Invariant sheaves (Q5934159)

An invariant sheaf on an \(n\)-manifold is the data consisting of a coherent \({\mathcal O}_X\)-module \(F_X\) for each smooth \(n\)-dimensional manifold \(X\) and an isomorphism \(i(f): f^*F_Y \to F_X\) for any étale morphism \(f: X \to Y\) with \(i(f)\) satisfying the chain condition \(i(g\circ f) = i(g)\circ i(f)\). Many canonically defined sheaves are invariant sheaves, e.g. tangent sheaves, sheaves of differential forms. The author shows that the category \(I(n)\) of invariant sheaves is equivalent to the category of \(G\)-modules, \(G\) being the group of formal transformations of an \(n\)-dimensional vector space which fix the origin. A relative version of this equivalence for invariant sheaves on \(S\)-schemes \(X\) is proved. NEWLINENEWLINENEWLINEThe infinite dimensional group \(G\) is a semi-direct product of \(GL(n)\) and a projective limit of finite dimensional unipotent groups. In particular, it contains \(\mathbb{G}_m\). A weight filtration for a \(G\)-module is defined using the weight modules for \(\mathbb{G}_m\). By the equivalence of categories, this gives a weight filtration for an invariant sheaf. It is shown that \(I(n)\) is a filtered rigid tensor category. Properties of invariant sheaves and their Ext groups are studied. Finally invariant sheaves on \(X\times X\) are considered.