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As a complex manifold $\mathbb{P}^n$ is locally the euclidean space $\mathbb{C}^n$, as a projective variety it is locally $\mathbb{C}^n$ with the Zariski topology, as a scheme it is locally $\text{Sp …

At the Brenner's introduction to the geometric view of the tight closure the author states that an affine scheme has a natural addition (this addition will be extended to the vector bundles). I wonder …

Not, necessarily. For example, in $K[x,y]$, $x^3,xy^2$ is not a $d$-sequence, but it generates an almost complete intersection. However, in some cases, the answer is positive. For example, see (5) of …

Let ,$R=\mathbb{Z}/3\mathbb{Z}[x,y,z,w]/(x^4+y^3+z^4)$. Then, $y\in (x,z)^F\subseteq (x,z)^*\subseteq \overline{(x,z)}$, and thence $(x,z)$ is not integrally closed, however $x,z$ is a regular sequenc …

Due to the comments to the previous answer (see the next answer) I give the following counterexample for the case where $d$-sequences are defined without considering permutations: Let $R=K[x,y]$ and …