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The Stacks project

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.)

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.

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.reference 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 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:

  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 (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.

  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 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

  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 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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