# Weakly regular ideals of noetherian rings

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

For an ideal \(I\) of a noetherian ring \(A\), weak regularity is a familiar condition: \(I\) is weakly regular exactly when, at every maximal ideal containing it, it is generated by a regular sequence. The cotangent complex detects the condition in degree two. With \(B=A/I\), the products of classes of degree one fill \(\operatorname{Tor}^A_2(B,B)\) up to a quotient isomorphic to \(H_2(L_{B/A})\), and over a noetherian local ring the vanishing of \(D_2(B/A,\kappa)\) at the residue field \(\kappa\) already forces \(I\) to be generated by a regular sequence. For algebras of finite type over a noetherian ring, it follows that the weak complete intersections are exactly the local complete intersections. The lesson proves:

- for an ideal of a ring of a topos, an exact sequence that identifies \(H_2(L_{B/A})\) with the cokernel of \(\theta\) in degree two;
- for elements \(f_1,\dots,f_r\) of a ring, an exact sequence linking the first Koszul homology \(H_1(f;A)\) with \(\operatorname{Tor}^A_2(B,B)\); it gives \(H_1(f;A)=0\) when the classes of the \(f_i\) form a basis of \(I/I^2\) and \(\theta\) is surjective in degree two;
- over any ring, \(I/I^2\) is projective and \(I\) is weakly regular if and only if \(\operatorname{Ext}^q_B(L_{B/A},M)=0\) for every \(B\)-module \(M\) and every \(q\geq2\);
- over a noetherian local ring, \(I\) is generated by a regular sequence if and only if \(D_2(B/A,\kappa)=0\), and then every minimal set of generators of \(I\) is a regular sequence;
- over a noetherian ring, ten equivalent conditions: weak regularity of \(I\); projectivity of \(I/I^2\) together with surjectivity of \(\theta\) in degree two, or with \(H_2(L_{B/A})=0\), or with \(S_B(I/I^2)\cong\operatorname{gr}_I(A)\); generation by regular sequences at the maximal ideals; regularity and Koszul regularity of the ideal; vanishing of \(D_2(B/A,M)\), or of \(\operatorname{Ext}^2_B(L_{B/A},M)\), for finitely generated \(M\); vanishing of \(D_2\) at the residue fields of the maximal ideals containing \(I\);
- for an algebra of finite type over a noetherian ring, local complete intersection, weak complete intersection and the vanishing of \(D_2\) agree; for schemes, the same holds for morphisms locally of finite type over a locally noetherian base, and a closed subscheme of a locally noetherian scheme is regularly immersed exactly when its ideal sheaf is weakly regular.

**Prerequisites.** [*Weakly regular ideals*](weakly-regular-ideals.md) (Section 2, Theorem 3.3, Proposition 4.2 and its proof, Exercise 5.2); [*Connectivity of powers of simplicial ideals*](connectivity-of-powers-of-simplicial-ideals.md) (Corollary 2.2); [*The fundamental spectral sequence*](the-fundamental-spectral-sequence.md) (Proposition 1.1, Lemma 4.1, Theorem 4.2, Exercise 5.1); [*Weak complete intersections*](weak-complete-intersections.md) (Definition 2.1, Proposition 2.2, Theorem 4.2); [*Non-additive functors of complexes in a ringed topos*](non-additive-functors-of-complexes-in-a-ringed-topos.md) (Lemma 4.2); [*The cotangent complex*](course:AG-HP/the-cotangent-complex#2-independence-of-the-resolution) (Theorem 2.2, Theorem 4.1, Proposition 5.1). From the AI-integrated Stacks project: regular sequences [[Stacks, Tag 00LF](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-definition-regular-sequence)], regular sequences in a neighbourhood [[Tag 061L](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-regular-sequence-in-neighbourhood)], quasi-regular sequences [[Tag 061P](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-definition-quasi-regular-sequence)] and their regularity over noetherian local rings [[Tag 061S](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-quasi-regular-regular)]; Koszul-regular and \(H_1\)-regular sequences [[Tag 062E](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-definition-koszul-regular-sequence)], regular sequences are Koszul-regular [[Tag 062F](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-lemma-regular-koszul-regular)], Koszul homology is killed by the ideal [[Tag 0663](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-lemma-homotopy-koszul)], the four regularity notions agree over noetherian local rings [[Tag 09CC](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-lemma-noetherian-finite-all-equivalent)]; regular and Koszul-regular ideals [[Tag 07CV](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-definition-regular-ideal)]; local complete intersection homomorphisms [[Tag 07D0](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-definition-local-complete-intersection)] and morphisms [[Tag 069F](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-morphisms.html#more-morphisms-definition-lci), [Tag 07DB](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-morphisms.html#more-morphisms-lemma-affine-lci)]; regular ideal sheaves and regular immersions [[Tag 063D](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/divisors.html#divisors-definition-regular-ideal-sheaf), [Tag 063G](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/divisors.html#divisors-lemma-regular-ideal-sheaf-scheme), [Tag 063J](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/divisors.html#divisors-definition-regular-immersion), [Tag 063L](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/divisors.html#divisors-lemma-regular-immersion-noetherian)]; Nakayama's lemma [[Tag 00DV](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-NAK)]; surjective endomorphisms of finite modules [[Tag 05G8](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-fun)]; finite projective modules [[Tag 00NX](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-finite-projective)]; flatness and localization [[Tag 00HT](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-flat-localization)]; local nature of vanishing and exactness [[Tag 00HN](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-characterize-zero-local)]; Tor of finite modules over noetherian rings [[Tag 0AZ4](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-tor-noetherian)]; Hilbert's basis theorem [[Tag 00FN](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-Noetherian-permanence)]; Ext groups in a derived category [[Tag 06XQ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-definition-ext)] and their vanishing below the gap [[Tag 06XS](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-negative-exts)]; the long exact sequences of Hom functors [[Tag 0149](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-representable-homological)]; the truncation triangle [[Tag 08J5](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-remark-truncation-distinguished-triangle)]; bounded above complexes of flat modules are K-flat [[Tag 06YQ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-lemma-bounded-flat-K-flat)]; perfect complexes [[Tag 0657](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-definition-perfect)]; the standard resolution and inverse images of the cotangent complex [[Tag 08SR](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/cotangent.html#cotangent-definition-standard-resolution-sheaves-rings), [Tag 08SV](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/cotangent.html#cotangent-lemma-pullback-cotangent-morphism-topoi)] and its stalks [[Tag 08T0](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/cotangent.html#cotangent-lemma-stalk-cotangent-complex)]; flatness through stalks [[Tag 05NE](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/modules.html#modules-lemma-flat-stalks-flat)]; stalks of skyscraper sheaves [[Tag 009B](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sheaves.html#sheaves-lemma-skyscraper-stalks)].

## Conventions

\(T\) is a topos and rings are commutative. Section 1 works with rings of \(T\). From Section 2 on, rings are rings in the usual sense, the rings of the punctual topos, to which the earlier lessons apply. For an ideal \(I\) of a ring \(A\), \(B=A/I\) and \(N=I/I^2\). The Tor algebra \(\operatorname{Tor}^A_\bullet(B,B)\), the conormal isomorphism \(\partial\colon\operatorname{Tor}^A_1(B,B)\to N\) and the homomorphism of graded \(B\)-algebras \(\theta\colon\Lambda_B(N)\to\operatorname{Tor}^A_\bullet(B,B)\) extending \(\partial^{-1}\) are those of the lesson on weakly regular ideals; \(\theta_q\) is its component of degree \(q\), and \(I\) is *weakly regular* if \(N\) is flat over \(B\) and \(\theta\) is an isomorphism.

The cotangent complex \(L_{B/A}\) is a complex of flat \(B\)-modules with chain degree \(q\) in cohomological degree \(-q\); \(H_q(L_{B/A})=H^{-q}(L_{B/A})\), and \(D_q(B/A,M)=H_q(L_{B/A}\otimes_BM)\) for a \(B\)-module \(M\) (conventions of the lesson on the fundamental spectral sequence). For \(B=A/I\), \(H_0(L_{B/A})=0\) and \(\partial_L\colon H_1(L_{B/A})\to N\) is an isomorphism (Proposition 1.1 (3) there). For an object \(E\) of \(D(B)\), a \(B\)-module \(M\) and \(q\in\mathbb Z\), \(\operatorname{Ext}^q_B(E,M)=\operatorname{Hom}_{D(B)}(E,M[q])\) [Tag 06XQ]; for modules these are the usual Ext groups. A *finite* module is a finitely generated one.

For elements \(f_1,\dots,f_r\) of a ring \(A\), \(K(f)\) is the Koszul complex of the lesson on weakly regular ideals, the exterior algebra \(\Lambda_A(A^r)\) on \(e_1,\dots,e_r\) with \(de_i=f_i\), and \(H_q(f;M)=H_q(K(f)\otimes_AM)\) for an \(A\)-module \(M\). The sequence is *Koszul-regular* if \(H_q(f;A)=0\) for \(q\neq0\), and *\(H_1\)-regular* if \(H_1(f;A)=0\) [Tag 062E]. It is *regular* if each \(f_i\) is a nonzerodivisor on \(A/(f_1,\dots,f_{i-1})\) and \(A/(f_1,\dots,f_r)\neq0\) [Tag 00LF]. It is *quasi-regular* if, with \(J=(f_1,\dots,f_r)\), the homomorphism from the polynomial ring \((A/J)[X_1,\dots,X_r]\) to \(\operatorname{gr}_J(A)\) mapping \(X_i\) to the class of \(f_i\) is an isomorphism [Tag 061P]. An ideal \(I\) is *regular*, respectively *Koszul-regular*, if every prime \(\mathfrak p\supseteq I\) has an element \(g\notin\mathfrak p\) such that \(I_g\) is generated by a regular, respectively Koszul-regular, sequence of \(A_g\) [Tag 07CV]; an ideal generated by one Koszul-regular sequence is weakly regular by Proposition 4.2 of the lesson on weakly regular ideals. In a local ring \((A,\mathfrak m,\kappa)\), a *minimal set of generators* of a finite ideal \(I\) is a set of generators whose classes form a basis of \(I/\mathfrak mI\) over \(\kappa\); by Nakayama's lemma [Tag 00DV], every lift of a basis of \(I/\mathfrak mI\) is one. For a prime \(\mathfrak p\), \(\kappa(\mathfrak p)=A_{\mathfrak p}/\mathfrak pA_{\mathfrak p}\).

## 1. Products in degree two

**Proposition 1.1.** Let \(A\) be a ring of \(T\), \(I\) an ideal, \(B=A/I\) and \(N=I/I^2\). There is an exact sequence of \(B\)-modules
\[
\begin{aligned}&H_3(L_{B/A})\to\Lambda^2_BN\xrightarrow{\ \theta_2\ }\operatorname{Tor}^A_2(B,B)\\&\quad\to H_2(L_{B/A})\to0.\end{aligned}
\]

**Proof.** Let \(P\to B\) be the resolution of Section 2 of the lesson on weakly regular ideals, \(Q=P\otimes_AB\), and \(J\) the kernel of the augmentation \(Q\to B\). The exact sequence \(0\to J^2\to J\to J/J^2\to0\) of simplicial \(B\)-modules gives the exact sequence
\[
\begin{aligned}&\pi_3(J/J^2)\to\pi_2(J^2)\to\pi_2(J)\\&\quad\to\pi_2(J/J^2)\to\pi_1(J^2),\end{aligned}
\]
in which \(\pi_1(J^2)=0\), \(J^2\) being \(2\)-connective (Corollary 2.2 (1) of the lesson on connectivity of powers of simplicial ideals). By Proposition 1.1 (1) of the lesson on the fundamental spectral sequence, \(J/J^2\cong L_{B/A}\), so \(\pi_q(J/J^2)=H_q(L_{B/A})\). The exact sequence \(0\to J\to Q\to B\to0\), in which \(B\) is constant, shows that \(\pi_2(J)\to\pi_2(Q)=\operatorname{Tor}^A_2(B,B)\) is an isomorphism; through it, \(\pi_2(J^2)\to\pi_2(J)\) becomes the map \(\iota_2\colon\pi_2(J^2)\to\pi_2(Q)\) induced by the inclusion. By Lemma 4.1 of the lesson on the fundamental spectral sequence, \(\theta_2=\iota_2\circ\lambda_2\circ\Lambda^2(\partial^{-1})\), where \(\lambda_2\colon\Lambda^2_B(\pi_1J)\to\pi_2(J^2)\) and \(\Lambda^2(\partial^{-1})\colon\Lambda^2_BN\to\Lambda^2_B(\pi_1J)\), with \(\pi_1(J)=\pi_1(Q)\), are isomorphisms. Replacing \(\pi_2(J^2)\) by \(\Lambda^2_BN\) through the isomorphism \(\lambda_2\circ\Lambda^2(\partial^{-1})\) gives the sequence of the statement. \(\square\)

**Corollary 1.2.** In the situation of Proposition 1.1, the cokernel of \(\theta_2\) is isomorphic to \(H_2(L_{B/A})\). In particular \(\theta_2\) is surjective if and only if \(H_2(L_{B/A})=0\). \(\square\)

## 2. Koszul complexes and the Tor algebra

In this section \(A\) is a ring, \(f_1,\dots,f_r\in A\), \(I=(f_1,\dots,f_r)\), \(B=A/I\), and \(\phi\colon B^r\to N\) maps \(e_i\) to the class \(\bar f_i\) of \(f_i\). The differential of \(K(f)\otimes_AB=\Lambda_B(B^r)\) is zero, since the \(f_i\) vanish in \(B\); so \(H_q(f;B)=\Lambda^q_B(B^r)\). Let \(\rho\colon H_1(f;A)\to H_1(f;B)=B^r\) be the map induced by \(K(f)\to K(f)\otimes_AB\).

**Lemma 2.1.** The ideal \(I\) annihilates \(H_q(f;M)\) for every \(A\)-module \(M\) and every \(q\).

**Proof.** The complex \(K(f)\otimes_AM\) is a differential graded module over \(K(f)\), and \(d(e_i\omega)=f_i\omega-e_i\,d\omega\) for \(\omega\in K(f)\otimes_AM\). So multiplication by \(f_i\) is homotopic to zero, with homotopy \(\omega\mapsto e_i\omega\) [Tag 0663]. \(\square\)

**Proposition 2.2.** The sequence
\[
\begin{aligned}&\Lambda^2_B(B^r)\xrightarrow{\ \theta_2\circ\Lambda^2\phi\ }\operatorname{Tor}^A_2(B,B)\\
&\quad\to H_1(f;A)\xrightarrow{\ \rho\ }B^r\\
&\quad\xrightarrow{\ \partial^{-1}\circ\phi\ }\operatorname{Tor}^A_1(B,B)\to0\end{aligned}
\]
is exact. If \(\phi\) is an isomorphism and \(\theta_2\) is surjective, then \(H_1(f;A)=0\): the sequence \(f_1,\dots,f_r\) is \(H_1\)-regular.

**Proof.** Let \(P\to B\) be the resolution of Section 2 of the lesson on weakly regular ideals and \(Q=P\otimes_AB\). The construction at the start of the proof of Proposition 4.2 there does not use Koszul regularity: there are \(x_i\in N_1P\) with \(dx_i=f_i\) and a homomorphism of differential graded \(A\)-algebras \(\Phi\colon K(f)\to NP\), \(e_i\mapsto x_i\), equal to the structure map \(A\to P_0\) in degree \(0\); and \(NP\otimes_AB=N(Q)\). So \(H(\Phi\otimes_AB)\), from \(\Lambda_B(B^r)\) to \(\pi_\bullet(Q)=\operatorname{Tor}^A_\bullet(B,B)\), is a homomorphism of graded \(B\)-algebras, and its second paragraph shows \(\partial\circ H_1(\Phi\otimes_AB)=\phi\). Two homomorphisms of graded \(B\)-algebras out of \(\Lambda_B(B^r)\) that agree in degree one are equal; hence \(H(\Phi\otimes_AB)=\theta\circ\Lambda(\phi)\).

Let \(C\) be the cone of \(\Phi\): \(C_q=K(f)_{q-1}\oplus N_qP\), with \(d(k,p)=(-dk,\Phi(k)+dp)\). The terms of \(K(f)\) are free, and \(N_qP\) is a direct summand of the free \(A\)-module \(P_q\), so \(C\) is a complex of flat \(A\)-modules in degrees \(\geq0\). The sequence \(0\to NP\to C\to K(f)[1]\to0\), split in each degree, gives the long exact sequence
\[
\begin{aligned}&\cdots\to H_q(NP)\to H_q(C)\\&\quad\to H_{q-1}(K(f))\xrightarrow{\ \pm H(\Phi)\ }H_{q-1}(NP)\\&\quad\to\cdots.\end{aligned}
\]
Since \(H_q(NP)=0\) for \(q\geq1\) and \(H_0(\Phi)\) is the identity of \(B\), we get \(H_0(C)=H_1(C)=0\), and the projection \(C_2\to K(f)_1\) induces an isomorphism \(H_2(C)\cong H_1(f;A)\).

Let \(M\) be an \(A\)-module. We claim that \(H_1(C\otimes_AM)=0\) and that \(H_2(C)\otimes_AM\to H_2(C\otimes_AM)\) is an isomorphism. Let \(Z_1=\ker(C_1\to C_0)\) and \(Z_2=\ker(C_2\to C_1)\). As \(H_0(C)=0\), \(0\to Z_1\to C_1\to C_0\to0\) is exact, so \(Z_1\) is flat; as \(H_1(C)=0\), \(0\to Z_2\to C_2\to Z_1\to0\) is exact, so \(Z_2\) is flat. Let \(C'\subseteq C\) be the subcomplex with \(C'_q=C_q\) for \(q\geq3\), \(C'_2=Z_2\) and \(C'_q=0\) for \(q\leq1\). The quotient \(C/C'\) is the exact complex \(0\to Z_1\to C_1\to C_0\to0\) of flat modules, so \((C/C')\otimes_AM\) is exact, and \(0\to C'\otimes_AM\to C\otimes_AM\to\) \((C/C')\otimes_AM\to0\) is exact because the terms of \(C/C'\) are flat. Hence \(H(C'\otimes_AM)\to H(C\otimes_AM)\) is an isomorphism. The complex \(C'\otimes_AM\) vanishes in degrees \(\leq1\), and by right exactness of \(\otimes_AM\), \(H_2(C'\otimes_AM)\) is the cokernel of \(C_3\otimes_AM\to Z_2\otimes_AM\), which is \(H_2(C)\otimes_AM\). This proves the claim.

The sequence \(0\to NP\otimes_AB\to C\otimes_AB\to\) \((K(f)\otimes_AB)[1]\to0\) is exact, being split in each degree, and its long exact sequence contains
\[
\begin{aligned}&H_2(f;B)\to\pi_2(Q)\to H_2(C\otimes_AB)\\&\quad\to H_1(f;B)\to\pi_1(Q)\\&\quad\to H_1(C\otimes_AB)=0,\end{aligned}
\]
whose maps out of \(H_q(f;B)\) are \(\pm H_q(\Phi\otimes_AB)\), that is \(\pm\theta_2\circ\Lambda^2\phi\) and \(\pm\partial^{-1}\circ\phi\). By Lemma 2.1, \(I\) annihilates \(H_2(C)\cong H_1(f;A)\), so \(H_2(C)\otimes_AB=H_2(C)\), and the claim gives \(H_2(C\otimes_AB)\cong H_1(f;A)\). Under this isomorphism the map \(H_2(C\otimes_AB)\to H_1(f;B)\) is \(\rho\), both being induced by the projection onto \(K(f)_1\). Signs do not affect exactness, so the sequence of the statement is exact.

If \(\phi\) is an isomorphism, \(\partial^{-1}\circ\phi\) is injective, so \(\rho=0\) and \(\operatorname{Tor}^A_2(B,B)\to H_1(f;A)\) is surjective. If moreover \(\theta_2\) is surjective, so is \(\theta_2\circ\Lambda^2\phi\), and the map \(\operatorname{Tor}^A_2(B,B)\to H_1(f;A)\) is zero. Hence \(H_1(f;A)=0\). \(\square\)

If \(f_1,\dots,f_r\) is Koszul-regular, then \(H_1(f;A)=0\), and \(\phi\) and \(\theta\) are isomorphisms (Proposition 4.2 of the lesson on weakly regular ideals). Exercise 5.1 computes the sequence for two elements that generate a principal ideal.

## 3. Noetherian rings

**Lemma 3.1.** Let \(A\) be a ring, \(I\) an ideal, \(B=A/I\), \(N=I/I^2\) and \(S\subseteq A\) a multiplicative subset.

1. The homomorphism of graded rings \(\operatorname{Tor}^A_\bullet(B,B)\to\) \(\operatorname{Tor}^{S^{-1}A}_\bullet(S^{-1}B,S^{-1}B)\) of the square from \(A\to B\) to \(S^{-1}A\to S^{-1}B\) induces an isomorphism \(S^{-1}\operatorname{Tor}^A_\bullet(B,B)\cong\) \(\operatorname{Tor}^{S^{-1}A}_\bullet(S^{-1}B,S^{-1}B)\). With \(S^{-1}I/(S^{-1}I)^2=S^{-1}N\), it carries \(S^{-1}\theta_{A,I}\) to \(\theta_{S^{-1}A,S^{-1}I}\).
2. If \(I\) is weakly regular, so is \(S^{-1}I\).
3. \(I\) is weakly regular if and only if \(I_{\mathfrak m}\) is a weakly regular ideal of \(A_{\mathfrak m}\) for every maximal ideal \(\mathfrak m\supseteq I\).
4. If \(g_1,\dots,g_n\in A\) generate the unit ideal, \(I\) is weakly regular if and only if every \(I_{g_j}\) is a weakly regular ideal of \(A_{g_j}\).

**Proof.** (1) The homomorphism is that of Proposition 2.2 (3) of the lesson on weakly regular ideals; on modules it is the map of Tor groups induced by the square. Its target is an \(S^{-1}A\)-module, so it extends to an \(S^{-1}A\)-linear homomorphism of graded rings out of \(S^{-1}\operatorname{Tor}^A_\bullet(B,B)\). To see that the extension is bijective, compute with a resolution \(F\to B\) by free \(A\)-modules: \(S^{-1}A\) is flat over \(A\) [Tag 00HT], so \(S^{-1}F\to S^{-1}B\) is a free resolution over \(S^{-1}A\), the map of resolutions \(F\to S^{-1}F\) computes the map of Tor groups, and \(S^{-1}F\otimes_{S^{-1}A}S^{-1}B=S^{-1}(F\otimes_AB)\). As localization is exact, the homology of \(S^{-1}(F\otimes_AB)\) is \(S^{-1}H(F\otimes_AB)\), and the extension is the identity of it. Finally \(\theta\) is natural (Section 2 of the lesson on weakly regular ideals): \(\theta_{S^{-1}A,S^{-1}I}\circ\Lambda(N\to S^{-1}N)\) equals \(\theta_{A,I}\) followed by the homomorphism of Tor algebras. Both \(\theta_{S^{-1}A,S^{-1}I}\) and the localization of \(\theta_{A,I}\) are \(S^{-1}A\)-linear on \(\Lambda(S^{-1}N)=S^{-1}\Lambda(N)\) and agree on the image of \(\Lambda(N)\), so they agree.

(2) follows from (1), since \(S^{-1}N\) is flat over \(S^{-1}B\) when \(N\) is flat over \(B\).

(3) The maximal ideals of \(B\) are the \(\mathfrak m/I\) with \(\mathfrak m\supseteq I\) maximal in \(A\), and the localization of \(B\) at \(\mathfrak m/I\) is \(B_{\mathfrak m}=A_{\mathfrak m}/I_{\mathfrak m}\). A \(B\)-module is flat if and only if its localizations at these maximal ideals are flat [Tag 00HT], and a homomorphism of \(B\)-modules is bijective if and only if its localizations at them are [Tag 00HN]. Apply this to \(N\) and to \(\theta\), using (1) with \(S=A\smallsetminus\mathfrak m\).

(4) Every maximal ideal \(\mathfrak m\supseteq I\) misses some \(g_j\), and \(A_{\mathfrak m}\) is a localization of \(A_{g_j}\). So the condition implies, by (2), that every \(I_{\mathfrak m}\) is weakly regular, and (3) applies. The converse is (2). \(\square\)

**Lemma 3.2.** Let \(A\) be a ring, \(I\) an ideal, \(B=A/I\), \(N=I/I^2\) and \(M\) a \(B\)-module. There are exact sequences
\[
\begin{aligned}&\operatorname{Tor}^B_2(N,M)\to H_2(L_{B/A})\otimes_BM\\&\quad\to D_2(B/A,M)\to\operatorname{Tor}^B_1(N,M)\to0,\end{aligned}
\]
\[
\begin{aligned}&0\to\operatorname{Ext}^1_B(N,M)\to\operatorname{Ext}^2_B(L_{B/A},M)\\&\quad\to\operatorname{Hom}_B(H_2(L_{B/A}),M)\\&\quad\to\operatorname{Ext}^2_B(N,M).\end{aligned}
\]

**Proof.** Write \(L=L_{B/A}\) and \(X=\tau^{\leq-2}L\). The truncation triangle [Tag 08J5] is \(X\to L\to\tau^{\geq-1}L\to X[1]\). As \(H^0(L)=0\) and \(H^{-1}(L)\cong N\), \(\tau^{\geq-1}L\cong N[1]\). The cohomology of \(X\) vanishes in degrees \(>-2\), and \(H^{-2}(X)=H_2(L)\).

By Lemma 4.2 (1) of the lesson on non-additive functors, applied to \(X[-2]\), \(X\) is represented by a complex of flat modules vanishing in degrees \(>-2\), which is K-flat [Tag 06YQ]. So \(X\otimes^{\mathbf L}_BM\) is represented by a complex vanishing in degrees \(>-2\); its cohomology vanishes in degree \(-1\), and by right exactness of \(\otimes_BM\) it is \(H_2(L)\otimes_BM\) in degree \(-2\). Also \(H^{-q}(N[1]\otimes^{\mathbf L}_BM)=\operatorname{Tor}^B_{q-1}(N,M)\), and \(H^{-2}(L\otimes^{\mathbf L}_BM)=D_2(B/A,M)\), since \(L\) has flat terms. The long exact cohomology sequence of the triangle \(X\otimes^{\mathbf L}_BM\to L\otimes^{\mathbf L}_BM\to N[1]\otimes^{\mathbf L}_BM\to\) in degrees \(-3\) to \(-1\) is the first sequence.

The functor \(\operatorname{Hom}_{D(B)}(-,M[2])\) is cohomological [Tag 0149]; on the triangle it gives the exact sequence
\[
\begin{aligned}&\operatorname{Hom}(X[1],M[2])\to\operatorname{Hom}(N[1],M[2])\\&\quad\to\operatorname{Hom}(L,M[2])\to\operatorname{Hom}(X,M[2])\\&\quad\to\operatorname{Hom}(N,M[2]).\end{aligned}
\]
Here \(\operatorname{Hom}(X[1],M[2])=\operatorname{Ext}^1_B(X,M)=0\) and \(\operatorname{Ext}^2_B(X,M)=\operatorname{Hom}_B(H^{-2}(X),M)\) [Tag 06XS], while the second and last terms are \(\operatorname{Ext}^1_B(N,M)\) and \(\operatorname{Ext}^2_B(N,M)\). This is the second sequence. \(\square\)

**Proposition 3.3.** Let \(A\) be a ring, \(I\) an ideal, \(B=A/I\) and \(N=I/I^2\). The following are equivalent:

1. \(N\) is projective and \(I\) is weakly regular;
2. \(N\) is projective and \(\partial_L\) induces an isomorphism \(L_{B/A}\cong N[1]\) in \(D(B)\);
3. \(\operatorname{Ext}^q_B(L_{B/A},M)=0\) for every \(B\)-module \(M\) and every \(q\geq2\).

**Proof.** (1) and (2) are equivalent by Theorem 4.2 of the lesson on the fundamental spectral sequence.

(2) implies (3): \(\operatorname{Ext}^q_B(N[1],M)=\operatorname{Ext}^{q-1}_B(N,M)=0\) for \(q\geq2\), as \(N\) is projective.

(3) implies (2): by the second sequence of Lemma 3.2, \(\operatorname{Ext}^1_B(N,M)=0\) for every \(M\), so \(N\) is projective. We show \(H_k(L_{B/A})=0\) by induction on \(k\geq2\). Assume \(H_j(L_{B/A})=0\) for \(2\leq j<k\), and let \(X=\tau^{\leq-k}L_{B/A}\). Then \(\tau^{\geq-k+1}L_{B/A}\cong N[1]\), and the cohomological functor \(\operatorname{Hom}(-,M[k])\) on the triangle \(X\to L_{B/A}\to N[1]\to X[1]\) gives the exact sequence
\[
\begin{aligned}&\operatorname{Ext}^{k-1}_B(N,M)\to\operatorname{Ext}^k_B(L_{B/A},M)\\&\quad\to\operatorname{Ext}^k_B(X,M)\to\operatorname{Ext}^k_B(N,M).\end{aligned}
\]
Its outer terms vanish, \(N\) being projective and \(k-1\geq1\), and its second term vanishes by (3). So \(\operatorname{Hom}_B(H_k(L_{B/A}),M)=\) \(\operatorname{Ext}^k_B(X,M)=0\) for every \(M\) [Tag 06XS]; for \(M=H_k(L_{B/A})\) this gives \(H_k(L_{B/A})=0\). Hence \(L_{B/A}\to\tau^{\geq-1}L_{B/A}\cong N[1]\) is an isomorphism, the last one given by \(\partial_L\). \(\square\)

**Proposition 3.4.** Let \((A,\mathfrak m,\kappa)\) be a noetherian local ring, \(I\subseteq\mathfrak m\) an ideal, \(B=A/I\) and \(N=I/I^2\). The following are equivalent:

1. \(I\) is generated by a regular sequence;
2. every minimal set of generators of \(I\) is a regular sequence;
3. \(I\) is weakly regular;
4. \(N\) is free and \(H_2(L_{B/A})=0\);
5. \(D_2(B/A,\kappa)=0\).

**Proof.** (2) implies (1), as \(I\) is finite and so has a minimal set of generators.

(1) implies (3): a regular sequence is Koszul-regular [Tag 062F], and an ideal generated by a Koszul-regular sequence is weakly regular (Proposition 4.2 of the lesson on weakly regular ideals).

(3) implies (4): \(N\) is a finite flat module over the noetherian local ring \(B\), so it is finite projective, hence free [Tag 00NX]; and \(H_2(L_{B/A})=0\) by Theorem 4.2 of the lesson on the fundamental spectral sequence.

(4) implies (2): let \(f_1,\dots,f_r\) be a minimal set of generators of \(I\). By Corollary 1.2, \(\theta_2\) is surjective. As \(I\subseteq\mathfrak m\), \(N\otimes_B\kappa=I/(I^2+\mathfrak mI)=I/\mathfrak mI\), with basis the classes of the \(f_i\); so the free module \(N\) has rank \(r\). The map \(\phi\colon B^r\to N\), \(e_i\mapsto\bar f_i\), is surjective, since the \(f_i\) generate \(I\); composed with an isomorphism \(N\cong B^r\) it becomes a surjective endomorphism of \(B^r\), which is bijective [Tag 05G8]. So \(\phi\) is an isomorphism, and Proposition 2.2 gives \(H_1(f;A)=0\). Over the noetherian local ring \(A\), an \(H_1\)-regular sequence in \(\mathfrak m\) is regular [Tag 09CC].

(4) implies (5): the first sequence of Lemma 3.2 for \(M=\kappa\) has \(\operatorname{Tor}^B_1(N,\kappa)=0\) and \(H_2(L_{B/A})\otimes_B\kappa=0\).

(5) implies (4): by the same sequence, \(\operatorname{Tor}^B_1(N,\kappa)=0\). Lift a basis of \(N\otimes_B\kappa\) to elements of \(N\); they generate \(N\) [Tag 00DV], giving an exact sequence \(0\to K\to B^s\to N\to0\) in which \(K\) is finite, \(B\) being noetherian. In the exact sequence \(\operatorname{Tor}^B_1(N,\kappa)\to K\otimes_B\kappa\to\) \(\kappa^s\to N\otimes_B\kappa\to0\) the third map is bijective, so \(K\otimes_B\kappa=0\) and \(K=0\) by Nakayama's lemma: \(N\) is free. Then \(\operatorname{Tor}^B_2(N,\kappa)=0\), and the sequence gives \(H_2(L_{B/A})\otimes_B\kappa=0\). By Proposition 1.1, \(H_2(L_{B/A})\) is a quotient of \(\operatorname{Tor}^A_2(B,B)\), a finite \(A\)-module [Tag 0AZ4]; so it is a finite \(B\)-module, and it vanishes by Nakayama's lemma. \(\square\)

**Theorem 3.5.** Let \(A\) be a noetherian ring, \(I\) an ideal, \(B=A/I\) and \(N=I/I^2\). The following are equivalent:

1. \(I\) is weakly regular;
2. \(N\) is projective and \(\theta_2\colon\Lambda^2_BN\to\operatorname{Tor}^A_2(B,B)\) is surjective;
3. \(N\) is projective and \(H_2(L_{B/A})=0\);
4. \(N\) is projective and the multiplication \(S_B(N)\to\operatorname{gr}_I(A)\) is an isomorphism;
5. for every maximal ideal \(\mathfrak m\supseteq I\), \(I_{\mathfrak m}\) is generated by a regular sequence of \(A_{\mathfrak m}\);
6. \(I\) is a regular ideal;
7. \(I\) is a Koszul-regular ideal;
8. \(\operatorname{Ext}^2_B(L_{B/A},M)=0\) for every finite \(B\)-module \(M\);
9. \(D_2(B/A,M)=0\) for every finite \(B\)-module \(M\);
10. \(D_2(B/A,\kappa(\mathfrak m))=0\) for every maximal ideal \(\mathfrak m\supseteq I\).

When they hold, \(L_{B/A}\cong N[1]\), and \(D_q(B/A,M)=0\) and \(\operatorname{Ext}^q_B(L_{B/A},M)=0\) for every \(B\)-module \(M\) and every \(q\geq2\).

**Proof.** For a maximal ideal \(\mathfrak m\supseteq I\), \(A_{\mathfrak m}\) is a noetherian local ring, \(I_{\mathfrak m}\subseteq\mathfrak mA_{\mathfrak m}\), \(B_{\mathfrak m}=A_{\mathfrak m}/I_{\mathfrak m}\) and \(I_{\mathfrak m}/I_{\mathfrak m}^2=N_{\mathfrak m}\). By Proposition 5.1 of *The cotangent complex*, \(L_{B_{\mathfrak m}/A_{\mathfrak m}}\cong L_{B/A}\otimes_BB_{\mathfrak m}\). So \(H_q(L_{B_{\mathfrak m}/A_{\mathfrak m}})=H_q(L_{B/A})_{\mathfrak m}\) and, \(\kappa(\mathfrak m)\) being a \(B_{\mathfrak m}\)-module, \(D_2(B_{\mathfrak m}/A_{\mathfrak m},\kappa(\mathfrak m))=D_2(B/A,\kappa(\mathfrak m))\). The ideal \(I\) is finite, so \(N\) is a finite \(B\)-module.

(1) implies (2): \(N\) is finite and flat over the noetherian ring \(B\), hence projective [Tag 00NX]; and \(\theta\) is an isomorphism.

(2) and (3) are equivalent by Corollary 1.2.

(3) implies (5): for a maximal ideal \(\mathfrak m\supseteq I\), \(N_{\mathfrak m}\) is a finite projective module over the local ring \(B_{\mathfrak m}\), hence free [Tag 00NX], and \(H_2(L_{B_{\mathfrak m}/A_{\mathfrak m}})=H_2(L_{B/A})_{\mathfrak m}=0\). Proposition 3.4 applies to \(I_{\mathfrak m}\).

(1) implies (4): \(N\) is projective as in (1) implies (2), and Theorem 3.3 (1) of the lesson on weakly regular ideals gives the isomorphism.

(4) implies (5): let \(\mathfrak m\supseteq I\) be maximal. Localization is exact and commutes with symmetric powers, so \(N_{\mathfrak m}\) is free and \(S(N_{\mathfrak m})\to\operatorname{gr}_{I_{\mathfrak m}}(A_{\mathfrak m})\) is an isomorphism. Let \(f_1,\dots,f_r\) be a minimal set of generators of \(I_{\mathfrak m}\). As in the proof of Proposition 3.4, the classes of the \(f_i\) form a basis of \(N_{\mathfrak m}\). So \(B_{\mathfrak m}[X_1,\dots,X_r]\to S(N_{\mathfrak m})\to\operatorname{gr}_{I_{\mathfrak m}}(A_{\mathfrak m})\), \(X_i\mapsto\bar f_i\), is an isomorphism: the sequence is quasi-regular [Tag 061P], hence regular [Tag 061S].

(5) implies (6): let \(\mathfrak p\supseteq I\) be prime and \(\mathfrak m\supseteq\mathfrak p\) maximal. A regular sequence generating \(I_{\mathfrak m}\) can be multiplied by units of \(A_{\mathfrak m}\) without changing either property; so there are \(f_1,\dots,f_r\in I\) whose images form a regular sequence of \(A_{\mathfrak m}\) generating \(I_{\mathfrak m}\). By [Tag 061L] there is \(g\notin\mathfrak m\) such that the images in \(A_g\) form a regular sequence. The finite module \(I/(f_1,\dots,f_r)\) vanishes after localization at \(\mathfrak m\), so it is killed by some \(h\notin\mathfrak m\). In \(A_{gh}\) the \(f_i\) generate \(I_{gh}\), and they form a regular sequence: \(A_{gh}\) is flat over \(A_g\), so each \(f_i\) remains a nonzerodivisor modulo the earlier ones, and \(A_{gh}/I_{gh}=B_{gh}\neq0\) because \(B_{\mathfrak p}\neq0\) is a localization of it. As \(gh\notin\mathfrak p\), \(I\) is a regular ideal [Tag 07CV].

(6) implies (7): regular sequences are Koszul-regular [Tag 062F].

(7) implies (1): for every prime \(\mathfrak p\supseteq I\) choose \(g_{\mathfrak p}\notin\mathfrak p\) such that \(I_{g_{\mathfrak p}}\) is generated by a Koszul-regular sequence of \(A_{g_{\mathfrak p}}\); then \(I_{g_{\mathfrak p}}\) is weakly regular. For \(h\in I\), \(I_h=A_h\) is weakly regular, as \(B_h=0\). The elements \(g_{\mathfrak p}\) and the elements of \(I\) generate the unit ideal: a prime containing \(I\) misses \(g_{\mathfrak p}\) for itself, and a prime not containing \(I\) misses some \(h\in I\). Finitely many of them generate the unit ideal, and Lemma 3.1 (4) applies.

(1) implies (8): \(N\) is projective, as in (1) implies (2); Proposition 3.3 gives \(\operatorname{Ext}^q_B(L_{B/A},M)=0\) for every \(M\) and \(q\geq2\).

(8) implies (3): by the second sequence of Lemma 3.2, \(\operatorname{Ext}^1_B(N,M)=0\) for every finite \(M\). Choose an exact sequence \(0\to K\to B^s\to N\to0\); \(K\) is finite, \(B\) being noetherian. In the exact sequence \(\operatorname{Hom}_B(B^s,K)\to\operatorname{Hom}_B(K,K)\to\) \(\operatorname{Ext}^1_B(N,K)=0\), the identity of \(K\) extends to a retraction \(B^s\to K\); so the sequence splits and \(N\) is projective. Then \(\operatorname{Ext}^2_B(N,M)=0\), and Lemma 3.2 gives \(\operatorname{Hom}_B(H_2(L_{B/A}),M)=0\) for every finite \(M\). As in the proof of Proposition 3.4, \(H_2(L_{B/A})\) is finite; for \(M=H_2(L_{B/A})\) we get \(H_2(L_{B/A})=0\).

(1) implies (9): Theorem 4.2 of the lesson on the fundamental spectral sequence.

(9) implies (10): \(\kappa(\mathfrak m)\) is a finite \(B\)-module.

(10) implies (5): for a maximal ideal \(\mathfrak m\supseteq I\), \(D_2(B_{\mathfrak m}/A_{\mathfrak m},\kappa(\mathfrak m))=\) \(D_2(B/A,\kappa(\mathfrak m))=0\), and Proposition 3.4 applies to \(I_{\mathfrak m}\).

The implications \(1\Rightarrow2\Rightarrow3\Rightarrow5\Rightarrow6\Rightarrow7\Rightarrow1\), \(1\Rightarrow4\Rightarrow5\), \(1\Rightarrow8\Rightarrow3\) and \(1\Rightarrow9\Rightarrow10\Rightarrow5\) prove the equivalence. The last assertions follow from Theorem 4.2 of the lesson on the fundamental spectral sequence and Proposition 3.3. \(\square\)

Projectivity of \(N\) alone does not suffice. For a field \(k\), \(A=k[t]/(t^2)\) and \(I=(t)\), \(N=kt\) is free of rank one over \(B=k\), while \(H_2(L_{k/A})\cong k\) (Exercise 5.1 of the lesson on the fundamental spectral sequence). Accordingly \(\theta_2\) is not surjective: \(\Lambda^2_kN=0\) and \(\operatorname{Tor}^A_2(k,k)\cong k\) (Exercise 5.2 of the lesson on weakly regular ideals).

## 4. Algebras of finite type and schemes

**Theorem 4.1.** Let \(A\) be a noetherian ring and \(B\) an \(A\)-algebra of finite type. The following are equivalent:

1. \(A\to B\) is a local complete intersection homomorphism [Tag 07D0];
2. \(B\) is a weak complete intersection over \(A\);
3. \(L_{B/A}\) is perfect with tor amplitude in \([-1,0]\);
4. for some presentation \(B=P/I\), \(P=A[x_1,\dots,x_n]\), the ideal \(I\) is weakly regular;
5. for every such presentation, \(I\) is weakly regular;
6. \(D_2(B/A,M)=0\) for every finite \(B\)-module \(M\);
7. \(\operatorname{Ext}^2_B(L_{B/A},M)=0\) for every finite \(B\)-module \(M\);
8. \(D_2(B/A,\kappa(\mathfrak n))=0\) for every maximal ideal \(\mathfrak n\) of \(B\).

In that case, for every presentation as in (4), \(I/I^2\) is finite projective and \(L_{B/A}\) is isomorphic in \(D(B)\) to the complex \(I/I^2\to\Omega_{P/A}\otimes_PB\), \(\bar f\mapsto df\otimes1\), in degrees \(-1\) and \(0\).

**Proof.** Fix a presentation \(B=P/I\). The ring \(P\) is noetherian [Tag 00FN], and \(L_{P/A}\to\Omega_{P/A}[0]\) is a quasi-isomorphism onto a finite free module (Theorem 2.2 of *The cotangent complex*). In the transitivity triangle \(X\to L_{B/A}\to L_{B/P}\to X[1]\) of \(A\to P\to B\) (Theorem 4.1 there), \(X=L_{P/A}\otimes^{\mathbf L}_PB\cong\Omega_{P/A}\otimes_PB\) is a finite free \(B\)-module in degree \(0\). For a \(B\)-module \(M\), \(H^i(X\otimes^{\mathbf L}_BM)=0\) for \(i\neq0\), so the long exact sequence gives \(D_2(B/A,M)\cong D_2(B/P,M)\). The cohomological functor \(\operatorname{Hom}(-,M[2])\) gives the exact sequence
\[
\begin{aligned}&\operatorname{Ext}^1_B(X,M)\to\operatorname{Ext}^2_B(L_{B/P},M)\\&\quad\to\operatorname{Ext}^2_B(L_{B/A},M)\to\operatorname{Ext}^2_B(X,M),\end{aligned}
\]
whose outer terms vanish; so \(\operatorname{Ext}^2_B(L_{B/A},M)\cong\operatorname{Ext}^2_B(L_{B/P},M)\). The maximal ideals of \(B\) are the \(\mathfrak m/I\) with \(\mathfrak m\supseteq I\) maximal in \(P\), with the same residue fields. Theorem 3.5, applied to the ideal \(I\) of \(P\), therefore shows that (6), (7), (8) and the weak regularity of \(I\) are equivalent.

By Theorem 4.2 of the lesson on weak complete intersections, \(I\) is weakly regular if and only if \(B\) is a weak complete intersection over \(A\). Condition (2) does not involve the presentation; so (2), (4) and (5) are equivalent.

By [Tag 07D0], \(A\to B\) is a local complete intersection if and only if \(I\) is a Koszul-regular ideal, for one presentation or equivalently for all. By Theorem 3.5 for \(P\), this holds if and only if \(I\) is weakly regular. So (1) is equivalent to (4).

If \(I\) is weakly regular, Theorem 4.2 of the lesson on weak complete intersections gives the two-term complex, \(I/I^2\) is finite projective by Theorem 3.5, and \(\Omega_{P/A}\otimes_PB\) is finite free; so \(L_{B/A}\) is perfect [Tag 0657] with tor amplitude in \([-1,0]\), and (2) implies (3). Finally (3) implies (2) by definition. \(\square\)

**Remark 4.2.** The AI-integrated Stacks project proves that a local complete intersection homomorphism has a perfect cotangent complex with tor amplitude in \([-1,0]\) [[Tag 08SL](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/cotangent.html#cotangent-proposition-cotangent-complex-local-complete-intersection)], and, for a homomorphism of noetherian rings of finite tor dimension, the converse for the naive cotangent complex [[Tag 0FJT](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/dpa.html#dpa-lemma-perfect-NL-lci)]. Theorem 4.1 assumes nothing about the tor dimension of \(B\) over \(A\): the cotangent complex in degree two takes its place.

**Corollary 4.3.** Let \(f\colon X\to Y\) be a morphism of schemes, locally of finite type, with \(Y\) locally noetherian. Then \(f\) is a local complete intersection morphism [Tag 069F] if and only if it is a weak complete intersection.

**Proof.** A local complete intersection morphism is a weak complete intersection by Proposition 2.2 (3) of the lesson on weak complete intersections. Conversely, let \(f\) be a weak complete intersection, and let \(U=\operatorname{Spec}B\subseteq X\) and \(V=\operatorname{Spec}A\subseteq Y\) be affine opens with \(f(U)\subseteq V\); \(A\) is noetherian and \(B\) is of finite type over \(A\). Restriction to \(U\) is exact and every \(\mathcal O_U\)-module is the restriction of an \(\mathcal O_X\)-module, its extension by zero, so \(L_{U/V}=L_{X/Y}|_U\) [Tag 08SV] has tor amplitude in \([-1,0]\). Let \(\mathfrak n\) be a maximal ideal of \(B\), \(u\in U\) the corresponding closed point and \(\mathfrak p\) the prime of \(A\) under \(\mathfrak n\). By [Tag 08T0], the stalk of \(L_{U/V}\) at \(u\) is \(L_{B_{\mathfrak n}/A_{\mathfrak p}}\). Taking stalks at \(u\) is exact and commutes with tensor products, and the skyscraper sheaf at the closed point \(u\) with value \(\kappa(\mathfrak n)\) has stalk \(\kappa(\mathfrak n)\) at \(u\) [Tag 009B]; so \(H^{-2}(L_{B_{\mathfrak n}/A_{\mathfrak p}}\otimes^{\mathbf L}\kappa(\mathfrak n))\) is the stalk at \(u\) of a sheaf that vanishes. By Proposition 5.1 of *The cotangent complex*, \(L_{B_{\mathfrak n}/A_{\mathfrak p}}\cong L_{B/A}\otimes_BB_{\mathfrak n}\), so \(D_2(B/A,\kappa(\mathfrak n))=0\). By Theorem 4.1, \(A\to B\) is a local complete intersection homomorphism, so \(U\to V\) is a local complete intersection morphism [Tag 07DB]. Such opens \(U\) cover \(X\), and the condition of [Tag 069F] is imposed point by point. \(\square\)

**Corollary 4.4.** Let \(X\) be a locally noetherian scheme and \(Z\subseteq X\) a closed subscheme with ideal sheaf \(\mathcal I\). The following are equivalent:

1. \(\mathcal I\) is a weakly regular ideal of the ring \(\mathcal O_X\) of the Zariski topos of \(X\);
2. for every \(x\in Z\), \(\mathcal I_x\) is generated by a regular sequence of \(\mathcal O_{X,x}\);
3. for every affine open \(U=\operatorname{Spec}A\) of \(X\), the ideal \(\mathcal I(U)\) of \(A\) is weakly regular;
4. \(Z\to X\) is a regular immersion [Tag 063J].

**Proof.** (1) and (2) are equivalent. The resolution \(P\) of \(\mathcal O_Z\) over \(\mathcal O_X\) of the lesson on weakly regular ideals is the standard resolution [Tag 08SR], built from free algebras on sheaves of sets, whose formation commutes with inverse images along morphisms of topoi, as in [Tag 08SV]; in particular its stalk at \(x\) is the resolution of \(\mathcal O_{Z,x}\) over \(\mathcal O_{X,x}\). Taking stalks is exact and commutes with tensor products, so the stalks of \(\operatorname{Tor}^{\mathcal O_X}_\bullet(\mathcal O_Z,\mathcal O_Z)\), with its product, of \(\partial\) and of \(\theta\) at \(x\) are those of the ideal \(\mathcal I_x\) of \(\mathcal O_{X,x}\). A homomorphism of sheaves is an isomorphism if and only if its stalks are, and \(\mathcal I/\mathcal I^2\) is flat if and only if its stalks are [Tag 05NE]. So \(\mathcal I\) is weakly regular if and only if every \(\mathcal I_x\) is. For \(x\notin Z\), \(\mathcal I_x=\mathcal O_{X,x}\) is weakly regular; for \(x\in Z\), \(\mathcal O_{X,x}\) is a noetherian local ring and \(\mathcal I_x\) lies in its maximal ideal, and Proposition 3.4 applies.

(2) and (3) are equivalent: for \(U=\operatorname{Spec}A\), \(I=\mathcal I(U)\) and \(x\in U\cap Z\) with prime \(\mathfrak p\supseteq I\), \(\mathcal I_x=I_{\mathfrak p}\subseteq A_{\mathfrak p}=\mathcal O_{X,x}\). If (3) holds, \(I_{\mathfrak p}\) is weakly regular by Lemma 3.1 (2), and generated by a regular sequence by Proposition 3.4. If (2) holds, Theorem 3.5 ((5) implies (1)) applies to \(I\).

(2) implies (4): for \(x\in Z\) choose an affine open \(U=\operatorname{Spec}A\ni x\), with \(I=\mathcal I(U)\) and prime \(\mathfrak p\). The argument of (5) implies (6) in Theorem 3.5, with \(\mathfrak p\) in place of \(\mathfrak m\), gives \(g\notin\mathfrak p\) such that \(I_g\) is generated by a regular sequence of \(A_g\). So \(\mathcal I\) is a regular ideal sheaf [Tag 063D], and \(Z\to X\) is a regular immersion.

(4) implies (2): by [Tag 063G], every \(x\in Z\) has an affine open neighbourhood \(U=\operatorname{Spec}A\) on which \(\mathcal I\) is generated by a regular sequence \(f_1,\dots,f_r\) of \(A\). With \(\mathfrak p\) the prime of \(x\), \(A_{\mathfrak p}\) is flat over \(A\), so each \(f_i\) remains a nonzerodivisor modulo the earlier ones in \(A_{\mathfrak p}\), and \(A_{\mathfrak p}/(f_1,\dots,f_r)=\mathcal O_{Z,x}\neq0\). \(\square\)

For locally noetherian \(X\), regular, Koszul-regular, \(H_1\)-regular and quasi-regular immersions coincide [Tag 063L], so condition (4) may be read with any of them.

## 5. Exercises

**Exercise 5.1.** Let \(k\) be a field, \(A=k[x,y]\), \(f_1=x\) and \(f_2=xy\), so that \(I=(x)\) and \(B=k[y]\). Compute \(H_1(f;A)\), \(\operatorname{Tor}^A_1(B,B)\) and \(\operatorname{Tor}^A_2(B,B)\), and check the exact sequence of Proposition 2.2. Which hypothesis of its last assertion fails?

**Exercise 5.2.** Let \(k\) be a field, \(A=k[x,y]\), \(I=(x^2,xy)\), \(B=A/I\) and \(N=I/I^2\).

1. Show that \(N\otimes_B\kappa(\mathfrak p)\) has dimension \(2\) for \(\mathfrak p=(x,y)\) and dimension \(1\) for \(\mathfrak p=(x,y-1)\). Conclude that \(N\) is not projective and that \(I\) is not weakly regular.
2. Show that \(D_2(B/A,\kappa(\mathfrak m))\neq0\) for \(\mathfrak m=(x,y)\), and that \(D_2(B/A,\kappa(\mathfrak m))=0\) for every other maximal ideal \(\mathfrak m\supseteq I\).

**Exercise 5.3.** Let \(k\) be a field, \(P=k[x,y,z]\), \(I=(xy,yz,zx)\) and \(B=P/I\), the union of the three coordinate axes. Show that \(D_2(B/k,\kappa(\mathfrak n))\neq0\) for the maximal ideal \(\mathfrak n\) of the origin and \(D_2(B/k,\kappa(\mathfrak n))=0\) for every other maximal ideal \(\mathfrak n\) of \(B\). Conclude that \(B\) is not a local complete intersection over \(k\).

**Exercise 5.4.** Let \(A\) be a noetherian ring, \(I\) an ideal and \(B=A/I\). Show that \(I\) is weakly regular if and only if \(\operatorname{Ext}^2_B(L_{B/A},\kappa(\mathfrak m))=0\) for every maximal ideal \(\mathfrak m\supseteq I\).

## 6. Complete solutions

**Solution 5.1.** The Koszul complex is \(0\to Ae_1e_2\to Ae_1\oplus Ae_2\to A\), with \(de_1=x\), \(de_2=xy\) and \(d(e_1e_2)=xe_2-xye_1=xz\), where \(z=-ye_1+e_2\). A \(1\)-cycle \(ae_1+be_2\) satisfies \(ax+bxy=0\), that is \(a=-by\), as \(A\) is a domain; so the cycles are the multiples of \(z\), the boundaries are the multiples of \(xz\), and \(H_1(f;A)\cong A/xA=B\), generated by the class of \(z\). As \(x\) is a nonzerodivisor, \(0\to A\xrightarrow{x}A\to B\to0\) is a free resolution; tensored with \(B\) it has zero differential, so \(\operatorname{Tor}^A_1(B,B)\cong B\), \(\operatorname{Tor}^A_2(B,B)=0\), and \(N=xA/x^2A\) is free on \(\bar x\). The map \(\phi\) sends \(ae_1+be_2\) to \((a+by)\bar x\); its kernel is generated by \(-ye_1+e_2=\rho(z)\). So the sequence of Proposition 2.2 is \(B\to0\to B\xrightarrow{\rho}B^2\to\operatorname{Tor}^A_1(B,B)\to0\), exact with \(\rho\) injective. Here \(\theta_2\) is surjective onto \(0\), but \(\phi\) is not injective: two elements generate the ideal \(I\), whose conormal module has rank one; accordingly \(H_1(f;A)\neq0\).

**Solution 5.2.** (1) For a prime \(\mathfrak p\supseteq I\), \(N\otimes_B\kappa(\mathfrak p)=I_{\mathfrak p}/\mathfrak pI_{\mathfrak p}\), because \(I^2\subseteq\mathfrak pI\). For \(\mathfrak p=(x,y)\): \(I\) is generated by \(x^2\) and \(xy\) in degree \(2\), and \(\mathfrak pI\) is spanned by monomials of degree \(\geq3\), so \(I/\mathfrak pI\) has basis \(x^2\), \(xy\); localizing at \(\mathfrak p\) does not change this \(\kappa(\mathfrak p)\)-vector space, and its dimension is \(2\). For \(\mathfrak p=(x,y-1)\): \(y\notin\mathfrak p\), so \(y\) is a unit of \(A_{\mathfrak p}\) and \(I_{\mathfrak p}=x(x,y)A_{\mathfrak p}=xA_{\mathfrak p}\); as \(x\) is a nonzerodivisor, \(I_{\mathfrak p}/\mathfrak pI_{\mathfrak p}\cong A_{\mathfrak p}/\mathfrak pA_{\mathfrak p}\) has dimension \(1\). The ideal \(I\) has radical \((x)\), so \(\operatorname{Spec}B\) is homeomorphic to \(\operatorname{Spec}k[y]\), which is irreducible and so connected. A finite projective module has locally constant rank [Tag 00NX], hence constant rank on \(\operatorname{Spec}B\); so \(N\) is not projective. By Theorem 3.5, \(I\) is not weakly regular.

(2) Let \(\mathfrak m\supseteq I\) be maximal with \(\mathfrak m\neq(x,y)\). Since \(x\in\mathfrak m\) and \(\mathfrak m\) does not contain the maximal ideal \((x,y)\), \(y\notin\mathfrak m\); so \(I_{\mathfrak m}=xA_{\mathfrak m}\), generated by the regular sequence \(x\) (\(A_{\mathfrak m}/xA_{\mathfrak m}\neq0\)). By Proposition 3.4, \(D_2(B_{\mathfrak m}/A_{\mathfrak m},\kappa(\mathfrak m))=0\), and this group is \(D_2(B/A,\kappa(\mathfrak m))\), by the first paragraph of the proof of Theorem 3.5. For \(\mathfrak m=(x,y)\), the elements \(x^2\), \(xy\) form a minimal set of generators of \(I_{\mathfrak m}\), by (1). They do not form a regular sequence: \(x\cdot xy=y\cdot x^2\) lies in \(x^2A_{\mathfrak m}\), but \(x\) does not, since \(x=x^2a/s\) with \(s\notin\mathfrak m\) would give \(s=xa\in\mathfrak m\). By Proposition 3.4, \(D_2(B/A,\kappa(\mathfrak m))\neq0\).

**Solution 5.3.** By the proof of Theorem 4.1, with the presentation \(B=P/I\), \(D_2(B/k,\kappa(\mathfrak n))=D_2(B/P,\kappa(\mathfrak n))\), which is \(D_2(B_{\mathfrak m}/P_{\mathfrak m},\kappa(\mathfrak m))\) for the maximal ideal \(\mathfrak m\supseteq I\) of \(P\) with \(\mathfrak n=\mathfrak m/I\) (first paragraph of the proof of Theorem 3.5). Let \(\mathfrak m\neq(x,y,z)\). Then one of \(x\), \(y\), \(z\) lies outside \(\mathfrak m\), say \(x\), the other cases being symmetric. As \(xy,zx\in\mathfrak m\), \(y,z\in\mathfrak m\), and \(I_{\mathfrak m}=(y,z)P_{\mathfrak m}\), since \(y=x^{-1}\cdot xy\), \(z=x^{-1}\cdot zx\) and \(yz\in(y)\). The sequence \(y,z\) is regular in \(P_{\mathfrak m}\): \(P_{\mathfrak m}\) and \(P_{\mathfrak m}/yP_{\mathfrak m}\), a localization of the domain \(k[x,z]\), are domains, \(y\neq0\), \(z\) is nonzero in the second, and \(P_{\mathfrak m}/(y,z)\neq0\). By Proposition 3.4, \(D_2=0\) there. At \(\mathfrak m=(x,y,z)\), \(I\) is spanned in degree \(2\) by the monomials \(xy\), \(yz\), \(zx\) and \(\mathfrak mI\) by monomials of degree \(\geq3\); so \(xy\), \(yz\), \(zx\) form a minimal set of generators of \(I_{\mathfrak m}\). They do not form a regular sequence: \(x\cdot yz=z\cdot xy\in xyP_{\mathfrak m}\), while \(x\notin xyP_{\mathfrak m}\), since \(x=xya/s\) would give \(s=ya\in\mathfrak m\). By Proposition 3.4, \(D_2\neq0\) there. By Theorem 4.1 ((1) implies (8)), \(B\) is not a local complete intersection over \(k\).

**Solution 5.4.** If \(I\) is weakly regular, \(\operatorname{Ext}^2_B(L_{B/A},M)=0\) for every \(M\) by Theorem 3.5. Conversely, let \(\mathfrak m\supseteq I\) be maximal and \(\kappa=\kappa(\mathfrak m)\). The second sequence of Lemma 3.2 gives \(\operatorname{Ext}^1_B(N,\kappa)=0\) and an injection \(\operatorname{Hom}_B(H_2(L_{B/A}),\kappa)\to\operatorname{Ext}^2_B(N,\kappa)\). Let \(F\to N\) be a resolution by finite free \(B\)-modules, which exists as \(B\) is noetherian and \(N\) finite. Since \(\kappa\) is a \(B_{\mathfrak m}\)-module, \(\operatorname{Hom}_B(F_i,\kappa)=\operatorname{Hom}_{B_{\mathfrak m}}(F_{i,\mathfrak m},\kappa)\), and \(F_{\mathfrak m}\to N_{\mathfrak m}\) is a free resolution; so \(\operatorname{Ext}^i_B(N,\kappa)=\operatorname{Ext}^i_{B_{\mathfrak m}}(N_{\mathfrak m},\kappa)\). Lift a basis of \(N_{\mathfrak m}\otimes\kappa\) to an exact sequence \(0\to K\to B_{\mathfrak m}^s\to N_{\mathfrak m}\to0\) with \(K\) finite. In the exact sequence
\[
\begin{aligned}&0\to\operatorname{Hom}(N_{\mathfrak m},\kappa)\to\operatorname{Hom}(B_{\mathfrak m}^s,\kappa)\\&\quad\to\operatorname{Hom}(K,\kappa)\to\operatorname{Ext}^1(N_{\mathfrak m},\kappa)=0,\end{aligned}
\]
the first map is an injection between \(\kappa\)-vector spaces of dimension \(s\), hence bijective; so \(\operatorname{Hom}(K,\kappa)=\operatorname{Hom}_\kappa(K\otimes\kappa,\kappa)=0\), \(K\otimes\kappa=0\) and \(K=0\) by Nakayama's lemma. So \(N_{\mathfrak m}\) is free, \(\operatorname{Ext}^2_B(N,\kappa)=0\), and \(\operatorname{Hom}_\kappa(H_2(L_{B/A})\otimes_B\kappa,\kappa)=0\). As \(H_2(L_{B/A})\) is finite (proof of Proposition 3.4), \(H_2(L_{B/A})_{\mathfrak m}=0\) by Nakayama's lemma. This holds for every maximal ideal \(\mathfrak m\supseteq I\); so \(N\) is projective [Tag 00NX] and \(H_2(L_{B/A})=0\) [Tag 00HN], and Theorem 3.5 applies.

## Sources

The characterizations of weakly regular ideals of noetherian rings by the cotangent complex in degree two, and of local complete intersections by \(D_2\), are due to Quillen, in Sections 8 and 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); the exact sequence of Proposition 2.2 follows his argument with the Koszul complex of a minimal set of generators, there obtained from a spectral sequence. That the converse of the amplitude property of local complete intersections holds for morphisms locally of finite type over a locally noetherian base is recorded by Illusie, in his work on the cotangent complex. The criteria for noetherian local rings by \(D_2\) at the residue field are also recorded, with references to André's work on the homology of commutative algebras, in Remark 1.6 of L. Avramov, *Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology*, Ann. of Math. 150 (1999), [arXiv:math/9909192](https://arxiv.org/abs/math/9909192).