Conductor (ring theory)

In ring theory, a branch of mathematics, the conductor is a measurement of how far apart a commutative ring and an extension ring are. Most often, the larger ring is a domain integrally closed in its field of fractions, and then the conductor measures the failure of the smaller ring to be integrally closed.
The conductor is of great importance in the study of non-maximal orders in the ring of integers of an algebraic number field. One interpretation of the conductor is that it measures the failure of unique factorization into prime ideals.
Definition.
Let "A" and "B" be commutative rings, and assume . The conductor of "A" in "B" is the ideal
Here is viewed as a quotient of "A"-modules, and denotes the annihilator. More concretely, the conductor is the set
Because the conductor is defined as an annihilator, it is an ideal of "A".
If "B" is an integral domain, then the conductor may be rewritten as
where formula_4 is considered as a subset of the fraction field of "B". That is, if "a" is non-zero and in the conductor, then every element of "B" may be written as a fraction whose numerator is in "A" and whose denominator is "a". Therefore the non-zero elements of the conductor are those that suffice as common denominators when writing elements of "B" as quotients of elements of "A".
Suppose "R" is a ring containing "B". For example, "R" might equal "B", or "B" might be a domain and "R" its field of fractions. Then, because , the conductor is also equal to
Elementary properties.
The conductor is the whole ring "A" if and only if it contains and, therefore, if and only if . Otherwise, the conductor is a proper ideal of "A".
If the index is finite, then , so formula_6. In this case, the conductor is non-zero. This applies in particular when "B" is the ring of integers in an algebraic number field and "A" is an order (a subring for which is finite).
The conductor is also an ideal of "B", because, for any and any formula_7, . In fact, an ideal "J" of "B" is contained in "A" if and only if "J" is contained in the conductor. Indeed, for such a "J", , so by definition "J" is contained in formula_8. Conversely, the conductor is an ideal of "A", so any ideal contained in it is contained in "A". This fact implies that formula_8 is the largest ideal of "A" which is also an ideal of "B". (It can happen that there are ideals of "A" contained in the conductor which are not ideals of "B".)
Suppose that "S" is a multiplicative subset of "A". Then
with equality in the case that "B" is a finitely generated "A"-module.
Conductors of Dedekind domains.
Some of the most important applications of the conductor arise when "B" is a Dedekind domain and is finite. For example, "B" can be the ring of integers of a number field and "A" a non-maximal order. Or, "B" can be the affine coordinate ring of a smooth projective curve over a finite field and "A" the affine coordinate ring of a singular model. The ring "A" does not have unique factorization into prime ideals, and the failure of unique factorization is measured by the conductor formula_8.
Ideals coprime to the conductor share many of pleasant properties of ideals in Dedekind domains. Furthermore, for these ideals there is a tight correspondence between ideals of "B" and ideals of "A":
All of these properties fail in general for ideals not coprime to the conductor. To see some of the difficulties that may arise, assume that "J" is a non-zero ideal of both "A" and "B" (in particular, it is contained in, hence not coprime to, the conductor). Then "J" cannot be an invertible fractional ideal of "A" unless . Because "B" is a Dedekind domain, "J" is invertible in "B", and therefore
since we may multiply both sides of the equation by "J"−1. If "J" is also invertible in "A", then the same reasoning applies. But the left-hand side of the above equation makes no reference to "A" or "B", only to their shared fraction field, and therefore . Therefore being an ideal of both "A" and "B" implies non-invertibility in "A".
Conductors of quadratic number fields.
Let "K" be a quadratic extension of Q, and let be its ring of integers. By extending to a Z-basis, we see that every order "O" in "K" has the form for some positive integer "c". The conductor of this order equals the ideal "cO""K". Indeed, it is clear that "cO""K" is an ideal of "O""K" contained in "O", so it is contained in the conductor. On the other hand, the ideals of "O" containing "cO""K" are the same as ideals of the quotient ring . The latter ring is isomorphic to by the second isomorphism theorem, so all such ideals of "O" are the sum of "cO""K" with an ideal of Z. Under this isomorphism, the conductor annihilates , so it must be .
In this case, the index is also equal to "c", so for orders of quadratic number fields, the index may be identified with the conductor. This identification fails for higher degree number fields.