## 52.28 Application to Lefschetz theorems

In this section we discuss the relation between coherent sheaves on a projective scheme $P$ and coherent modules on formal completion along an ample divisor $Q$.

Let $k$ be a field. Let $P$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ P$-module. Let $s \in \Gamma (P, \mathcal{L})$ be a section1 and let $Q = Z(s)$ be the zero scheme, see Divisors, Definition 31.14.8. For all $n \geq 1$ we denote $Q_ n = Z(s^ n)$ the $n$th infinitesimal neighbourhood of $Q$. If $\mathcal{F}$ is a coherent $\mathcal{O}_ P$-module, then we denote $\mathcal{F}_ n = \mathcal{F}|_{Q_ n}$ the restriction, i.e., the pullback of $\mathcal{F}$ by the closed immersion $Q_ n \to P$.

Proposition 52.28.1. In the situation above assume for all points $p \in P \setminus Q$ we have

$\text{depth}(\mathcal{F}_ p) + \dim (\overline{\{ p\} }) > s$

Then the map

$H^ i(P, \mathcal{F}) \longrightarrow \mathop{\mathrm{lim}}\nolimits H^ i(Q_ n, \mathcal{F}_ n)$

is an isomorphism for $0 \leq i < s$.

Proof. We will use More on Morphisms, Lemma 37.51.1 and we will use the notation used and results found More on Morphisms, Section 37.51 without further mention; this proof will not make sense without at least understanding the statement of the lemma. Observe that in our case $A = \bigoplus _{m \geq 0} \Gamma (P, \mathcal{L}^{\otimes m})$ is a finite type $k$-algebra all of whose graded parts are finite dimensional $k$-vector spaces, see Cohomology of Schemes, Lemma 30.16.1.

We may and do think of $s$ as an element $f \in A_1 \subset A$, i.e., a homogeneous element of degree $1$ of $A$. Denote $Y = V(f) \subset X$ the closed subscheme defined by $f$. Then $U \cap Y = (\pi |_ U)^{-1}(Q)$ scheme theoretically. Recall the notation $\mathcal{F}_ U = \pi ^*\mathcal{F}|_ U = (\pi |_ U)^*\mathcal{F}$. This is a coherent $\mathcal{O}_ U$-module. Choose a finite $A$-module $M$ such that $\mathcal{F}_ U = \widetilde{M}|_ U$ (for existence see Local Cohomology, Lemma 51.8.2). We claim that $H^ i_ Z(M)$ is annihilated by a power of $f$ for $i \leq s + 1$.

To prove the claim we will apply Local Cohomology, Proposition 51.10.1. Translating into geometry we see that it suffices to prove for $u \in U$, $u \not\in Y$ and $z \in \overline{\{ u\} } \cap Z$ that

$\text{depth}(\mathcal{F}_{U, u}) + \dim (\mathcal{O}_{\overline{\{ u\} }, z}) > s + 1$

This requires only a small amount of thought.

Observe that $Z = \mathop{\mathrm{Spec}}(A_0)$ is a finite set of closed points of $X$ because $A_0$ is a finite dimensional $k$-algebra. (The reader who would like $Z$ to be a singleton can replace the finite $k$-algebra $A_0$ by $k$; it won't affect anything else in the proof.)

The morphism $\pi : L \to P$ and its restriction $\pi |_ U : U \to P$ are smooth of relative dimension $1$. Let $u \in U$, $u \not\in Y$ and $z \in \overline{\{ u\} } \cap Z$. Let $p = \pi (u) \in P \setminus Q$ be its image. Then either $u$ is a generic point of the fibre of $\pi$ over $p$ or a closed point of the fibre. If $u$ is a generic point of the fibre, then $\text{depth}(\mathcal{F}_{U, u}) = \text{depth}(\mathcal{F}_ p)$ and $\dim (\overline{\{ u\} }) = \dim (\overline{\{ p\} }) + 1$. If $u$ is a closed point of the fibre, then $\text{depth}(\mathcal{F}_{U, u}) = \text{depth}(\mathcal{F}_ p) + 1$ and $\dim (\overline{\{ u\} }) = \dim (\overline{\{ p\} })$. In both cases we have $\dim (\overline{\{ u\} }) = \dim (\mathcal{O}_{\overline{\{ u\} }, z})$ because every point of $Z$ is closed. Thus the desired inequality follows from the assumption in the statement of the lemma.

Let $A'$ be the $f$-adic completion of $A$. So $A \to A'$ is flat by Algebra, Lemma 10.97.2. Denote $U' \subset X' = \mathop{\mathrm{Spec}}(A')$ the inverse image of $U$ and similarly for $Y'$ and $Z'$. Let $\mathcal{F}'$ on $U'$ be the pullback of $\mathcal{F}_ U$ and let $M' = M \otimes _ A A'$. By flat base change for local cohomology (Local Cohomology, Lemma 51.5.7) we have

$H^ i_{Z'}(M') = H^ i_ Z(M) \otimes _ A A'$

and we find that for $i \leq s + 1$ these are annihilated by a power of $f$. Consider the diagram

$\xymatrix{ & H^ i(U, \mathcal{F}_ U) \ar[ld] \ar[d] \ar[r] & \mathop{\mathrm{lim}}\nolimits H^ i(U, \mathcal{F}_ U/f^ n\mathcal{F}_ U) \ar@{=}[d] \\ H^ i(U, \mathcal{F}_ U) \otimes _ A A' \ar@{=}[r] & H^ i(U', \mathcal{F}') \ar[r] & \mathop{\mathrm{lim}}\nolimits H^ i(U', \mathcal{F}'/f^ n\mathcal{F}') }$

The lower horizontal arrow is an isomorphism for $i < s$ by Lemma 52.13.2 and the torsion property we just proved. The horizontal equal sign is flat base change (Cohomology of Schemes, Lemma 30.5.2) and the vertical equal sign is because $U \cap Y$ and $U' \cap Y'$ as well as their $n$th infinitesimal neighbourhoods are mapped isomorphically onto each other (as we are completing with respect to $f$).

Applying More on Morphisms, Equation (37.51.0.2) we have compatible direct sum decompositions

$\mathop{\mathrm{lim}}\nolimits H^ i(U, \mathcal{F}_ U/f^ n\mathcal{F}_ U) = \mathop{\mathrm{lim}}\nolimits \left( \bigoplus \nolimits _{m \in \mathbf{Z}} H^ i(Q_ n, \mathcal{F}_ n \otimes \mathcal{L}^{\otimes m}) \right)$

and

$H^ i(U, \mathcal{F}_ U) = \bigoplus \nolimits _{m \in \mathbf{Z}} H^ i(P, \mathcal{F} \otimes \mathcal{L}^{\otimes m})$

Thus we conclude by Algebra, Lemma 10.98.4. $\square$

Lemma 52.28.2. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$ with $n$th infinitesimal neighbourhood $Y_ n = Z(s^ n)$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume that for all $x \in X \setminus Y$ we have

$\text{depth}(\mathcal{F}_ x) + \dim (\overline{\{ x\} }) > 1$

Then $\Gamma (V, \mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{F}|_{Y_ n})$ is an isomorphism for any open subscheme $V \subset X$ containing $Y$.

Proof. By Proposition 52.28.1 this is true for $V = X$. Thus it suffices to show that the map $\Gamma (V, \mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{F}|_{Y_ n})$ is injective. If $\sigma \in \Gamma (V, \mathcal{F})$ maps to zero, then its support is disjoint from $Y$ (details omitted; hint: use Krull's intersection theorem). Then the closure $T \subset X$ of $\text{Supp}(\sigma )$ is disjoint from $Y$. Whence $T$ is proper over $k$ (being closed in $X$) and affine (being closed in the affine scheme $X \setminus Y$, see Morphisms, Lemma 29.43.18) and hence finite over $k$ (Morphisms, Lemma 29.44.11). Thus $T$ is a finite set of closed points of $X$. Thus $\text{depth}(\mathcal{F}_ x) \geq 2$ is at least $1$ for $x \in T$ by our assumption. We conclude that $\Gamma (V, \mathcal{F}) \to \Gamma (V \setminus T, \mathcal{F})$ is injective and $\sigma = 0$ as desired. $\square$

Example 52.28.3. Let $k$ be a field and let $X$ be a proper variety over $k$. Let $Y \subset X$ be an effective Cartier divisor such that $\mathcal{O}_ X(Y)$ is ample and denote $Y_ n$ its $n$th infinitesimal neighbourhood. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. Here are some special cases of Proposition 52.28.1.

1. If $X$ is a curve, we don't learn anything.

2. If $X$ is a Cohen-Macaulay (for example normal) surface, then

$H^0(X, \mathcal{E}) \to \mathop{\mathrm{lim}}\nolimits H^0(Y_ n, \mathcal{E}|_{Y_ n})$

is an isomorphism.

3. If $X$ is a Cohen-Macaulay threefold, then

$H^0(X, \mathcal{E}) \to \mathop{\mathrm{lim}}\nolimits H^0(Y_ n, \mathcal{E}|_{Y_ n}) \quad \text{and}\quad H^1(X, \mathcal{E}) \to \mathop{\mathrm{lim}}\nolimits H^1(Y_ n, \mathcal{E}|_{Y_ n})$

are isomorphisms.

Presumably the pattern is clear. If $X$ is a normal threefold, then we can conclude the result for $H^0$ but not for $H^1$.

Before we prove the next main result, we need a lemma.

Lemma 52.28.4. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Assume

1. $A$ is a graded ring, $\mathfrak a = A_+$, and $I$ is a homogeneous ideal,

2. $(\mathcal{F}_ n) = (\widetilde{M_ n}|_ U)$ where $(M_ n)$ is an inverse system of graded $A$-modules, and

3. $(\mathcal{F}_ n)$ extends canonically to $X$.

Then there is a finite graded $A$-module $N$ such that

1. the inverse systems $(N/I^ nN)$ and $(M_ n)$ are pro-isomorphic in the category of graded $A$-modules modulo $A_+$-power torsion modules, and

2. $(\mathcal{F}_ n)$ is the completion of of the coherent module associated to $N$.

Proof. Let $(\mathcal{G}_ n)$ be the canonical extension as in Lemma 52.16.8. The grading on $A$ and $M_ n$ determines an action

$a : \mathbf{G}_ m \times X \longrightarrow X$

of the group scheme $\mathbf{G}_ m$ on $X$ such that $(\widetilde{M_ n})$ becomes an inverse system of $\mathbf{G}_ m$-equivariant quasi-coherent $\mathcal{O}_ X$-modules, see Groupoids, Example 39.12.3. Since $\mathfrak a$ and $I$ are homogeneous ideals the closed subschemes $Z$, $Y$ and the open subscheme $U$ are $\mathbf{G}_ m$-invariant closed and open subschemes. The restriction $(\mathcal{F}_ n)$ of $(\widetilde{M_ n})$ is an inverse system of $\mathbf{G}_ m$-equivariant coherent $\mathcal{O}_ U$-modules. In other words, $(\mathcal{F}_ n)$ is a $\mathbf{G}_ m$-equivariant coherent formal module, in the sense that there is an isomorphism

$\alpha : (a^*\mathcal{F}_ n) \longrightarrow (p^*\mathcal{F}_ n)$

over $\mathbf{G}_ m \times U$ satisfying a suitable cocycle condition. Since $a$ and $p$ are flat morphisms of affine schemes, by Lemma 52.16.9 we conclude that there exists a unique isomorphism

$\beta : (a^*\mathcal{G}_ n) \longrightarrow (p^*\mathcal{G}_ n)$

over $\mathbf{G}_ m \times X$ restricting to $\alpha$ on $\mathbf{G}_ m \times U$. The uniqueness guarantees that $\beta$ satisfies the corresponding cocycle condition. In this way each $\mathcal{G}_ n$ becomes a $\mathbf{G}_ m$-equivariant coherent $\mathcal{O}_ X$-module in a manner compatible with transition maps.

By Groupoids, Lemma 39.12.5 we see that $\mathcal{G}_ n$ with its $\mathbf{G}_ m$-equivariant structure corresponds to a graded $A$-module $N_ n$. The transition maps $N_{n + 1} \to N_ n$ are graded module maps. Note that $N_ n$ is a finite $A$-module and $N_ n = N_{n + 1}/I^ n N_{n + 1}$ because $(\mathcal{G}_ n)$ is an object of $\textit{Coh}(X, I\mathcal{O}_ X)$. Let $N$ be the finite graded $A$-module foud in Algebra, Lemma 10.98.3. Then $N_ n = N/I^ nN$, whence $(\mathcal{G}_ n)$ is the completion of the coherent module associated to $N$, and a fortiori we see that (b) is true.

To see (a) we have to unwind the situation described above a bit more. First, observe that the kernel and cokernel of $M_ n \to H^0(U, \mathcal{F}_ n)$ is $A_+$-power torsion (Local Cohomology, Lemma 51.8.2). Observe that $H^0(U, \mathcal{F}_ n)$ comes with a natural grading such that these maps and the transition maps of the system are graded $A$-module map; for example we can use that $(U \to X)_*\mathcal{F}_ n$ is a $\mathbf{G}_ m$-equivariant module on $X$ and use Groupoids, Lemma 39.12.5. Next, recall that $(N_ n)$ and $(H^0(U, \mathcal{F}_ n))$ are pro-isomorphic by Definition 52.16.7 and Lemma 52.16.8. We omit the verification that the maps defining this pro-isomorphism are graded module maps. Thus $(N_ n)$ and $(M_ n)$ are pro-isomorphic in the category of graded $A$-modules modulo $A_+$-power torsion modules. $\square$

Let $k$ be a field. Let $P$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ P$-module. Let $s \in \Gamma (P, \mathcal{L})$ be a section and let $Q = Z(s)$ be the zero scheme, see Divisors, Definition 31.14.8. Let $\mathcal{I} \subset \mathcal{O}_ P$ be the ideal sheaf of $Q$. We will use $\textit{Coh}(P, \mathcal{I})$ to denote the category of coherent formal modules introduced in Cohomology of Schemes, Section 30.23.

Proposition 52.28.5. In the situation above let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(P, \mathcal{I})$. Assume for all $q \in Q$ and for all primes $\mathfrak p \in \mathcal{O}_{P, q}^\wedge$, $\mathfrak p \not\in V(\mathcal{I}_ q^\wedge )$ we have

$\text{depth}((\mathcal{F}_ q^\wedge )_\mathfrak p) + \dim (\mathcal{O}_{P, q}^\wedge /\mathfrak p) + \dim (\overline{\{ q\} }) > 2$

Then $(\mathcal{F}_ n)$ is the completion of a coherent $\mathcal{O}_ P$-module.

Proof. By Cohomology of Schemes, Lemma 30.23.6 to prove the lemma, we may replace $(\mathcal{F}_ n)$ by an object differing from it by $\mathcal{I}$-torsion (see below for more precision). Let $T' = \{ q \in Q \mid \dim (\overline{\{ q\} }) = 0\}$ and $T = \{ q \in Q \mid \dim (\overline{\{ q\} }) \leq 1\}$. The assumption in the proposition is exactly that $Q \subset P$, $(\mathcal{F}_ n)$, and $T' \subset T \subset Q$ satisfy the conditions of Lemma 52.21.2 with $d = 1$; besides trivial manipulations of inequalities, use that $V(\mathfrak p) \cap V(\mathcal{I}^\wedge _ y) = \{ \mathfrak m^\wedge _ y\} \Leftrightarrow \dim (\mathcal{O}_{P, q}^\wedge /\mathfrak p) = 1$ as $\mathcal{I}_ y^\wedge$ is generated by $1$ element. Combining these two remarks, we may replace $(\mathcal{F}_ n)$ by the object $(\mathcal{H}_ n)$ of $\textit{Coh}(P, \mathcal{I})$ found in Lemma 52.21.2. Thus we may and do assume $(\mathcal{F}_ n)$ is pro-isomorphic to an inverse system $(\mathcal{F}_ n'')$ of coherent $\mathcal{O}_ P$-modules such that $\text{depth}(\mathcal{F}''_{n, q}) + \dim (\overline{\{ q\} }) \geq 2$ for all $q \in Q$.

We will use More on Morphisms, Lemma 37.51.1 and we will use the notation used and results found More on Morphisms, Section 37.51 without further mention; this proof will not make sense without at least understanding the statement of the lemma. Observe that in our case $A = \bigoplus _{m \geq 0} \Gamma (P, \mathcal{L}^{\otimes m})$ is a finite type $k$-algebra all of whose graded parts are finite dimensional $k$-vector spaces, see Cohomology of Schemes, Lemma 30.16.1.

By Cohomology of Schemes, Lemma 30.23.9 the pull back by $\pi |_ U : U \to P$ is an object $(\pi |_ U^*\mathcal{F}_ n)$ of $\textit{Coh}(U, f\mathcal{O}_ U)$ which is pro-isomorphic to the inverse system $(\pi |_ U^*\mathcal{F}_ n'')$ of coherent $\mathcal{O}_ U$-modules. We claim

$\text{depth}(\pi |_ U^*\mathcal{F}''_{n, y}) + \delta _ Z^ Y(y) \geq 3$

for all $y \in U \cap Y$. Since all the points of $Z$ are closed, we see that $\delta _ Z^ Y(y) \geq \dim (\overline{\{ y\} })$ for all $y \in U \cap Y$, see Lemma 52.18.1. Let $q \in Q$ be the image of $y$. Since the morphism $\pi : U \to P$ is smooth of relative dimension $1$ we see that either $y$ is a closed point of a fibre of $\pi$ or a generic point. Thus we see that

$\text{depth}(\pi ^*\mathcal{F}''_{n, y}) + \delta _ Z^ Y(y) \geq \text{depth}(\pi ^*\mathcal{F}''_{n, y}) + \dim (\overline{\{ y\} }) = \text{depth}(\mathcal{F}''_{n, q}) + \dim (\overline{\{ q\} }) + 1$

because either the depth goes up by $1$ or the dimension. This proves the claim.

By Lemma 52.22.1 we conclude that $(\pi |_ U^*\mathcal{F}_ n)$ canonically extends to $X$. Observe that

$M_ n = \Gamma (U, \pi |_ U^*\mathcal{F}_ n) = \bigoplus \nolimits _{m \in \mathbf{Z}} \Gamma (P, \mathcal{F}_ n \otimes _{\mathcal{O}_ P} \mathcal{L}^{\otimes m})$

is canonically a graded $A$-module, see More on Morphisms, Equation (37.51.0.2). By Properties, Lemma 28.18.2 we have $\pi |_ U^*\mathcal{F}_ n = \widetilde{M_ n}|_ U$. Thus we may apply Lemma 52.28.4 to find a finite graded $A$-module $N$ such that $(M_ n)$ and $(N/I^ nN)$ are pro-isomorphic in the category of graded $A$-modules modulo $A_+$-torsion modules. Let $\mathcal{F}$ be the coherent $\mathcal{O}_ P$-module associated to $N$, see Cohomology of Schemes, Proposition 30.15.3. The same proposition tells us that $(\mathcal{F}/\mathcal{I}^ n\mathcal{F})$ is pro-isomorphic to $(\mathcal{F}_ n)$. Since both are objects of $\textit{Coh}(P, \mathcal{I})$ we win by Lemma 52.15.3. $\square$

Example 52.28.6. Let $k$ be a field and let $X$ be a proper variety over $k$. Let $Y \subset X$ be an effective Cartier divisor such that $\mathcal{O}_ X(Y)$ is ample and denote $\mathcal{I} \subset \mathcal{O}_ X$ the corresponding sheaf of ideals. Let $(\mathcal{E}_ n)$ an object of $\textit{Coh}(X, \mathcal{I})$ with $\mathcal{E}_ n$ finite locally free. Here are some special cases of Proposition 52.28.5.

1. If $X$ is a curve or a surface, we don't learn anything.

2. If $X$ is a Cohen-Macaulay threefold, then $(\mathcal{E}_ n)$ is the completion of a coherent $\mathcal{O}_ X$-module $\mathcal{E}$.

3. More generally, if $\dim (X) \geq 3$ and $X$ is $(S_3)$, then $(\mathcal{E}_ n)$ is the completion of a coherent $\mathcal{O}_ X$-module $\mathcal{E}$.

Of course, if $\mathcal{E}$ exists, then $\mathcal{E}$ is finite locally free in an open neighbourhood of $Y$.

Proposition 52.28.7. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module and let $s \in \Gamma (X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$ and denote $\mathcal{I} \subset \mathcal{O}_ X$ the corresponding sheaf of ideals. Let $\mathcal{V}$ be the set of open subschemes of $X$ containing $Y$ ordered by reverse inclusion. Assume that for all $x \in X \setminus Y$ we have

$\text{depth}(\mathcal{O}_{X, x}) + \dim (\overline{\{ x\} }) > 2$

Then the completion functor

$\mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{Coh}(\mathcal{O}_ V) \longrightarrow \textit{Coh}(X, \mathcal{I})$

is an equivalence on the full subcategories of finite locally free objects.

Proof. To prove fully faithfulness it suffices to prove that

$\mathop{\mathrm{colim}}\nolimits _\mathcal {V} \Gamma (V, \mathcal{L}^{\otimes m}) \longrightarrow \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{L}^{\otimes m}|_{Y_ n})$

is an isomorphism for all $m$, see Lemma 52.15.2. This follows from Lemma 52.28.2.

Essential surjectivity. Let $(\mathcal{F}_ n)$ be a finite locally free object of $\textit{Coh}(X, \mathcal{I})$. Then for $y \in Y$ we have $\mathcal{F}_ y^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_{n, y}$ is is a finite free $\mathcal{O}_{X, y}^\wedge$-module. Let $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ be a prime with $\mathfrak p \not\in V(\mathcal{I}_ y^\wedge )$. Then $\mathfrak p$ lies over a prime $\mathfrak p_0 \subset \mathcal{O}_{X, y}$ which corresponds to a specialization $x \leadsto y$ with $x \not\in Y$. By Local Cohomology, Lemma 51.11.3 and some dimension theory (see Varieties, Section 33.20) we have

$\text{depth}((\mathcal{O}_{X, y}^\wedge )_\mathfrak p) + \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) = \text{depth}(\mathcal{O}_{X, x}) + \dim (\overline{\{ x\} }) - \dim (\overline{\{ y\} })$

Thus our assumptions imply the assumptions of Proposition 52.28.5 are satisfied and we find that $(\mathcal{F}_ n)$ is the completion of a coherent $\mathcal{O}_ X$-module $\mathcal{F}$. It then follows that $\mathcal{F}_ y$ is finite free for all $y \in Y$ and hence $\mathcal{F}$ is finite locally free in an open neighbourhood $V$ of $Y$. This finishes the proof. $\square$

[1] We do not require $s$ to be a regular section. Correspondingly, $Q$ is only a locally principal closed subscheme of $P$ and not necessarily an effective Cartier divisor.

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