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Wheat sheaves near King's Somborne.Here the individual sheaves have been put together into a stook ("stooked") to dry. A sheaf of grain on a plaque Sheafing machine. A sheaf (/ ʃ iː f /; pl.: sheaves) is a bunch of cereal-crop stems bound together after reaping, traditionally by sickle, later by scythe or, after its introduction in 1872, by a mechanical reaper-binder.
This observation is used to construct another example which is crucial in algebraic geometry, namely quasi-coherent sheaves. Here the topological space in question is the spectrum of a commutative ring R {\displaystyle R} , whose points are the prime ideals p {\displaystyle {\mathfrak {p}}} in R {\displaystyle R} .
Here we use the definition of constructible étale sheaves from the book by Freitag and Kiehl referenced below. In what follows in this subsection, all sheaves F {\displaystyle {\mathcal {F}}} on schemes X {\displaystyle X} are étale sheaves unless otherwise noted.
[1] [2] In addition to cutting the small-grain crop, a binder also 'binds' the stems into bundles or sheaves. These sheaves are usually then 'shocked' into A-shaped conical stooks, resembling small tipis, to allow the grain to dry for several days before being picked up and threshed. Withington's original binder used wire to tie the bundles.
Sheaves of grain would be opened up and the stalks spread across the threshing floor. Pairs of donkeys or oxen (or sometimes cattle , or horses) would then be walked round and round, often dragging a heavy threshing board behind them, to tear the ears of grain from the stalks, and loosen the grain itself from the husks .
In sailing, a block is a single or multiple pulley. One or a number of sheaves are enclosed in an assembly between cheeks or chocks. In use, a block is fixed to the end of a line, to a spar, or to a surface. A line (rope) is reeved through the sheaves, and maybe through one or more matching blocks at some far end, to make up a tackle.
A pulley is a wheel on an axle or shaft enabling a taut cable or belt passing over the wheel to move and change direction, or transfer power between itself and a shaft. A sheave or pulley wheel is a pulley using an axle supported by a frame or shell (block) to guide a cable or exert force.
If dealing with sheaves of sets instead of sheaves of abelian groups, the same definition applies. Similarly, if f: (X, O X) → (Y, O Y) is a morphism of ringed spaces, we obtain a direct image functor f ∗: Sh(X,O X) → Sh(Y,O Y) from the category of sheaves of O X-modules to the category of sheaves of O Y-modules.