NOTES ON RESIDUES

MARKUS ROST

The purpose of this text is to define the residue map for differentials on a smooth curve. The method described here was inspired by the definition of the Grothendieck residues in [2]. For other approaches see [3, 4, 1, 5].

Remark. In the current version of this text, we consider just smooth rational points on a curve. It is not so difficult to reduce the invariance of the residue at arbitrary smooth closed points on a curve over k to that of k-rational points, by some ad hoc arguments. However it would be desirable to extend the material of this section to arbitrary smooth closed point from the very beginning. Moreover one should give a proof of (7) (see the next Lemma) in a less ad hoc fashion.

Let k be a field and let R be the local ring of a curve over k at a smooth k-rational point. The maximal ideal of R is denoted by 𝔪.

Fix n ≥ 0 and let I = 𝔪^{n}.

Let

Δ: R → R/I ⊗_{k} R
Δ(x) = x ⊗ 1 − 1 ⊗ x

One has

Δ(xy) = (x ⊗ 1)Δ(y) + Δ(x)(1 ⊗ y)

for x, y ∈ R.

Let

μ̃: R/I ⊗_{k} R → R/I

be the multiplication map. Obviously Δ(x) ∈ ker μ̃ for x ∈ R. If t is a prime element of R (i. e., 𝔪 = tR), then the ideal ker μ̃ is generated by Δ(t). Since Δ(t) is a not a zero divisor, the quotient

\frac{Δ(x)}{Δ(t)} ∈ R/I ⊗_{k} R

is defined.

We define

Δ_{t}(x) = \frac{Δ(x)}{Δ(t)} + R/I ⊗_{k} I ∈ R/I ⊗_{k} R/I

Date: July 19, 2001.

These “difference quotients”, defined for x ∈ R and prime elements t, have the following basic properties. Let t, t′ be prime elements of R and let x, y ∈ R. Then

(1)  Δ_{t′}(x) = Δ_{t}(x)Δ_{t′}(t)
(2)  Δ_{t}(t^{r}) = ∑_{i=0}^{r−1}(t^{i} ⊗ t^{r−1−i})
(3)  Δ_{t}(xy) = (x ⊗ 1)Δ_{t}(y) + (1 ⊗ y)Δ_{t}(x)

Further, let z, z′ ∈ I. From (3) we get

(4)  Δ_{t}(xz) = (x ⊗ 1)Δ_{t}(z) = (1 ⊗ x)Δ_{t}(z)
(5)  Δ_{t}(zz′) = 0

Let

μ: R/I ⊗_{k} R/I → R/I

be the multiplication map. The ideal ker μ is generated by the elements x⊗1−1⊗x with x ∈ R. From (4) we get for z ∈ I:

(6)  (ker μ)Δ_{t}(z) = 0

The annihilator of the ideal ker μ consists exactly of the elements

Δ_{t}(z) (z ∈ I)

(we don’t need this in the following).

Let Ω_{R} be the module of differentials of R/k with derivation

d: R → Ω_{R}
x ↦ dx

The R-module Ω_{R} is free with basis dt for any prime element t of R. We denote by d_{I}x the image of dx in Ω_{R}/IΩ_{R}.

Lemma. For x ∈ R and a prime element t of R one has

(7)  d_{I}x = μ(Δ_{t}(x))d_{I}t

Proof. This holds for x = t^{r} since dt^{r} = rt^{r−1}dt and by (2). Moreover it holds for x ∈ I^{2} since d_{I}I^{2} = 0 and by (5). □

Proposition. There exist a (R/I ⊗_{k} R/I)-linear homomorphism

φ: I/I^{2} → R/I ⊗_{k} Ω_{R}/IΩ_{R}

with

φ(z + I^{2}) = Δ_{t}(z)(1 ⊗ d_{I}t)

for z ∈ I and every prime element t of R. Here we consider I/I^{2} as an (R/I ⊗_{k} R/I)-module via μ.

Proof. We first show that

Δ_{t}(z)(1 ⊗ d_{I}t)

with z ∈ I does not depend on the choice of t. Indeed, let t, t′ be prime elements of R. The elements

Δ_{t′}(t), 1 ⊗ μ(Δ_{t′}(t))

NOTES ON RESIDUES 3

have the same image under μ. Using (1), (6), and (7) one finds Δ_{t′}(z)(1⊗d_{I}t′) = Δ_{t}(z)Δ_{t′}(t)(1⊗d_{I}t′) = Δ_{t}(z)(1⊗μ(Δ_{t′}(t))d_{I}t′) = Δ_{t}(z)(1⊗d_{I}t)

The proposition follows now from (4) and (5). □ Let F be the fraction field of R and let Ω_{F} = F ⊗_{R} Ω_{R}. Let z be an element

of R with I = zR and let

\frac{1}{z}: Ω_{R}/IΩ_{R} → Ω_{F} /Ω_{R}

ω + IΩ_{R} ↦ \frac{ω}{z} + Ω_{R}

We define Φ_{I}: Hom_{k}(R/I, k) → Ω_{F} /Ω_{R}

Φ_{I}(α) = (α ⊗ \frac{1}{z})(φ(z + I^{2}))

The proposition shows that Φ_{I} is R-linear and does not depend on the choice of z (with I = zR). Lemma. Let t be a prime element of R. Then

(8)  Φ_{I}(α) = ∑_{i=0}^{n−1} α(t^{i})\frac{dt}{t^{i+1}} + Ω_{R}

Proof. Choosing z = t^{n} we get

Φ_{I}(α) = (α ⊗ \frac{1}{t^{n}})(∑_{i=0}^{n−1}(t^{i} ⊗ t^{n−1−i})(1 ⊗ d_{I}t))

= \frac{1}{t^{n}} ∑_{i=0}^{n−1} α(t^{i})t^{n−1−i}dt + Ω_{R}

and the claim follows. □

Let

Hom_{k}^{c}(R, k) = \underrightarrow{lim}_{n} Hom_{k}(R/𝔪^{n}, k) ⊂ Hom_{k}(R, k)

be the “continuous k-dual” of R. By (8), the maps Φ_{I} for various I paste together and we get (using that t^{−i}, i > 0 is a k-basis of F/R): Local Residue Theorem. There exist a R-linear bijection

Φ: Hom_{k}^{c}(R, k) → Ω_{F} /Ω_{R}

with

(9)  Φ(α) = ∑_{i=0}^{∞} α(t^{i})\frac{dt}{t^{i+1}} + Ω_{R}

for every prime element t of R. □

One may call Φ the “local residue isomorphism”, cf. Proposition 7.2 and Proposition 8.4 in [2, Chapter III]. The residue map will now be defined as

Res: Ω_{F} → k

Res(ω) = (Φ^{−1}(ω + Ω_{R}))(1)

The map Res has indeed the characteristic properties

Res(Ω_{R}) = 0, Res(\frac{dt}{t^{n}}) = \begin{cases}1 & if n = 1 \\ 0 & if n > 1 \end{cases}

of the residue map (the second computation follows from (9)).

References

[1] P. Cartier, Une nouvelle opération sur les formes différentielles, C. R. Acad. Sci. Paris 244 (1957), 426–428.

[2] R. Hartshorne, Residues and duality, Lecture Notes in Mathematics, vol. 20, Springer-Verlag, Berlin, 1966.

[3] J.-P. Serre, Groupes algébriques et corps de classes, Publications de l’institut de mathématique de l’université de Nancago, VII. Hermann, Paris, 1959.

[4] , Algebraic groups and class fields, Springer-Verlag, New York, 1988, Translated from the French.

[5] J. Tate, Residues of differentials on curves, Ann. Sci. École Norm. Sup. (4) 1 (1968), 149–159.

Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA

   E-mail address: rost@math.ohio-state.edu
   URL: http://www.math.ohio-state.edu/~rost