Ulrich Ideals and Modules Over Two-Dimensional Rational Singularities

Abstract

The main aim of this paper is to classify Ulrich ideals and Ulrich modules over two-dimensional Gorenstein rational singularities (rational double points) from a geometric point of view. To achieve this purpose, we introduce the notion of (weakly) special Cohen-Macaulay modules with respect to ideals, and study the relationship between those modules and Ulrich modules with respect to good ideals.

Residually faithful modules and the Cohen–Macaulay type of idealizations

“…Ulrich modules play an important role in the representation theory of local and graded algebras. See [9,10] for a generalization of Ulrich modules, which later we shall be back to. Here, let us note that a MCM R-module M is an Ulrich R-module with respect to m if and only if mM = qM for some (hence, every) minimal reduction q of m, provided the residue class field R/m of R is infinite (see, e.g., [13,Proposition 2.2]).…”

“…Notice that when dim R = 1, every Ulrich ideal of R is an Ulrich R-module with respect itself. Ulrich modules and ideals are closely explored by [6,9,10,14], and it is known that they enjoy very specific properties. For instance, the syzygy modules Ω i R (R/I) (i ≥ d) for an Ulrich ideal I are Ulrich R-modules with respect to I.…”

“…In Section 4, we are concentrated in the latter case where r(R ⋉ M) = r(R) + r R (M), which is closely related to the theory of Ulrich modules ( [2,9,10,14]). In fact, the equality r(R ⋉ M) = r(R) + r R (M) is equivalent to saying that (q : R m)M = qM for some (and hence every) parameter ideal q of R, so that all the Ulrich modules and all the syzygy modules Ω i R (R/m) (i ≥ d) satisfy the above equality r(R ⋉ M) = r(R) + r R (M) (Theorems 4.1, 4.3), provided R is not a regular local ring (here Ω i R (R/m) is considered in a minimal free resolution of R/m).…”

“…

The Cohen-Macaulay type of idealizations of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings is explored. There are two extremal cases, one of which is closely related to the theory of Ulrich modules [2,9,10,14], and the other one is closely related to the theory of residually faithful modules and the theory of closed ideals [3].

…”

Ulrich ideals and 2-AGL rings

“…The maximal ideal of a Cohen-Macaulay local ring with minimal multiplicity is a typical example of Ulrich ideals, and the higher syzygy modules of Ulrich ideals are Ulrich modules. In [17,18], all the Ulrich ideals of Gorenstein local rings of finite CM-representation type and of dimension at most 2 are determined, by means of the classification in the representation theory. On the other hand, in [21], the first author, R. Takahashi, and the third author studied the structure of the complex RHom R (R/I, R) for Ulrich ideals I in a Cohen-Macaulay local ring R of arbitrary dimension, and proved that in a one-dimensional non-Gorenstein AGL ring (R, m), the only possible Ulrich ideal is the maximal ideal m ([21, Theorem 2.14 (1)]).…”