ramification index

1 Ramification in number fields

Definition 1 (First definition).

Let $L/K$ be an extension of number fields. Let ${\\mathfrak{p}}$ be a nonzeroprime ideal in the ring of integers $\\mathcal{O}_{K}$ of $K$ , and suppose theideal ${\\mathfrak{p}}\\mathcal{O}_{L}\\subset\\mathcal{O}_{L}$ factors as

{\\mathfrak{p}}\\mathcal{O}_{L}=\\prod_{i=1}^{n}{\\mathfrak{P}}_{i}^{e_{i}}

for some prime ideals ${\\mathfrak{P}}_{i}\\subset\\mathcal{O}_{L}$ and exponents $e_{i}\\in\\mathbb{N}$ . The natural number $e_{i}$ is called the ramification indexof ${\\mathfrak{P}}_{i}$ over ${\\mathfrak{p}}$ . It is often denoted $e({\\mathfrak{P}}_{i}/{\\mathfrak{p}})$ . If $e_{i}>1$ for any $i$ , then we say the ideal ${\\mathfrak{p}}$ ramifies in $L$ .

Likewise, if ${\\mathfrak{P}}$ is a nonzero prime ideal in $\\mathcal{O}_{L}$ , and ${\\mathfrak{p}}:={\\mathfrak{P}}\\cap\\mathcal{O}_{K}$ , then we say ${\\mathfrak{P}}$ ramifies over $K$ if theramification index $e({\\mathfrak{P}}/{\\mathfrak{p}})$ of ${\\mathfrak{P}}$ in the factorization of theideal ${\\mathfrak{p}}\\mathcal{O}_{L}\\subset\\mathcal{O}_{L}$ is greater than 1. That is, a prime ${\\mathfrak{p}}$ in $\\mathcal{O}_{K}$ ramifies in $L$ if at least one prime ${\\mathfrak{P}}$ dividing ${\\mathfrak{p}}\\mathcal{O}_{L}$ ramifies over $K$ . If $L/K$ is a Galois extension, then theramification indices of all the primes dividing ${\\mathfrak{p}}\\mathcal{O}_{L}$ are equal,since the Galois group is transitive on this set of primes.

1.1 The local view

The phenomenon of ramification has an equivalent interpretation interms of local rings. With $L/K$ as before, let ${\\mathfrak{P}}$ be a prime in $\\mathcal{O}_{L}$ with ${\\mathfrak{p}}:={\\mathfrak{P}}\\cap\\mathcal{O}_{K}$ . Then the induced map oflocalizations $(\\mathcal{O}_{K})_{\\mathfrak{p}}\\hookrightarrow(\\mathcal{O}_{L})_{\\mathfrak{P}}$ is a localhomomorphism of local rings (in fact, of discrete valuation rings),and the ramification index of ${\\mathfrak{P}}$ over ${\\mathfrak{p}}$ is the unique naturalnumber $e$ such that

{\\mathfrak{p}}(\\mathcal{O}_{L})_{\\mathfrak{P}}=({\\mathfrak{P}}(\\mathcal{O}_{L}%)_{\\mathfrak{P}})^{e}\\subset(\\mathcal{O}_{L})_{\\mathfrak{P}}.

An astute reader may notice that this formulation of ramificationindex does not require that $L$ and $K$ be number fields, or even thatthey play any role at all. We take advantage of this fact here to givea second, more general definition.

Definition 2 (Second definition).

Let $\\iota:A\\longrightarrow B$ be any ring homomorphism. Suppose ${\\mathfrak{P}}\\subset B$ is a prime ideal such that the localization $B_{\\mathfrak{P}}$ of $B$ at ${\\mathfrak{P}}$ is adiscrete valuation ring. Let ${\\mathfrak{p}}$ be the prime ideal $\\iota^{-1}({\\mathfrak{P}})\\subset A$ , so that $\\iota$ induces a local homomorphism $\\iota_{\\mathfrak{P}}:A_{\\mathfrak{p}}\\longrightarrow B_{\\mathfrak{P}}$ . Then the ramification index $e({\\mathfrak{P}}/{\\mathfrak{p}})$ isdefined to be the unique natural number such that

\\iota({\\mathfrak{p}})B_{\\mathfrak{P}}=({\\mathfrak{P}}B_{\\mathfrak{P}})^{e({%\\mathfrak{P}}/{\\mathfrak{p}})}\\subset B_{\\mathfrak{P}},

or $\\infty$ if $\\iota({\\mathfrak{p}})B_{\\mathfrak{P}}=(0)$ .

The reader who is not interested in local rings may assume that $A$ and $B$ are unique factorization domains, in which case $e({\\mathfrak{P}}/{\\mathfrak{p}})$ isthe exponent of ${\\mathfrak{P}}$ in the factorization of the ideal $\\iota({\\mathfrak{p}})B$ ,just as in our first definition (but without the requirement that therings $A$ and $B$ originate from number fields).

There is of course much more that can be said about ramificationindices even in this purely algebraic setting, but we limit ourselvesto the following remarks:

1.
Suppose $A$ and $B$ are themselves discrete valuation rings,with respective maximal ideals ${\\mathfrak{p}}$ and ${\\mathfrak{P}}$ . Let $\\hat{A}:=\\,\\underset{\\longleftarrow}{\\lim}\\,A/{\\mathfrak{p}}^{n}$ and $\\hat{B}:=\\,\\underset{\\longleftarrow}{\\lim}\\,B/{\\mathfrak{P}}^{n}$ be the completions of $A$ and $B$ with respect to ${\\mathfrak{p}}$ and ${\\mathfrak{P}}$ . Then
$e({\\mathfrak{P}}/{\\mathfrak{p}})=e({\\mathfrak{P}}\\hat{B}/{\\mathfrak{p}}\\hat{A}).$ (1)
In other words, the ramification index of ${\\mathfrak{P}}$ over ${\\mathfrak{p}}$ in the $A$ –algebra $B$ equals the ramification index in the completions of $A$ and $B$ with respect to ${\\mathfrak{p}}$ and ${\\mathfrak{P}}$ .
2.
Suppose $A$ and $B$ are Dedekind domains, with respectivefraction fields $K$ and $L$ . If $B$ equals the integral closure of $A$ in $L$ , then
$\\sum_{{\\mathfrak{P}}\\mid{\\mathfrak{p}}}e({\\mathfrak{P}}/{\\mathfrak{p}})f({%\\mathfrak{P}}/{\\mathfrak{p}})\\leq[L:K],$ (2)
where ${\\mathfrak{P}}$ ranges over all prime ideals in $B$ that divide ${\\mathfrak{p}}B$ , and $f({\\mathfrak{P}}/{\\mathfrak{p}}):=\\dim_{A/{\\mathfrak{p}}}(B/{\\mathfrak{P}})$ is the inertial degree of ${\\mathfrak{P}}$ over ${\\mathfrak{p}}$ . Equality holds in Equation (2) whenever $B$ isfinitely generated as an $A$ –module.

2 Ramification in algebraic geometry

The word “ramify” in English means “to divide into two or morebranches,” and we will show in this section that the mathematical termlives up to its common English meaning.

Definition 3 (Algebraic version).

Let $f:C_{1}\\longrightarrow C_{2}$ be a non–constant regular morphism of curves(by which we mean one dimensional nonsingular irreducible algebraicvarieties) over an algebraically closed field $k$ . Then $f$ has anonzero degree $n:=\\deg f$ , which can be defined in any of thefollowing ways:

•
The number of points in a generic fiber $f^{-1}(p)$ , for $p\\in C_{2}$
•
The maximum number of points in $f^{-1}(p)$ , for $p\\in C_{2}$
•
The degree of the extension $k(C_{1})/f^{*}k(C_{2})$ of functionfields

There is a finite set of points $p\\in C_{2}$ for which the inverseimage $f^{-1}(p)$ does not have size $n$ , and we call these points thebranch points or ramification points of $f$ . If $P\\in C_{1}$ with $f(P)=p$ , then the ramification index $e(P/p)$ of $f$ at $P$ is the ramification index obtained algebraically fromDefinition 2 by taking

•
$A=k[C_{2}]_{p}$ , the local ring consisting of all rationalfunctions in the function field $k(C_{2})$ which are regular at $p$ .
•
$B=k[C_{1}]_{P}$ , the local ring consisting of all rationalfunctions in the function field $k(C_{1})$ which are regular at $P$ .
•
${\\mathfrak{p}}={\\mathfrak{m}}_{p}$ , the maximal ideal in $A$ consisting of allfunctions which vanish at $p$ .
•
${\\mathfrak{P}}={\\mathfrak{m}}_{P}$ , the maximal ideal in $B$ consisting of allfunctions which vanish at $P$ .
•
$\\iota=f^{*}_{p}:k[C_{2}]_{p}\\hookrightarrow k[C_{1}]_{P}$ , the map onthe function fields induced by the morphism $f$ .

Example 4.

The picture in Figure 1 may be worth a thousand words. Let $k=\\mathbb{C}$ and $C_{1}=C_{2}=\\mathbb{C}=\\mathbb{A}^{1}_{\\mathbb{C}}$ . Take the map $f:\\mathbb{C}\\longrightarrow\\mathbb{C}$ given by $f(y)=y^{2}$ . Then $f$ is plainly a map of degree 2, and every pointin $C_{2}$ except for 0 has two preimages in $C_{1}$ . The point 0 is thusa ramification point of $f$ of index 2, and we have drawn the graph of $f$ near $0$ .

Figure 1: The function $f(y)=y^{2}$ near $y=0$ .

Note that we have only drawn the real locus of $f$ because that is allthat can fit into two dimensions. We see from the figure that atypical point on $C_{2}$ such as the point $x=1$ has two points in $C_{1}$ which map to it, but that the point $x=0$ has only onecorresponding point of $C_{1}$ which “branches” or “ramifies” into twodistinct points of $C_{1}$ whenever one moves away from 0.

2.1 Relation to the number field case

The relationship between Definition 2 andDefinition 3 is easiest to explain in the case where $f$ is a map between affine varieties. When $C_{1}$ and $C_{2}$ are affine,then their coordinate rings $k[C_{1}]$ and $k[C_{2}]$ are Dedekinddomains, and the points of the curve $C_{1}$ (respectively, $C_{2}$ )correspond naturally with the maximal ideals of the ring $k[C_{1}]$ (respectively, $k[C_{2}]$ ). The ramification points of the curve $C_{1}$ are then exactly the points of $C_{1}$ which correspond to maximalideals of $k[C_{1}]$ that ramify in the algebraic sense, with respect tothe map $f^{*}:k[C_{2}]\\longrightarrow k[C_{1}]$ of coordinate rings.

Equation (2) in this case says

\\sum_{P\\in f^{-1}(p)}e(P/p)=n,

and we see that the well known formula (2) in numbertheory is simply the algebraic analogue of the geometric fact that thenumber of points in the fiber of $f$ , counting multiplicities, isalways $n$ .

Example 5.

Let $f:\\mathbb{C}\\longrightarrow\\mathbb{C}$ be given by $f(y)=y^{2}$ as inExample 4. Since $C_{2}$ is just the affine line, thecoordinate ring $\\mathbb{C}[C_{2}]$ is equal to $\\mathbb{C}[X]$ , the polynomial ring inone variable over $\\mathbb{C}$ . Likewise, $\\mathbb{C}[C_{1}]=\\mathbb{C}[Y]$ , and the inducedmap $f^{*}:\\mathbb{C}[X]\\longrightarrow\\mathbb{C}[Y]$ is naturally given by $f^{*}(X)=Y^{2}$ . Wemay accordingly identify the coordinate ring $\\mathbb{C}[C_{2}]$ with thesubring $\\mathbb{C}[X^{2}]$ of $\\mathbb{C}[X]=\\mathbb{C}[C_{1}]$ .

Now, the ring $\\mathbb{C}[X]$ is a principal ideal domain, and the maximalideals in $\\mathbb{C}[X]$ are exactly the principal ideals of the form $(X-a)$ for any $a\\in\\mathbb{C}$ . Hence the nonzero prime ideals in $\\mathbb{C}[X^{2}]$ are of the form $(X^{2}-a)$ , and these factor in $\\mathbb{C}[X]$ as

(X^{2}-a)=(X-\\sqrt{a})(X+\\sqrt{a})\\subset\\mathbb{C}[X].

Note that the two prime ideals $(X-\\sqrt{a})$ and $(X+\\sqrt{a})$ of $\\mathbb{C}[X]$ are equal only when $a=0$ , so we see that the ideal $(X^{2}-a)$ in $\\mathbb{C}[X^{2}]$ , corresponding to the point $a\\in C_{2}$ , ramifiesin $C_{1}$ exactly when $a=0$ . We have therefore recovered ourprevious geometric characterization of the ramified points of $f$ ,solely in terms of the algebraic factorizations of ideals in $\\mathbb{C}[X]$ .

In the case where $f$ is a map between projective varieties,Definition 2 does not directly apply to thecoordinate rings of $C_{1}$ and $C_{2}$ , but only to those of open coversof $C_{1}$ and $C_{2}$ by affine varieties. Thus we do have an instance ofyet another new phenomenon here, and rather than keep the reader insuspense we jump straight to the final, most general definition oframification that we will give.

Definition 6 (Final form).

Let $f:(X,\\mathcal{O}_{X})\\longrightarrow(Y,\\mathcal{O}_{Y})$ be a morphism of locally ringedspaces. Let $p\\in X$ and suppose that the stalk $(\\mathcal{O}_{X})_{p}$ is adiscrete valuation ring. Write $\\phi_{p}:(\\mathcal{O}_{Y})_{f(p)}\\longrightarrow(\\mathcal{O}_{X})_{p}$ for the induced map of $f$ on stalks at $p$ . Then the ramification index of $p$ over $Y$ is the unique natural number $e$ ,if it exists (or $\\infty$ if it does not exist), such that

\\phi_{p}({\\mathfrak{m}}_{f(p)})(\\mathcal{O}_{X})_{p}={\\mathfrak{m}}_{p}^{e},

where ${\\mathfrak{m}}_{p}$ and ${\\mathfrak{m}}_{f(p)}$ are the respective maximal ideals of $(\\mathcal{O}_{X})_{p}$ and $(\\mathcal{O}_{Y})_{f(p)}$ . We say $p$ is ramified in $Y$ if $e>1$ .

Example 7.

A ring homomorphism $\\iota:A\\longrightarrow B$ corresponds functorially to amorphism $\\operatorname{Spec}(B)\\longrightarrow\\operatorname{Spec}(A)$ of locally ringed spaces from theprime spectrum of $B$ to that of $A$ , and the algebraic notion oframification from Definition 2 equals thesheaf–theoretic notion of ramification from Definition 6.

Example 8.

For any morphism of varieties $f:C_{1}\\longrightarrow C_{2}$ , there is an inducedmorphism $f^{\\#}$ on the structure sheaves of $C_{1}$ and $C_{2}$ , which arelocally ringed spaces. If $C_{1}$ and $C_{2}$ are curves, then the stalks are one dimensional regular local rings and therefore discrete valuation rings, so in this way we recover the algebraicgeometric definition (Definition 3) from the sheafdefinition (Definition 6).

3 Ramification in complex analysis

Ramification points or branch points in complex geometry are merely aspecial case of the high–flown terminology ofDefinition 6. However, they are important enough to merit aseparate mention here.

Definition 9 (Analytic version).

Let $f:M\\longrightarrow N$ be a holomorphic map of Riemann surfaces. For any $p\\in M$ , there exists local coordinate charts $U$ and $V$ around $p$ and $f(p)$ such that $f$ is locally the map $z\\mapsto z^{e}$ from $U$ to $V$ . The natural number $e$ is called the ramification indexof $f$ at $p$ , and $p$ is said to be a branch point or ramification point of $f$ if $e>1$ .

Example 10.

Take the map $f:\\mathbb{C}\\longrightarrow\\mathbb{C}$ , $f(y)=y^{2}$ ofExample 4. We study the behavior of $f$ near theunramified point $y=1$ and near the ramified point $y=0$ . Near $y=1$ ,take the coordinate $w=y-1$ on the domain and $v=x-1$ on therange. Then $f$ maps $w+1$ to $(w+1)^{2}$ , which in the $v$ coordinateis $(w+1)^{2}-1=2w+w^{2}$ . If we change coordinates to $z=2w+w^{2}$ on the domain, keeping $v$ on the range, then $f(z)=z$ , so theramification index of $f$ at $y=1$ is equal to 1.

Near $y=0$ , the function $f(y)=y^{2}$ is already in the form $z\\mapsto z^{e}$ with $e=2$ , so the ramification index of $f$ at $y=0$ isequal to 2.

3.1 Algebraic–analytic correspondence

Of course, the analytic notion of ramification given inDefinition 9 can be couched in terms of locally ringedspaces as well. Any Riemann surface together with its sheaf ofholomorphic functions is a locally ringed space. Furthermore the stalkat any point is always a discrete valuation ring, because germs ofholomorphic functions have Taylor expansions making the stalkisomorphic to the power series ring $\\mathbb{C}[[z]]$ . We can therefore applyDefinition 6 to any holomorphic map of Riemann surfaces, andit is not surprising that this process yields the same results asDefinition 9.

More generally, every map of algebraic varieties $f:V\\longrightarrow W$ can beinterpreted as a holomorphic map of Riemann surfaces in the usual way,and the ramification points on $V$ and $W$ under $f$ as algebraicvarieties are identical to their ramification points as Riemannsurfaces. It turns out that the analytic structure may be regarded ina certain sense as the “completion” of the algebraic structure, and inthis sense the algebraic–analytic correspondence between theramification points may be regarded as the geometric version of theequality (1) in number theory.

The algebraic–analytic correspondence of ramification points isitself only one manifestation of the wide ranging identificationbetween algebraic geometry and analytic geometry which is explained togreat effect in the seminal paper of Serre [6].

References

1 Robin Hartshorne, AlgebraicGeometry, Springer–Verlag, 1977 (GTM 52).
2 Gerald Janusz, Algebraic Number Fields, SecondEdition, American Mathematical Society, 1996 (GSM 7).
3 Jürgen Jost, Compact Riemann Surfaces,Springer–Verlag, 1997.
4 Dino Lorenzini, An Invitation to Arithmetic Geometry, American Mathematical Society, 1996 (GSM 9).
5 Jean–Pierre Serre, Local Fields,Springer–Verlag, 1979 (GTM 67).
6 Jean–Pierre Serre, “Géométrie algébraique etgéométrie analytique,” Ann. de L’Inst. Fourier 6 pp. 1–42,1955–56.
7 Joseph Silverman, The Arithmetic of EllipticCurves, Springer–Verlag, 1986 (GTM 106).