An explicit construction of the Grothendieck residue complex

An explicit construction of the Grothendieck residue complex

Amnon Yekutieli
An explicit construction of the Grothendieck residue complex
  • Année : 1992
  • Tome : 208
  • Format : Électronique, Papier
  • Langue de l'ouvrage :
  • Class. Math. : 14F10, 14B15, 12J10
  • Nb. de pages : 138
  • ISSN : 0303-1179
  • DOI : 10.24033/ast.150

Let $X$ be a scheme of finite type over a field $k$, with structural morphism $\pi $. The residue complex $K_X^{\displaystyle \cdot }$ on $X$ is the Cousin complex associated to $\pi ^!k$, as in “Residues and Duality”. We give an explicit construction of this complex when $X$ is reduced and $k$ is perfect. We begin with the theory of semi-topological rings. These rings admit many operations (e.g. limits, base change) and there is a differential calculus over them. This theory is used to treat topological local fields (TLFs), which are the high dimensional local fields of Parshin, endowed with suitable topologies. We investigate the structure of TLFs, and give an improved version of the Parshin-Lomadze residue functor on the category TLF($k$). Next we turn to the Beilinson completion functors, which we also topologize. These provide a link between the geometry of $X$ and TLFs. The Parshin residue map Res$_{\xi ,\tau }$ depends on a saturated chain $\xi = (x,\ldots ,y)$ in $X$ and a coefficient field $\tau $ for $y$. Definie $K(\tau ) := \mathrm {Hom}_{k(y)}^{\mathrm {cont}}(\widehat {O}_{X,y},\omega (y))$. The residue map Res$_{\xi ,\tau }$ gives rise to a coboundary homomorphism $\delta _{\xi ,\sigma /\tau } : K(\sigma )\to K(\tau )$. Using base change arguments we remove the dependence of $\delta _\xi $ on coefficient fields. Summing over all $x\in X$ we get our complex $(K_X^{\displaystyle \cdot }, \delta _X)$. We then proceed to show that this complex has the correct properties. In the appendix the canonical isomorphism $K_X^{\displaystyle \cdot }\simeq \pi ^!k$ is exhibited.

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