Quasicoherent and coherent sheaves

(Quasicoherent)

For an $A$-module $M$, let $\tilde{M}$ be the sheaf on $\Spec A$ with $\tilde{M}(D_f) = M_f$. An $\OO_X$-module $\mcf$ is quasicoherent if $X$ has a cover by affine opens $U_i = \Spec A_i$ with $\mcf|_{U_i} \cong \tilde{M_i}$ for $A_i$-modules $M_i$.

(Coherent module)

Let $A$ be a ring. An $A$-module $M$ is coherent if it is finitely generated and for every $n \geq 0$ and every $A$-linear map $\varphi \colon A^n \to M$, not necessarily surjective, the kernel of $\varphi$ is finitely generated.

(Coherent sheaf)

An $\OO_X$-module $\mcf$ is coherent if $X$ has a cover by affine opens $U_i = \Spec A_i$ with $\mcf|_{U_i} \cong \tilde{M_i}$ for coherent $A_i$-modules $M_i$.

For an $A$-module $M$: coherent $\implies$ finitely presented $\implies$ finitely generated. If $A$ is Noetherian, the three conditions coincide, so on a locally Noetherian scheme $X$ a quasicoherent $\mcf$ is coherent exactly when each $M_i$ in some affine cover is finitely generated.

On $\Spec A$, the functor $M \mapsto \tilde{M}$ is an exact equivalence from $A$-modules to quasicoherent sheaves, with inverse $\mcf \mapsto \mcf(X)$.

(An $\OO_X$-module that is not quasicoherent)

Let $A = k[x]_{(x)}$, so $X = \Spec A$ has two points: the generic point $\eta$ and the closed point $(x)$, and the open sets are $\emptyset$, $U = \{\eta\} = D(x)$ and $X$, with $\OO_X(U) = k(x)$. Define $\mcf(X) = k(x)$, an $A$-module by multiplication in the field $k(x) \supseteq A$, and $\mcf(U) = \mcf(\emptyset) = 0$, with zero restriction maps. This is a sheaf of $\OO_X$-modules: the only open cover of $X$ contains $X$ itself, so the sheaf conditions hold, and the module structures are compatible with restriction because $\mcf(U) = 0$. It is not quasicoherent: if $\mcf \cong \tilde{M}$, then $M \cong \mcf(X) = k(x)$ and $\mcf(D(x)) \cong M_x = k(x) \neq 0$, while $\mcf(U) = 0$.

The module structure matters. If instead $x$ acts by $0$ on $k(x)$, through the residue field $A/(x) = k$, then $M_x = 0$ and $\mcf = \tilde{M}$ is quasicoherent: it is the skyscraper sheaf at the closed point. The same holds for the skyscraper sheaf at the origin of $\AA^1_k = \Spec k[x]$ with value $k(x)$: with the multiplication action of $\OO(V) \subseteq k(x)$ on each open $V \ni 0$ it is an $\OO$-module that is not quasicoherent, and with the action through the residue field $k$ it is quasicoherent.

If $\mcf$ is quasicoherent, then $\mcf|_U\cong\widetilde{\mcf(U)}$ for every affine open $U\subseteq X$, not only for the members of one cover [@Har10a, Proposition II.5.4]. Quasicoherent sheaves on an affine scheme have vanishing higher cohomology.

An $\OO_X$-module $\mcf$ is quasicoherent if and only if every point has an open neighbourhood $U$ on which $\mcf$ has a presentation \[ \OO_U^{(I)} \to \OO_U^{(J)} \to \mcf|_U \to 0 . \]

What is a quasicoherent sheaf?

What does coherent add?

Which local presentation characterizes quasicoherence?

What is a coherent module over an arbitrary ring, and how is it related to finite presentation?

Give an $\OO_X$-module that is not quasicoherent.

Dependencies

D-VKR54$\Spec A$ as a locally ringed spaceuses

Backlinks

T-MODVBLocally free sheaves are vector bundlesrelated-to
D-MODOXSheaves of modules, and the operations on themrelated-to
D-MODGGGlobally generated sheavesuses
D-MODPULLPullback and pushforward of sheaves of modulesuses
D-SCHRELSPECPROJRelative Spec and relative Projuses
FE-MODZOOSeparating quasicoherent, coherent, and locally freeuses
D-MODIDEALThe ideal sheaf of a closed subschemeuses
D-MODGRMODThe sheaf associated to a graded module, and $\Gamma_*$uses
D-4GCH6The sheaf of differentials, and its quasicoherenceuses
D-CB9XSThe twisting sheaves $\OO(n)$, and $\Pic(\PP^n) = \ZZ$uses

What links to this

Quasicoherence and twisting