5 problems

Let $X$ be a smooth scheme over $S$ of dimension $d$. Write $X^{(a)}$ for the points of codimension $a$, $i_{*,x}$ for extension by zero from $x$, $\epsilon$ for the relevant morph…

Let $k$ be a perfect field of characteristic $p>0$, and let $X$ be a smooth $d$-dimensional $k$-scheme. Write $X^{(a)}$ for the points of codimension $a$, $i_{*,x}$ for extension b…

Let $X = \operatorname{Spec} A$, where $(A,\mathfrak{m}_A,L)$ is a regular local ring. For each $n \geq 0$, let $\mathcal{G}_n(X)$ be the Gersten complex associated with the codime…

Let $A$ be a Noetherian local $\mathbb Q$-algebra, let $I\subseteq A$ be an ideal, let $n\in\mathbb Z$, and consider an abstract blow-up square … Assume that $X$ is regular. Here…

Let $X=\operatorname{Spec}(R)$ be the spectrum of a regular local ring $R$. For each integer $l$, let $\mathrm{M}_l(X)$ be the category of coherent sheaves on $X$ whose supports ha…