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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. …

Let $S = \oplus_{d \geq 0} S_d$ be a graded (Noetherian) ring, let $I \subset S$ be a homogeneous ideal, and let $f \in S$ be a homogeneous element. Denote by $S_{(f)}$ the subring of degree-$0$ elem …

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 …