Multiplicative closure operations on ring extensions

Asymptotic for the number of star operations on one-dimensional Noetherian domains

“…The two quadratic functions that bound lim sup q→∞ θ(s, q) are rather close to the quadratic functions that give the limit of θ((n, 1), q) in Proposition 3.8: for example, if n is even, then the difference between the upper bound and the limit of the case (n, 1) is just n(n−2) The definition given above is slightly different from the definition given in [19,Section 4] since we are imposing that * I also closes k (as we want * I to be a star operation and not only a multiplicative operation). However, the two definitions are actually very close.…”

“…To unify the treatment of these classes of closure operations, the paper [19] introduced the concept of multiplicative operations: these are a class of closure operations that can be defined in any ring extension A ⊆ B over any set G of A-submodules of B, and their definition is flexible enough to cover (for suitable choices of A, B and G) all three classical cases. Furthermore, multiplicative operations enjoy some functorial properties: while these are usually a staple of many semiprime operations, for star and semistar operations they are very rare, especially due to the fact that quotienting an integral domain disrupts its quotient field.…”

“…(There are some limited exceptions: see [3] for properties of star operations along pullbacks and [20] for an application to Prüfer domains.) In the special case of local Noetherian domains of dimension 1, multiplicative operations allow to bypass this problem by considering only the submodules contained in the integral closure T of the starting domain D (which contain all the needed data), and then by quotienting T over the conductor (D : T ) (see [19,Section 6] and the discussion after Definition 3.4 below). In particular, this reduces the problem of finding all star operations on D to the study of the Artinian ring T /(D : T ).…”

“…In this paper, we continue the study initiated in Sections 6 and 7 of [19] on this case, in particular concentrating on the case where the conductor (D : T ) is equal to the maximal ideal m D of D; in the Artinian setting, this case correspond to the study of a particular set of multiplicative operations defined on a finite extension k ⊆ B, where k is a field. As we are interested in cardinality problems, we assume throughout the paper that k is a finite field of cardinality q: in particular, we are interested in what happens when the structure of B is "fixed" (see Section 3 for a more precise definition) while q changes, that is, we are interested in the cardinality of the set Star(D) of star operations on D as a function of q, and especially in understanding how fast the growth of |Star(D)| is.…”

“…Proof. Let v(I) be the multiplicative operation generated by I on (k, B, F 1 (k, B)), as in [19,Definition 4.2]. Then, * I is just the infimum of v(k) and v(I), and thus Hence, we need a stronger situation.…”