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Ulrich ideals and modules over two-dimensional rational singularities. (English) Zbl 1360.13028

In [“Ulrich ideals and modules,” to appear in Math. Proc. Cambridge Philos. Soc., arXiv:1206.3197], the authors established the general theory of Ulrich ideals and modules. While the classical version (maximally generated maximal Cohen-Macaulay modules) was introduced by B. Ulrich [Math. Z. 188, 23–32 (1984; Zbl 0573.13013)] and by Brennan-Herzog-Ulrich [J. P. Brennan et al., Math. Scand. 61, No. 2, 181–203 (1987; Zbl 0653.13015)], the notion of Ulrich modules is more general and describes a larger set. The broad goal is to classify all Ulrich ideals and modules. In the earlier paper the authors achieved this in the case of a one-dimensional Gorenstein local ring of finite CM-representation type. The main result of this paper is a classification of Ulrich ideals and Ulrich modules over two-dimensional Gorenstein rational singularities from a geometric point of view. To achieve this, the authors introduce and use the notion of weakly special Cohen-Macaulay modules. In the last section they consider two-dimensional non-Gorenstein rational singularities.

MSC:

13C14 Cohen-Macaulay modules
14B05 Singularities in algebraic geometry
14C25 Algebraic cycles
14E16 McKay correspondence
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