Gersten complex

Definition

Let $X$ be a Noetherian scheme of finite Krull dimension. For $p \ge 0$ let $X^{(p)}$ be the set of points $x \in X$ of codimension $p$, that is, with $\dim \mathcal{O}_{X,x} = p$. Let $k(x)$ be the residue field of $x$. Let $K_m$ be Quillen's algebraic K-groups, with $K_m = 0$ for $m < 0$. Let $G_m$ be the K-theory of coherent sheaves. The filtration of coherent sheaves by codimension of support gives Quillen's coniveau spectral sequence $$E_1^{p,q} = \bigoplus_{x \in X^{(p)}} K_{-p-q}(k(x)) \Rightarrow G_{-p-q}(X).$$ For $n \ge 0$, the Gersten complex of $X$ in degree $n$ is the row $q = -n$ of its $E_1$ page. It is the chain complex $$\bigoplus_{x \in X^{(0)}} K_n(k(x)) \xrightarrow{d_1} \bigoplus_{x \in X^{(1)}} K_{n-1}(k(x)) \xrightarrow{d_1} \cdots \xrightarrow{d_1} \bigoplus_{x \in X^{(n)}} K_0(k(x)).$$ The same construction applies to $\operatorname{Spec} R$ for a Noetherian local ring $R$. Applied to each open subset of $X$, it gives a complex of sheaves on $X$.