Knowlpedia

Definition

Direct image of a sheaf

The sheaf on the target whose sections are sections over inverse images of open sets.

Definition

Let f:X→Yf:X\to Y be a and let F\mathcal F be a on XX. The direct image, or pushforward, of F\mathcal F is the sheaf f∗Ff_*\mathcal F on YY defined by

(f∗F)(V):=F(f−1(V))(f_*\mathcal F)(V):=\mathcal F(f^{-1}(V))

for every open subset V⊆YV\subseteq Y. Its restriction maps are those of F\mathcal F, applied to inclusions f−1(W)⊆f−1(V)f^{-1}(W)\subseteq f^{-1}(V).

Thus the direct image does not alter the sections themselves; it regards sections on inverse images in XX as sections over open sets of YY. For a sheaf of rings, f∗Ff_*\mathcal F is again a sheaf of rings.