11 problems

Let $R$ be a regular Noetherian local ring, let $K_j^M(R)$ denote Milnor $K$-theory, and write $\widehat K_j^M(R)$ for the improved Milnor $K$-group. For each codimension $i$, writ…

The Gersten conjecture. These complexes are exact whenever $R$ is a regular local ring. Here $K$ is the fraction field of $R$, and $k(q)$ is the residue field at $q$. The conjectur…

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 $X$ 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 $S=\operatorname{Spec}(\mathcal O_K)$ be the spectrum of a discrete valuation ring with local parameter $\pi$ and residue field $k$, let $X$ be a scheme of finite type over $S$…

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…

Gersten's conjecture. If $A$ is a regular local ring, then this map is injective for all $n\geq 0$. This is a theorem when $A$ contains a field; the paper studies analogues of this…

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…

Gersten exactness conjecture. This complex is exact for every closed point $x$ of $\mathcal{X}$.

Let $A$ be a one-dimensional complete local ring with maximal ideal $\mathfrak m$, and define its completed $K$-groups by … Write $\operatorname{Frac} A$ for the product of the loc…