Asymptotic depth of twisted higher direct image sheaves

Abstract

Let $ \pi:X \rightarrow X_{0}$ be a projective morphism of schemes, such that $ X_{0}$ is Noetherian and essentially of finite type over a field $ K$. Let $ i \in \mathbb{N}_{0}$, let $ {\mathcal{F}}$ be a coherent sheaf of $ {\mathcal{O}}_{X}$-modules and let $ {\mathcal{L}}$ be an ample invertible sheaf over $ X$. Let $ Z_{0} \subseteq X_{0}$ be a closed set. We show that the depth of the higher direct image sheaf $ {\mathcal{R}}^{i}\pi_{*}({\mathcal{L}}^{n} \otimes_{{\mathcal{O}}_{X}} {\mathcal{F}})$ along $ Z_{0}$ ultimately becomes constant as $ n$ tends to $ -\infty$, provided $ X_{0}$ has dimension $ \leq 2$. There are various examples which show that the mentioned asymptotic stability may fail if $ \dim(X_{0}) \geq 3$. To prove our stability result, we show that for a finitely generated graded module $ M$ over a homogeneous Noetherian ring $ R=\bigoplus_{n \geq 0}R_{n}$ for which $ R_{0}$ is essentially of finite type over a field and an ideal $ \mathfrak{a}_{0} \subseteq R_{0}$, the $ \mathfrak{a}_{0}$-depth of the $ n$-th graded component $ H^{i}_{R_{+}}(M)_{n}$ of the $ i$-th local cohomology module of $ M$ with respect to $ R_{+}:=\bigoplus_{k>0}R_{k}$ ultimately becomes constant in codimension $ \leq 2$ as $ n$ tends to $ -\infty$.