1 Ramification in number fields
Definition 1 (First definition).
Likewise, if is a nonzero prime ideal in , and , then we say ramifies over if the ramification index of in the factorization of the ideal is greater than 1. That is, a prime in ramifies in if at least one prime dividing ramifies over . If is a Galois extension, then the ramification indices of all the primes dividing 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 in terms of local rings. With as before, let be a prime in with . Then the induced map of localizations is a local homomorphism of local rings (in fact, of discrete valuation rings), and the ramification index of over is the unique natural number such that
An astute reader may notice that this formulation of ramification index does not require that and be number fields, or even that they play any role at all. We take advantage of this fact here to give a second, more general definition.
Definition 2 (Second definition).
Let be any ring homomorphism. Suppose is a prime ideal such that the localization of at is a discrete valuation ring. Let be the prime ideal , so that induces a local homomorphism . Then the ramification index is defined to be the unique natural number such that
or if .
The reader who is not interested in local rings may assume that and are unique factorization domains, in which case is the exponent of in the factorization of the ideal , just as in our first definition (but without the requirement that the rings and originate from number fields).
There is of course much more that can be said about ramification indices even in this purely algebraic setting, but we limit ourselves to the following remarks:
2 Ramification in algebraic geometry
The word “ramify” in English means “to divide into two or more branches,” and we will show in this section that the mathematical term lives up to its common English meaning.
Definition 3 (Algebraic version).
Let be a non–constant regular morphism of curves (by which we mean one dimensional nonsingular irreducible algebraic varieties) over an algebraically closed field . Then has a nonzero degree , which can be defined in any of the following ways:
There is a finite set of points for which the inverse image does not have size , and we call these points the branch points or ramification points of . If with , then the ramification index of at is the ramification index obtained algebraically from Definition 2 by taking
, the local ring consisting of all rational functions in the function field which are regular at .
, the maximal ideal in consisting of all functions which vanish at .
, the maximal ideal in consisting of all functions which vanish at .
, the map on the function fields induced by the morphism .
The picture in Figure 1 may be worth a thousand words. Let and . Take the map given by . Then is plainly a map of degree 2, and every point in except for 0 has two preimages in . The point 0 is thus a ramification point of of index 2, and we have drawn the graph of near .
Note that we have only drawn the real locus of because that is all that can fit into two dimensions. We see from the figure that a typical point on such as the point has two points in which map to it, but that the point has only one corresponding point of which “branches” or “ramifies” into two distinct points of whenever one moves away from 0.
2.1 Relation to the number field case
The relationship between Definition 2 and Definition 3 is easiest to explain in the case where is a map between affine varieties. When and are affine, then their coordinate rings and are Dedekind domains, and the points of the curve (respectively, ) correspond naturally with the maximal ideals of the ring (respectively, ). The ramification points of the curve are then exactly the points of which correspond to maximal ideals of that ramify in the algebraic sense, with respect to the map of coordinate rings.
Equation (2) in this case says
Let be given by as in Example 4. Since is just the affine line, the coordinate ring is equal to , the polynomial ring in one variable over . Likewise, , and the induced map is naturally given by . We may accordingly identify the coordinate ring with the subring of .
Note that the two prime ideals and of are equal only when , so we see that the ideal in , corresponding to the point , ramifies in exactly when . We have therefore recovered our previous geometric characterization of the ramified points of , solely in terms of the algebraic factorizations of ideals in .
In the case where is a map between projective varieties, Definition 2 does not directly apply to the coordinate rings of and , but only to those of open covers of and by affine varieties. Thus we do have an instance of yet another new phenomenon here, and rather than keep the reader in suspense we jump straight to the final, most general definition of ramification that we will give.
Definition 6 (Final form).
Let be a morphism of locally ringed spaces. Let and suppose that the stalk is a discrete valuation ring. Write for the induced map of on stalks at . Then the ramification index of over is the unique natural number , if it exists (or if it does not exist), such that
where and are the respective maximal ideals of and . We say is ramified in if .
For any morphism of varieties , there is an induced morphism on the structure sheaves of and , which are locally ringed spaces. If and are curves, then the stalks are one dimensional regular local rings and therefore discrete valuation rings, so in this way we recover the algebraic geometric definition (Definition 3) from the sheaf definition (Definition 6).
3 Ramification in complex analysis
Definition 9 (Analytic version).
Let be a holomorphic map of Riemann surfaces. For any , there exists local coordinate charts and around and such that is locally the map from to . The natural number is called the ramification index of at , and is said to be a branch point or ramification point of if .
Take the map , of Example 4. We study the behavior of near the unramified point and near the ramified point . Near , take the coordinate on the domain and on the range. Then maps to , which in the coordinate is . If we change coordinates to on the domain, keeping on the range, then , so the ramification index of at is equal to 1.
Near , the function is already in the form with , so the ramification index of at is equal to 2.
3.1 Algebraic–analytic correspondence
Of course, the analytic notion of ramification given in Definition 9 can be couched in terms of locally ringed spaces as well. Any Riemann surface together with its sheaf of holomorphic functions is a locally ringed space. Furthermore the stalk at any point is always a discrete valuation ring, because germs of holomorphic functions have Taylor expansions making the stalk isomorphic to the power series ring . We can therefore apply Definition 6 to any holomorphic map of Riemann surfaces, and it is not surprising that this process yields the same results as Definition 9.
More generally, every map of algebraic varieties can be interpreted as a holomorphic map of Riemann surfaces in the usual way, and the ramification points on and under as algebraic varieties are identical to their ramification points as Riemann surfaces. It turns out that the analytic structure may be regarded in a certain sense as the “completion” of the algebraic structure, and in this sense the algebraic–analytic correspondence between the ramification points may be regarded as the geometric version of the equality (1) in number theory.
The algebraic–analytic correspondence of ramification points is itself only one manifestation of the wide ranging identification between algebraic geometry and analytic geometry which is explained to great effect in the seminal paper of Serre .
- 1 Robin Hartshorne, Algebraic Geometry, Springer–Verlag, 1977 (GTM 52).
- 2 Gerald Janusz, Algebraic Number Fields, Second Edition, 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 et géométrie analytique,” Ann. de L’Inst. Fourier 6 pp. 1–42, 1955–56.
- 7 Joseph Silverman, The Arithmetic of Elliptic Curves, Springer–Verlag, 1986 (GTM 106).
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