Stalks and exactness · Qual Corpus

Sheaves of abelian groups on $X$ form an abelian category, and a sequence of them is exact exactly when it is exact on every stalk.

Exactness is checked on stalks, and global sections are only left exact (Tag PR-C9ZEK)

Sheaf exactness is stalkwise; global sections preserve kernels but can lose surjectivity.Suggest a slogan

A sequence of sheaves $\mcf \to \mcg \to \mch$ is exact exactly when \[ \mcf_p \to \mcg_p \to \mch_p \] is exact for every $p \in X$. For $0 \to \mcf \to \mcg \to \mch \to 0$ exact, the sequence of global sections \[ 0 \to \mcf(X) \to \mcg(X) \to \mch(X) \] is exact, but the last map need not be surjective.

Surjectivity of $\mcg \to \mch$ means every germ lifts, which is a statement about some neighbourhood of each point. A global section of $\mch$ therefore lifts over each member of some cover, and the lifts need not agree on overlaps. The obstruction to correcting them is a Čech $1$-cocycle, and it is $H^1(X, \mcf)$ that measures whether it can be corrected.

The failure occurs on overlaps: local lifts differ by a Čech $1$-cocycle in the kernel, and its class in $H^1(X,\mcf)$ is the obstruction to gluing them to a global lift.

How do you check that a sequence of sheaves is exact?

Is a surjection of sheaves surjective on sections?

Where does the failure of right exactness go?

A surjection of sheaves need not be surjective on global sections.

A surjection of sheaves that is not surjective on sections (Tag FE-Y12XB)

On $X = \PP^1$ take two distinct points $p \neq q$ and the evaluation sequence \[ 0 \to \OO(-p-q) \to \OO_X \to \OO_p \oplus \OO_q \to 0 . \] The right map is surjective: at $p$ it is the surjection $\OO_{X,p} \to k$, at $q$ likewise, and at every other point the target stalk is $0$.

On global sections it is the map $k \to k^2$ sending a constant $c$ to $(c,c)$, which is not surjective: a regular function on $\PP^1$ taking the value $0$ at $p$ and $1$ at $q$ would be a nonconstant global regular function. The cokernel is one-dimensional, and indeed $H^1(\PP^1, \OO(-2)) \cong k$.

A nonzero map $\OO_{\PP^1}(-2) \to \OO_{\PP^1}$ is multiplication by a nonzero section $s \in H^0(\OO(2))$, whose zero scheme $Z$ has degree $2$. Its cokernel is $\OO_Z$: either $\OO_p \oplus \OO_q$ when $s$ has two distinct zeros $p, q$, as above, or $\OO_{\PP^1,p}/\mfm_p^2$, of length $2$ at $p$, when $s$ vanishes to order $2$ at $p$. In both cases $H^0$ of the cokernel is $2$-dimensional.

The example makes the failure numerical: right exactness on global sections fails by exactly the dimension of $H^1$ of the kernel.

The same shape recurs as the classical exponential sequence on a complex manifold, where the failure of $\OO \to \OO^*$ on sections is the existence of line bundles, and $H^1(X,\OO^*) = \Pic(X)$.

Give a surjection of sheaves which is not surjective on global sections.

On $\PP^1$, find the cokernel of a map $\OO(-2) \to \OO$.

The exponential sequence, and the logarithm that does not glue (Tag FE-DNTT6)

On $X = \CC^*$ take the exponential sequence \[ 0 \to \ul{\ZZ} \xrightarrow{\ 2\pi i\ } \OO_X \xrightarrow{\ \exp\ } \OO_X^* \to 0 . \] It is exact as a sequence of sheaves, because it is exact on stalks: near any point a nonvanishing holomorphic function has a logarithm.

It is not exact as a sequence of presheaves. Cover $\CC^*$ by contractible opens $U_i$, on each of which $\log$ exists, so $\id_{U_i} \in \OO^*(U_i)$ lies in the image presheaf. These sections agree on overlaps and glue to $\id_{\CC^*}$, which is not in the image presheaf: a global logarithm on $\CC^*$ does not exist.

So the image presheaf fails the gluing axiom, and sheafifying it is exactly what supplies the missing section.

The same sequence, taken in cohomology, is where the Picard group comes from: \[ H^1(X, \OO) \to H^1(X, \OO^*) \to H^2(X, \ZZ) , \] with $H^1(X,\OO^*) = \Pic(X)$ and the right-hand map the first Chern class. The failure to glue here and the failure of surjectivity on global sections in PR-C9ZEK are the same phenomenon counted twice.

Give an example showing the image presheaf is not a sheaf.

Sheaves with isomorphic stalks, and pushforwards of locally constant sheaves (Tag FE-SHFISOSTALKS)

Let $X = Y = S^1 \subset \CC$ be the unit circle, and let $f \colon X \to Y$ be the two-sheeted covering map $f(z) = z^2$. Let $\mcf = \ul{\ZZ}_X$ be the constant sheaf with value $\ZZ$ on $X$.

Every $y \in Y$ has arbitrarily small connected open neighbourhoods $V$ with $f^{-1}(V) = U_1 \amalg U_2$, each $U_i$ connected and mapped homeomorphically onto $V$. So $(f_* \mcf)(V) = \mcf(U_1) \oplus \mcf(U_2) \cong \ZZ^2$, the restriction maps between such neighbourhoods are isomorphisms, and $(f_* \mcf)_y \cong \ZZ^2$.
Let $\mcg = \ul{\ZZ^2}_Y$ be the constant sheaf. Its stalks are also $\ZZ^2$ at every point.
$\Gamma(Y, \mcg) \cong \ZZ^2$ because $Y$ is connected, while $\Gamma(Y, f_* \mcf) = \mcf(X) \cong \ZZ$ because $X$ is connected. Since $\ZZ \not\cong \ZZ^2$, the sheaves $f_* \mcf$ and $\mcg$ are not isomorphic, although all their stalks are isomorphic.

By step 1, $f_* \mcf$ restricts to a constant sheaf on each such $V$, so it is a locally constant sheaf. It is not constant: going once around $Y$ exchanges the two sheets, and its global sections are the invariants $(\ZZ^2)^{\ZZ/2} \cong \ZZ$. So this example shows that stalks do not determine a sheaf; it does not show that pushforward destroys local constancy, because the pushforward of a locally constant sheaf along a finite covering map is locally constant.

A morphism of sheaves $\phi \colon \mcf \to \mcg$ is an isomorphism exactly when every stalk map $\phi_x$ is an isomorphism. The hypothesis is a single morphism $\phi$ inducing the stalk isomorphisms; isomorphisms between stalks chosen separately at each point do not suffice.

(A pushforward of a locally constant sheaf that is not locally constant)

Let $j \colon U = (0, \infty) \hookrightarrow \RR$ be the inclusion and $\mcf = \ul{\ZZ}_U$. For an open interval $I \subseteq \RR$, $(j_* \mcf)(I) = \mcf(I \cap (0, \infty))$, which is $\ZZ$ if $I$ meets $(0,\infty)$ and $0$ otherwise. So $(j_* \mcf)_x = \ZZ$ for $x \geq 0$ and $(j_* \mcf)_x = 0$ for $x < 0$. Every neighbourhood of $0$ contains points of both kinds, and a locally constant sheaf has isomorphic stalks at all points of a connected neighbourhood, so $j_* \mcf$ is not locally constant.

Give an example of two sheaves with isomorphic stalks at every point that are not isomorphic.

Is the pushforward of a locally constant sheaf always locally constant?

Exactness of pushforward, and pullback to a point (Tag PR-SHFPUSHEX)

Pushforward is left exact, closed-embedding pushforward is exact, and pulling a sheaf back to a point gives its stalk.Suggest a slogan

Let $f \colon X \to Y$ be continuous and $0 \to \mcf' \to \mcf \to \mcf''$ an exact sequence of sheaves of abelian groups on $X$.

$0 \to f_* \mcf' \to f_* \mcf \to f_* \mcf''$ is exact.
If $f$ is a homeomorphism onto a closed subset of $Y$, then $f_*$ is exact.
For $i \colon \{x\} \hookrightarrow X$ the inclusion of a point, $i^{-1} \mcf \cong \mcf_x$.

Without the closed-image hypothesis part 2 fails: for $j \colon \CC^* \hookrightarrow \CC$ the exponential sequence $0 \to 2\pi i \ZZ \to \OO \to \OO^* \to 0$ on $\CC^*$ is exact, but $j_* \OO \to j_* \OO^*$ is not surjective on stalks at $0$, since $z$ has no logarithm on any punctured disc.

Show that pushforward is left exact, and exact along a closed embedding.

For $i \colon \{x\} \hookrightarrow X$, show that $i^{-1} \mathcal{F} \cong \mathcal{F}_x$.

The obstruction to a global lift

Lifting a global section of $\mch$ is possible over each member of some open cover, by the definition of surjectivity. The lifts differ on overlaps by sections of $\mcf$, and those differences form a Čech $1$-cocycle. The section lifts globally exactly when that cocycle is a coboundary, so the obstruction lives in $H^1(X, \mcf)$ and the long exact sequence \[ 0 \to \mcf(X) \to \mcg(X) \to \mch(X) \to H^1(X, \mcf) \to \cdots \] records it; see cohomology.

Locally constant and constructible sheaves

Local systems and monodromy (Tag D-SHFLOCSYS)

Let $A$ be an abelian group with the discrete topology. The constant sheaf $\ul{A}$ on a topological space $X$ has $\ul{A}(U)$ the group of continuous maps $U \to A$, with restriction of functions. If $U$ is open and connected, then $\ul{A}(U) = A$; more generally, if every connected component of the open set $U$ is open, then $\ul{A}(U) = \prod_{\pi_0(U)} A$, one factor for each connected component. [@Har10a, §II.1]

(Locally constant sheaf and local system)

A sheaf $\mathcal{L}$ of abelian groups on a topological space $X$ is locally constant if every point has an open neighbourhood $U$ with $\mathcal{L}|_U$ isomorphic to a constant sheaf. A local system is a locally constant sheaf of abelian groups, or of $k$-vector spaces, whose stalks are finitely generated, respectively finite-dimensional.

(Monodromy)

Let $X$ be connected, locally path-connected and semilocally simply connected, with base point $x$. Taking the stalk at $x$ with its action by parallel transport along loops gives an equivalence between local systems of $k$-vector spaces on $X$ and finite-dimensional representations $\pi_1(X, x) \to \GL(\mathcal{L}_x)$. The constant local systems correspond to the trivial representations, and $\Gamma(X, \mathcal{L}) = \mathcal{L}_x^{\pi_1(X,x)}$.

For the double cover $f \colon S^1 \to S^1$, $z \mapsto z^2$, the pushforward $f_* \ul{\QQ}$ is the local system of rank $2$ whose monodromy sends the generator of $\pi_1(S^1) = \ZZ$ to the swap of the two coordinates of $\QQ^2$. It decomposes as the constant local system $\ul{\QQ}$ plus the rank-one local system with monodromy $-1$, which has no nonzero global sections.

What is a local system?

Constructible sheaves (Tag D-SHFCONSTR)

Let $X$ be a complex algebraic variety with its analytic topology. A sheaf $\mathcal{F}$ of $k$-vector spaces on $X$ is constructible if there is a finite partition $X = \coprod_i X_i$ into Zariski locally closed subvarieties such that each restriction $\mathcal{F}|_{X_i}$ is a local system.

The same definition with the Zariski or étale topology on a Noetherian scheme, and locally constant sheaves with finite stalks in place of local systems, gives constructible étale sheaves. Constructible sheaves are preserved by $f^{-1}$, by $f_*$ and $f_!$ for morphisms of varieties, and by $\otimes$ and $\mathcal{H}om$, in the derived sense: for a morphism of complex varieties, the six operations (D-SHFSIX) carry bounded complexes with constructible cohomology sheaves to bounded complexes with constructible cohomology sheaves.

Let $j \colon U = \AA^1 \setminus \{0\} \hookrightarrow \AA^1$ and $i \colon \{0\} \hookrightarrow \AA^1$ over $\CC$. The sheaf $j_! \ul{\CC}_U$ has stalk $\CC$ on $U$ and $0$ at $0$, and $i_* \CC$ has stalk $\CC$ at $0$ and $0$ elsewhere. Both are constructible with respect to the partition $\AA^1 = U \amalg \{0\}$, and neither is a local system on $\AA^1$.

What is a constructible sheaf?

Support

Support of sections and sheaves (Tag D-UDIVH)

Let $\mcf$ be a sheaf of abelian groups on a topological space $X$. For an open set $U\subseteq X$ and a section $s\in\mcf(U)$, the support of the section is \[ \supp(s)\coloneqq\ts{x\in U\st s_x\ne0\text{ in }\mcf_x}. \] The support of the sheaf is \[ \supp(\mcf)\coloneqq\ts{x\in X\st\mcf_x\ne0}. \] Here $s_x$ is the germ of $s$ in the stalk $\mcf_x$.

For every open set $U\subseteq X$ and $s\in\mcf(U)$, the subset $\supp(s)$ is closed in $U$.

If $s_x=0$, the definition of a germ gives an open neighborhood $x\in V\subseteq U$ with $s|_V=0$. Every point of $V$ therefore lies outside $\supp(s)$. Thus $U\setminus\supp(s)$ is open.

(A sheaf with nonclosed support)

Let $p$ be a prime number and put $X=\Spec\ZZ_{(p)}$. Its points are the generic point $\eta$ and the closed point $x=(p)$, and its open sets are $\varnothing$, $\{\eta\}$, and $X$. Define a sheaf of abelian groups $\mcf$ by \[ \mcf(X)=0,\qquad\mcf(\{\eta\})=\ZZ,\qquad\mcf(\varnothing)=0, \] with zero restriction maps for proper inclusions. Every open cover of a nonempty open set contains that open set itself, so the sheaf axioms hold. The only neighborhood of $x$ is $X$, whereas $\{\eta\}$ is a neighborhood of $\eta$. Consequently $\mcf_x=0$ and $\mcf_\eta\cong\ZZ$, giving $\supp(\mcf)=\{\eta\}$, which is not closed in $X$. For the open inclusion $j:\{\eta\}\hookrightarrow X$, this sheaf is the extension by zero $j_!\ul{\ZZ}$.

Let $X$ be a scheme and $\mcf$ a quasi-coherent $\OO_X$-module of finite type. Then $\supp(\mcf)$ is closed. For an affine open $U=\Spec A$ with $\mcf|_U\cong\widetilde M$, \[ \supp(\mcf)\cap U=V(\Ann_A M). \] For $M$ generated by $m_1,\ldots,m_r$, a prime $\mfp$ has $M_\mfp=0$ if and only if some $a\notin\mfp$ annihilates every $m_i$, that is, if and only if $\mfp\not\supseteq\Ann_AM$ (P-AGH256SUPPORT).

Let $i:Z\to X$ be a closed immersion of schemes, and let $\mcf$ be a sheaf of abelian groups on $Z$. Then \[ \supp(i_*\mcf)=i(\supp\mcf). \] In particular, $\supp(i_*\mcf)$ is closed in $X$ if and only if $\supp\mcf$ is closed in $Z$.

For $x\notin i(Z)$, the open set $X\setminus i(Z)$ is a neighborhood on which $i_*\mcf$ is zero, so $(i_*\mcf)_x=0$. For $x=i(z)$, the sets $i^{-1}(U)$, with $U$ an open neighborhood of $x$ in $X$, form a neighborhood basis at $z$ in $Z$ because $i$ is a homeomorphism onto its image. Taking the colimit of $(i_*\mcf)(U)=\mcf(i^{-1}(U))$ gives $(i_*\mcf)_{i(z)}\cong\mcf_z$. The support equality follows. The closedness equivalence follows because $i$ identifies $Z$ with a closed subspace of $X$.

The identity $i:X\to X$ is a closed immersion. For $X=\Spec\ZZ_{(p)}$ and the sheaf with $\mcf_\eta\cong\ZZ$ and $\mcf_x=0$, its direct image has support $\{\eta\}$, which is not closed.

Define the support of a section, and of a sheaf.

Why is $\supp(s)$ closed?

Give a sheaf whose support is not closed.

The support of a sheaf need not be closed: for an open immersion $j\colon U\hookrightarrow X$, extension by zero gives $\supp(j_!\ZZ_U)=U$. For a coherent sheaf on a noetherian scheme, the support is closed.

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