"Nice" definition of discriminant as alluded to in an answer of Qing Liu

Question

In his answer here Qing Liu mentioned "the 'discriminant' of X which measures the defect of a functorial isomorphism which involves powers of the relative dualizing sheaf of X/R."

Could somebody give a reference for or explain this?

I ask because there is a natural definition of conductor of a proper morphism to a normal scheme (for example as in Serre's "Algebraic groups and class fields"), and some relationship between the conductor and discriminant, but the definitions of discriminants I have seen always seem situation-specific and ad-hoc.

Thanks.

Answer 1

The "magistral paper of Takeshi Saito" mentioned by Qing Liu is the following: 'Conductor, Discriminant, and the Noether formula of Arithmetic surfaces', Duke Math Journal, vol. 57, no. 1. See here if you have a subscription to Duke: projecteuclid.org/euclid.dmj/1077306852 The definition of the discriminant is at the top of the second page. Or see section three of Liu's notes which he referenced in his reply to which you provided a link.

But for what it's worth, here is a quick answer (basically copied from Saito's paper):

Let $T$ be a nice scheme (spectrum of a field is enough for us), and $g:Y\to T$ a relative curve (i.e. geometrically connected, proper, relative dimension one). When $Y$ is smooth over $T$, there is a functorial isomorphism $\Delta:\det Rg_*(\omega_{Y/T}^{\otimes 2})\to(\det Rg_*\omega_{Y/T})^{\otimes 13}$. This is due to Deligne (from this unpublished letter to Quillen, link thanks to D. Eriksson's answer).

Now let $\mathcal{O}_K$ be a Henselian discrete valuation ring with algebraically closed residue field, put $S=\mbox{Spec}\mathcal{O}_K$, and let $X\to S$ be a relative curve which is regular. We have an invertible $\mathcal{O}_K$-module

$V=\mbox{Hom} (\det Rg_*(\omega_{X/S}^{\otimes 2}),(\det Rg_*\omega_{X/S})^{\otimes 13})$.

and Deligne's result, applied to the generic fibre of $X$, produces a natural element $\Delta\in V\otimes_{\mathcal{O}_K} K$. The discriminant of $X$ is defined to be the order of $\Delta$ (that is, the unique $d\in\mathbb{Z}$ such that $\mathcal{O}_K\Delta=\mathfrak{p}^d V$).

I do not know if there is a similar notion of discriminant for higher dimensional $S$-schemes.

Answer 2

It seems I don't have enough reputation to make comments, so I'll write a comment here instead: The "Lettre à Quillen" mentioned in Matthew's answer is now available on Deligne's webpage:

http://publications.ias.edu/sites/default/files/Lettre-a-Quillen.pdf

Answer 3

I want to mention that there is a published version of Deligne's letter online, namely the famous paper "Le déterminant de la cohomologie". See here:

https://publications.ias.edu/node/404

I am trying to type and translate the paper on my own, but so far I have only translated bits of it here and there as it is lengthy. I heard from Soule that the exponent $12$, is ultimately coming from the second term contribution of the expansion of the Todd class.

In light of Brian Conrad's comment ("the oracle status of the Deligne reference is an obstruction to saying more"), I want to point out a possible mistake in Deligne's paper. Here I quote from Gerard Freixas i Montplet and Anna von Pippich's paper (see footnote of page 2):

" To be rigorous, Deligne mistakenly applies Bismut–Freed’s curvature theorem for Quillen connections. The subtle point is the compatibility of the Quillen connection with the holomorphic structure of the Knudsen–Mumford determinant. Nevertheless, the Riemann–Roch isometry he claims can be established by appealing instead to the results of Bismut–Gillet–Soule, where this compatibility is addressed. Notice however these results came later in time."

I am not sure if the issue was noticed by other people, but this may be a subtle point worth mentioning as the paper is a canonical reference in Arakelov intersection theory.