzero element, which is a nonzerodivisor of the domain \(P\) and lies in the maximal ideal; in either case it is generated by a regular sequence. So \(\hat A\) is a complete intersection, and so is \(A\).

## Sources

The characterizations of regular local rings by \(D_2(k/A)\) and of complete intersections by \(D_3(k/A)\), and the reading of \(D_1\) and \(D_2\) as numbers of generators and relations, are due to Quillen, in Section 10 of his MIT notes on the homology of commutative rings (1968), [available from Haynes Miller's page](https://math.mit.edu/~hrm/manuscripts/quillen-commutative-algebra.pdf). Avramov's paper cited in Remark 2.8 records these criteria, with references to André's work on the homology of commutative algebras (Remark 1.6 there).