Let R be an augmented regular local ring over a field $k$ with maximal ideal m. There is the Grothendieck residue symbol:

$$Res: H^n_m(\Omega^n) \to k$$

If $k=\mathbb{C}$ and $R$ is affine space, this map has a well-known interpretation by way of an integral formula.

If X is a projective scheme over $\mathbb{C}$, there is the Serre-duality trace $$ tr: H^n(\omega) \to \mathbb{C}$$ which at least when $X$ is smooth can be represented similarly in terms of integrals. My question is whether there is a way to combine these two ideas namely suppose $X$ is a smooth quasi-projective variety over $\mathbb{C}$ and $Z$ a proper subscheme, can we define a trace:

$$ tr/Res: R^n\Gamma_{Z}(\Omega_X^n) \to \mathbb{C} $$

Is there a description for this map in terms of integrals?

Most of the literature on residues, Grothendieck duality and the like is way too heavy for me. Where is this written up in a friendly way?

Edit: Proposition 2.3.2 or Remark 2.3.8 of http://www.math.purdue.edu/~lipman/papers/formal-duality.pdf seem close to what I want (or maybe are exactly what I would like). Still, I would be happy if there is a more simple minded reference with some analytic treatment.

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    $\begingroup$ What does augmanted mean in the context of regular local rings? $\endgroup$ Jul 11, 2013 at 21:44
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    $\begingroup$ I have edited it to indicate that it's also a k-algebra. Then augmented just means that the residue field is also $k$. $\endgroup$
    – user36931
    Jul 12, 2013 at 0:39
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    $\begingroup$ In the last displayed equation I think it should be $$R^n \Gamma_Z(\Omega_X^n)$$ not $R\Gamma_Z^n(\Omega_X^n)$, right? $\endgroup$ Jul 12, 2013 at 8:46
  • $\begingroup$ Yes, corrected. $\endgroup$
    – user36931
    Jul 12, 2013 at 16:39

1 Answer 1


Perhaps you should look at Lipman's book:

"Dualizing sheaves, differentials and residues on algebraic varieties", Asterisque 117

It gives a treatment of the topic you asked for. In brief, there is a commutative triangle:

$\require{AMScd}$ \begin{CD} \bigoplus_{P \in X} H^n_P(\Omega^n) @>can>> H^n(X,\Omega^n)\\ @V\oplus_{P \in X}res_PVV \swarrow \tiny{\int_X} \\ k \end{CD}

Expressing the fact that the integral is computed through residues. The treatment is very down to earth. In particular, he avoids derived categories.


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