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$\def\sO{\mathcal{O}} \def\frm{\mathfrak{m}} $If $(R,\frm)$ is a local ring, the field $R/\frm$ is called the residue field of $R$. More generally, let $(X,\sO_X)$ be a locally ringed space. The residue field of $X$ at $x$, denoted $\kappa(x)$, is the residue field of the local ring $\sO_{X,x}$. My question is: why is it called like that?

I guess this question may be answered in two possible directions:

  1. Who came up with the name in the first place? Do we know their reasons for this choice of terminology?

  2. What is a possible motivation that justifies why this name is a good fit for the idea behind the residue field?

In the particular cases where the locally ringed space $(X,\sO_X)$ is a smooth manifold, a complex manifold or a $k$-prevariety along with the corresponding structure sheaf of functions that are, respectively, real-valued smooth, complex-valued holomorphic, or $k$-valued regular, the homomorphisms $\sO_{X,x}\to\kappa(x)$ and $\sO(U)\to\kappa(x)$ (with $U\subset X$ an open neighborhood of $x$) are exactly evaluation at $x$. In this situation, I don't see what exactly the "residue" is. Neither I do in the case when $(X,\sO_X)$ is a variety (in the sense of schemes). The only residue field I know how to interpret in this case is the one at the generic point: It is the function field of the variety, but again I don't know what does this have to do with residues.

Elías Guisado Villalgordo
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  • This question has some discussion of the meaning of "residue field", but it's not evident whether it addresses the 'residue' terminology specifically. Myself I wonder if it might owe something to residues in complex analysis. – Semiclassical Feb 17 '23 at 17:53
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    I think you're thinking of residues from complex analysis. I suspect this comes from number theory. Integers modulo a prime $p$ is called the residue and gives the residue field $Z/pZ$. – Mohan Feb 17 '23 at 18:06
  • Residue, residual, reduction, remainder, this seems like random choices in a single lexical field. – reuns Feb 17 '23 at 18:46

1 Answers1

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According to Oxford Languages residue means

"a small amount of something that remains after the main part has gone or been taken or used."

Among the synonims they give remainder.

When you mod out $\mathbb{Z}$ by the ideal $N\mathbb{Z}$ the elements in the quotient are just the (classes of the) remainders of division by $N$.

By analogy, you can say that the elements if $R/{\frak m}$ are what is left from elements of $R$ "after the part in $\frak m$ has been removed" and this should be the justification for the name of residue field (or ring)

Andrea Mori
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