(Constructible sheaf)
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 \(\AA^1\).
Dependencies
D-SHFLOCSYSLocal systems and monodromyuses
Backlinks
T-LADICPROPERProper pushforward preserves constructible $\ell$-adic sheavesuses