Lemma 31.20.1. Let $X$ be a ringed space. Let $f_1, \ldots , f_ r \in \Gamma (X, \mathcal{O}_ X)$. We have the following implications $f_1, \ldots , f_ r$ is a regular sequence $\Rightarrow $ $f_1, \ldots , f_ r$ is a Koszul-regular sequence $\Rightarrow $ $f_1, \ldots , f_ r$ is an $H_1$-regular sequence $\Rightarrow $ $f_1, \ldots , f_ r$ is a quasi-regular sequence.
31.20 Regular ideal sheaves
In this section we generalize the notion of an effective Cartier divisor to higher codimension. Recall that a sequence of elements $f_1, \ldots , f_ r$ of a ring $R$ is a regular sequence if for each $i = 1, \ldots , r$ the element $f_ i$ is a nonzerodivisor on $R/(f_1, \ldots , f_{i - 1})$ and $R/(f_1, \ldots , f_ r) \not= 0$, see Algebra, Definition 10.68.1. There are three closely related weaker conditions that we can impose. The first is to assume that $f_1, \ldots , f_ r$ is a Koszul-regular sequence, i.e., that $H_ i(K_\bullet (f_1, \ldots , f_ r)) = 0$ for $i > 0$, see More on Algebra, Definition 15.30.1. The sequence is called an $H_1$-regular sequence if $H_1(K_\bullet (f_1, \ldots , f_ r)) = 0$. Another condition we can impose is that with $J = (f_1, \ldots , f_ r)$, the map
\[ R/J[T_1, \ldots , T_ r] \longrightarrow \bigoplus \nolimits _{n \geq 0} J^ n/J^{n + 1} \]
which maps $T_ i$ to $f_ i \bmod J^2$ is an isomorphism. In this case we say that $f_1, \ldots , f_ r$ is a quasi-regular sequence, see Algebra, Definition 10.69.1. Given an $R$-module $M$ there is also a notion of $M$-regular and $M$-quasi-regular sequence.
We can generalize this to the case of ringed spaces as follows. Let $X$ be a ringed space and let $f_1, \ldots , f_ r \in \Gamma (X, \mathcal{O}_ X)$. We say that $f_1, \ldots , f_ r$ is a regular sequence if for each $i = 1, \ldots , r$ the map
31.20.0.1
\begin{equation} \label{divisors-equation-map-regular} f_ i : \mathcal{O}_ X/(f_1, \ldots , f_{i - 1}) \longrightarrow \mathcal{O}_ X/(f_1, \ldots , f_{i - 1}) \end{equation}
is an injective map of sheaves. We say that $f_1, \ldots , f_ r$ is a Koszul-regular sequence if the Koszul complex
31.20.0.2
\begin{equation} \label{divisors-equation-koszul} K_\bullet (\mathcal{O}_ X, f_\bullet ), \end{equation}
see Modules, Definition 17.24.2, is acyclic in degrees $> 0$. We say that $f_1, \ldots , f_ r$ is a $H_1$-regular sequence if the Koszul complex $K_\bullet (\mathcal{O}_ X, f_\bullet )$ is exact in degree $1$. Finally, we say that $f_1, \ldots , f_ r$ is a quasi-regular sequence if the map
31.20.0.3
\begin{equation} \label{divisors-equation-map-quasi-regular} \mathcal{O}_ X/\mathcal{J}[T_1, \ldots , T_ r] \longrightarrow \bigoplus \nolimits _{d \geq 0} \mathcal{J}^ d/\mathcal{J}^{d + 1} \end{equation}
is an isomorphism of sheaves where $\mathcal{J} \subset \mathcal{O}_ X$ is the sheaf of ideals generated by $f_1, \ldots , f_ r$. (There is also a notion of $\mathcal{F}$-regular and $\mathcal{F}$-quasi-regular sequence for a given $\mathcal{O}_ X$-module $\mathcal{F}$ which we will introduce here if we ever need it.)
Proof. Since we may check exactness at stalks, a sequence $f_1, \ldots , f_ r$ is a regular sequence if and only if the maps
\[ f_ i : \mathcal{O}_{X, x}/(f_1, \ldots , f_{i - 1}) \longrightarrow \mathcal{O}_{X, x}/(f_1, \ldots , f_{i - 1}) \]
are injective for all $x \in X$. In other words, the image of the sequence $f_1, \ldots , f_ r$ in the ring $\mathcal{O}_{X, x}$ is a regular sequence for all $x \in X$. The other types of regularity can be checked stalkwise as well (details omitted). Hence the implications follow from More on Algebra, Lemmas 15.30.2, 15.30.3, and 15.30.6. $\square$
Definition 31.20.2. Let $X$ be a ringed space. Let $\mathcal{J} \subset \mathcal{O}_ X$ be a sheaf of ideals.
We say $\mathcal{J}$ is regular if for every $x \in \text{Supp}(\mathcal{O}_ X/\mathcal{J})$ there exists an open neighbourhood $x \in U \subset X$ and a regular sequence $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ such that $\mathcal{J}|_ U$ is generated by $f_1, \ldots , f_ r$.
We say $\mathcal{J}$ is Koszul-regular if for every $x \in \text{Supp}(\mathcal{O}_ X/\mathcal{J})$ there exists an open neighbourhood $x \in U \subset X$ and a Koszul-regular sequence $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ such that $\mathcal{J}|_ U$ is generated by $f_1, \ldots , f_ r$.
We say $\mathcal{J}$ is $H_1$-regular if for every $x \in \text{Supp}(\mathcal{O}_ X/\mathcal{J})$ there exists an open neighbourhood $x \in U \subset X$ and a $H_1$-regular sequence $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ such that $\mathcal{J}|_ U$ is generated by $f_1, \ldots , f_ r$.
We say $\mathcal{J}$ is quasi-regular if for every $x \in \text{Supp}(\mathcal{O}_ X/\mathcal{J})$ there exists an open neighbourhood $x \in U \subset X$ and a quasi-regular sequence $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ such that $\mathcal{J}|_ U$ is generated by $f_1, \ldots , f_ r$.
Many properties of this notion immediately follow from the corresponding notions for regular and quasi-regular sequences in rings.
Lemma 31.20.3. Let $X$ be a ringed space. Let $\mathcal{J}$ be a sheaf of ideals. We have the following implications: $\mathcal{J}$ is regular $\Rightarrow $ $\mathcal{J}$ is Koszul-regular $\Rightarrow $ $\mathcal{J}$ is $H_1$-regular $\Rightarrow $ $\mathcal{J}$ is quasi-regular.
Proof. The lemma immediately reduces to Lemma 31.20.1. $\square$
Lemma 31.20.4. Let $X$ be a locally ringed space. Let $\mathcal{J} \subset \mathcal{O}_ X$ be a sheaf of ideals. Then $\mathcal{J}$ is quasi-regular if and only if the following conditions are satisfied:
$\mathcal{J}$ is an $\mathcal{O}_ X$-module of finite type,
$\mathcal{J}/\mathcal{J}^2$ is a finite locally free $\mathcal{O}_ X/\mathcal{J}$-module, and
the canonical maps
are isomorphisms for all $n \geq 0$.
\[ \text{Sym}^ n_{\mathcal{O}_ X/\mathcal{J}}(\mathcal{J}/\mathcal{J}^2) \longrightarrow \mathcal{J}^ n/\mathcal{J}^{n + 1} \]
Proof. It is clear that if $U \subset X$ is an open such that $\mathcal{J}|_ U$ is generated by a quasi-regular sequence $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ then $\mathcal{J}|_ U$ is of finite type, $\mathcal{J}|_ U/\mathcal{J}^2|_ U$ is free with basis $f_1, \ldots , f_ r$, and the maps in (3) are isomorphisms because they are coordinate free formulation of the degree $n$ part of (31.20.0.3). Hence it is clear that being quasi-regular implies conditions (1), (2), and (3).
Conversely, suppose that (1), (2), and (3) hold. Pick a point $x \in \text{Supp}(\mathcal{O}_ X/\mathcal{J})$. Then there exists a neighbourhood $U \subset X$ of $x$ such that $\mathcal{J}|_ U/\mathcal{J}^2|_ U$ is free of rank $r$ over $\mathcal{O}_ U/\mathcal{J}|_ U$. After possibly shrinking $U$ we may assume there exist $f_1, \ldots , f_ r \in \mathcal{J}(U)$ which map to a basis of $\mathcal{J}|_ U/\mathcal{J}^2|_ U$ as an $\mathcal{O}_ U/\mathcal{J}|_ U$-module. In particular we see that the images of $f_1, \ldots , f_ r$ in $\mathcal{J}_ x/\mathcal{J}^2_ x$ generate. Hence by Nakayama's lemma (Algebra, Lemma 10.20.1) we see that $f_1, \ldots , f_ r$ generate the stalk $\mathcal{J}_ x$. Hence, since $\mathcal{J}$ is of finite type, by Modules, Lemma 17.9.4 after shrinking $U$ we may assume that $f_1, \ldots , f_ r$ generate $\mathcal{J}$. Finally, from (3) and the isomorphism $\mathcal{J}|_ U/\mathcal{J}^2|_ U = \bigoplus \mathcal{O}_ U/\mathcal{J}|_ U f_ i$ it is clear that $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ is a quasi-regular sequence. $\square$
Lemma 31.20.5. Let $(X, \mathcal{O}_ X)$ be a locally ringed space. Let $\mathcal{J} \subset \mathcal{O}_ X$ be a sheaf of ideals. Let $x \in X$ and $f_1, \ldots , f_ r \in \mathcal{J}_ x$ whose images give a basis for the $\kappa (x)$-vector space $\mathcal{J}_ x/\mathfrak m_ x\mathcal{J}_ x$.
If $\mathcal{J}$ is quasi-regular, then there exists an open neighbourhood such that $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ form a quasi-regular sequence generating $\mathcal{J}|_ U$.
If $\mathcal{J}$ is $H_1$-regular, then there exists an open neighbourhood such that $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ form an $H_1$-regular sequence generating $\mathcal{J}|_ U$.
If $\mathcal{J}$ is Koszul-regular, then there exists an open neighbourhood such that $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ form an Koszul-regular sequence generating $\mathcal{J}|_ U$.
Proof. First assume that $\mathcal{J}$ is quasi-regular. We may choose an open neighbourhood $U \subset X$ of $x$ and a quasi-regular sequence $g_1, \ldots , g_ s \in \mathcal{O}_ X(U)$ which generates $\mathcal{J}|_ U$. Note that this implies that $\mathcal{J}/\mathcal{J}^2$ is free of rank $s$ over $\mathcal{O}_ U/\mathcal{J}|_ U$ (see Lemma 31.20.4 and its proof) and hence $r = s$. We may shrink $U$ and assume $f_1, \ldots , f_ r \in \mathcal{J}(U)$. Thus we may write
\[ f_ i = \sum a_{ij} g_ j \]
for some $a_{ij} \in \mathcal{O}_ X(U)$. By assumption the matrix $A = (a_{ij})$ maps to an invertible matrix over $\kappa (x)$. Hence, after shrinking $U$ once more, we may assume that $(a_{ij})$ is invertible. Thus we see that $f_1, \ldots , f_ r$ give a basis for $(\mathcal{J}/\mathcal{J}^2)|_ U$ which proves that $f_1, \ldots , f_ r$ is a quasi-regular sequence over $U$.
Note that in order to prove (2) and (3) we may, because the assumptions of (2) and (3) are stronger than the assumption in (1), already assume that $f_1, \ldots , f_ r \in \mathcal{J}(U)$ and $f_ i = \sum a_{ij}g_ j$ with $(a_{ij})$ invertible as above, where now $g_1, \ldots , g_ r$ is a $H_1$-regular or Koszul-regular sequence. Since the Koszul complex on $f_1, \ldots , f_ r$ is isomorphic to the Koszul complex on $g_1, \ldots , g_ r$ via the matrix $(a_{ij})$ (see More on Algebra, Lemma 15.28.4) we conclude that $f_1, \ldots , f_ r$ is $H_1$-regular or Koszul-regular as desired. $\square$
Lemma 31.20.6. Any regular, Koszul-regular, $H_1$-regular, or quasi-regular sheaf of ideals on a scheme is a finite type quasi-coherent sheaf of ideals.
Proof. This follows as such a sheaf of ideals is locally generated by finitely many sections. And any sheaf of ideals locally generated by sections on a scheme is quasi-coherent, see Schemes, Lemma 26.10.1. $\square$
Lemma 31.20.7. Let $X$ be a scheme. Let $\mathcal{J}$ be a sheaf of ideals. Then $\mathcal{J}$ is regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) if and only if for every $x \in \text{Supp}(\mathcal{O}_ X/\mathcal{J})$ there exists an affine open neighbourhood $x \in U \subset X$, $U = \mathop{\mathrm{Spec}}(A)$ such that $\mathcal{J}|_ U = \widetilde{I}$ and such that $I$ is generated by a regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) sequence $f_1, \ldots , f_ r \in A$.
Proof. By assumption we can find an open neighbourhood $U$ of $x$ over which $\mathcal{J}$ is generated by a regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) sequence $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$. After shrinking $U$ we may assume that $U$ is affine, say $U = \mathop{\mathrm{Spec}}(A)$. Since $\mathcal{J}$ is quasi-coherent by Lemma 31.20.6 we see that $\mathcal{J}|_ U = \widetilde{I}$ for some ideal $I \subset A$. Now we can use the fact that
\[ \widetilde{\ } : \text{Mod}_ A \longrightarrow \mathit{QCoh}(\mathcal{O}_ U) \]
is an equivalence of categories which preserves exactness. For example the fact that the functions $f_ i$ generate $\mathcal{J}$ means that the $f_ i$, seen as elements of $A$ generate $I$. The fact that (31.20.0.1) is injective (resp. (31.20.0.2) is exact, (31.20.0.2) is exact in degree $1$, (31.20.0.3) is an isomorphism) implies the corresponding property of the map $A/(f_1, \ldots , f_{i - 1}) \to A/(f_1, \ldots , f_{i - 1})$ (resp. the complex $K_\bullet (A, f_1, \ldots , f_ r)$, the map $A/I[T_1, \ldots , T_ r] \to \bigoplus I^ n/I^{n + 1}$). Thus $f_1, \ldots , f_ r \in A$ is a regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) sequence of the ring $A$. $\square$
Lemma 31.20.8. Let $X$ be a locally Noetherian scheme. Let $\mathcal{J} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $x$ be a point of the support of $\mathcal{O}_ X/\mathcal{J}$. The following are equivalent
$\mathcal{J}_ x$ is generated by a regular sequence in $\mathcal{O}_{X, x}$,
$\mathcal{J}_ x$ is generated by a Koszul-regular sequence in $\mathcal{O}_{X, x}$,
$\mathcal{J}_ x$ is generated by an $H_1$-regular sequence in $\mathcal{O}_{X, x}$,
$\mathcal{J}_ x$ is generated by a quasi-regular sequence in $\mathcal{O}_{X, x}$,
there exists an affine neighbourhood $U = \mathop{\mathrm{Spec}}(A)$ of $x$ such that $\mathcal{J}|_ U = \widetilde{I}$ and $I$ is generated by a regular sequence in $A$, and
there exists an affine neighbourhood $U = \mathop{\mathrm{Spec}}(A)$ of $x$ such that $\mathcal{J}|_ U = \widetilde{I}$ and $I$ is generated by a Koszul-regular sequence in $A$, and
there exists an affine neighbourhood $U = \mathop{\mathrm{Spec}}(A)$ of $x$ such that $\mathcal{J}|_ U = \widetilde{I}$ and $I$ is generated by an $H_1$-regular sequence in $A$, and
there exists an affine neighbourhood $U = \mathop{\mathrm{Spec}}(A)$ of $x$ such that $\mathcal{J}|_ U = \widetilde{I}$ and $I$ is generated by a quasi-regular sequence in $A$,
there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_ U$ is regular, and
there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_ U$ is Koszul-regular, and
there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_ U$ is $H_1$-regular, and
there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_ U$ is quasi-regular.
In particular, on a locally Noetherian scheme the notions of regular, Koszul-regular, $H_1$-regular, or quasi-regular ideal sheaf all agree.
Proof. It follows from Lemma 31.20.7 that (5) $\Leftrightarrow $ (9), (6) $\Leftrightarrow $ (10), (7) $\Leftrightarrow $ (11), and (8) $\Leftrightarrow $ (12). It is clear that (5) $\Rightarrow $ (1), (6) $\Rightarrow $ (2), (7) $\Rightarrow $ (3), and (8) $\Rightarrow $ (4). We have (1) $\Rightarrow $ (5) by Algebra, Lemma 10.68.6. We have (9) $\Rightarrow $ (10) $\Rightarrow $ (11) $\Rightarrow $ (12) by Lemma 31.20.3. Finally, (4) $\Rightarrow $ (1) by Algebra, Lemma 10.69.6. Now all 12 statements are equivalent. $\square$
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