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Let $f: X \rightarrow X'$ be a morphism of schemes, and let $\mathcal{I}^{\bullet}$ be a complex of $\mathcal{O}_X$-modules. There are two spectral sequences (well, more than that, but these are the …

There is no maximal length. See "Some results on associated primes of local cohomology modules", J. Asadollahi and P. Schenzel, Japan J. Math. 29 (2003), 285--296. Proposition 2.2 in this paper es …

Let $(R, \mathfrak{m})$ be a local ring and $X = Spec(R)$. Let $Y = V(I)$ be a closed subscheme of $X$, defined by an ideal $I \subset R$, and let $P \in X$ (in fact, $P \in Y$) be the closed point. …

It is false! As mentioned in the edit, a positive answer to this question would imply the Lyubeznik numbers $\lambda_{i,j}$ of a projective scheme are independent of the defining projective embedding …

Suppose $Y$ is a projective variety over a field $k$. Fix an embedding $\iota: Y \hookrightarrow \mathbb{P}^n_k$ for some $n$, and consider the local cohomology sheaves $\mathcal{H}^j_Y(\omega_{\mathb …

Let $X$ be a nonsingular algebraic variety over a field $k$ of characteristic zero. (We may assume $k$ algebraically closed if need be, but I want to avoid specifically demanding $k = \mathbb{C}$.) Le …