The Stacks project

30.20 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 10.67.1. 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 15.29.1. 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 10.68.1. 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

30.20.0.1
\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

30.20.0.2
\begin{equation} \label{divisors-equation-koszul} K_\bullet (\mathcal{O}_ X, f_\bullet ), \end{equation}

see Modules, Definition 17.21.2, 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

30.20.0.3
\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.)

Lemma 30.20.1. 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.

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 15.29.2, 15.29.3, and 15.29.6. $\square$

Definition 30.20.2. Let $X$ be a ringed space. Let $\mathcal{J} \subset \mathcal{O}_ X$ be a sheaf of ideals.

  1. 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 $f_1, \ldots , f_ r$.

  2. 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 $f_1, \ldots , f_ r$.

  3. 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 $f_1, \ldots , f_ r$.

  4. 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 30.20.3. 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 30.20.1. $\square$

Lemma 30.20.4. 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:

  1. $\mathcal{J}$ is an $\mathcal{O}_ X$-module of finite type,

  2. $\mathcal{J}/\mathcal{J}^2$ is a finite locally free $\mathcal{O}_ X/\mathcal{J}$-module, and

  3. the canonical maps

    \[ \text{Sym}^ n_{\mathcal{O}_ X/\mathcal{J}}(\mathcal{J}/\mathcal{J}^2) \longrightarrow \mathcal{J}^ n/\mathcal{J}^{n + 1} \]

    are isomorphisms for all $n \geq 0$.

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 (30.20.0.3). 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 10.19.1) 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 30.20.5. 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 $\mathcal{J}_ x/\mathfrak m_ x\mathcal{J}_ x$.

  1. 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 $\mathcal{J}|_ U$.

  2. 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 $\mathcal{J}|_ U$.

  3. 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 30.20.4 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 30.20.6. 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 25.10.1. $\square$

Lemma 30.20.7. 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 30.20.6 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 (30.20.0.1) is injective (resp. (30.20.0.2) is exact, (30.20.0.2) is exact in degree $1$, (30.20.0.3) 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 30.20.8. 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

  1. $\mathcal{J}_ x$ is generated by a regular sequence in $\mathcal{O}_{X, x}$,

  2. $\mathcal{J}_ x$ is generated by a Koszul-regular sequence in $\mathcal{O}_{X, x}$,

  3. $\mathcal{J}_ x$ is generated by an $H_1$-regular sequence in $\mathcal{O}_{X, x}$,

  4. $\mathcal{J}_ x$ is generated by a quasi-regular sequence in $\mathcal{O}_{X, x}$,

  5. 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

  6. 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

  7. 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

  8. 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 $A$,

  9. there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_ U$ is regular, and

  10. there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_ U$ is Koszul-regular, and

  11. there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_ U$ is $H_1$-regular, and

  12. 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 30.20.7 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 10.67.6. We have (9) $\Rightarrow $ (10) $\Rightarrow $ (11) $\Rightarrow $ (12) by Lemma 30.20.3. Finally, (4) $\Rightarrow $ (1) by Algebra, Lemma 10.68.6. Now all 12 statements are equivalent. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 067M. Beware of the difference between the letter 'O' and the digit '0'.