2014/01/13 by Hailong Dao, Dao, Hailong, Eleonore Faber +3
Mathematics · #13C14 #14A22 #14B05 #14E15 #16E10 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AC #math.AG #math.RA #msc:13C14 #msc:14A22 #msc:14B05 #msc:14E15 #msc:16E10
paper · pdf · doi:10.48550/arxiv.1401.3000
v3: minor revision, final version to appear in Algebr. Represent. Theory; v2: Major revision, especially in Section 2. We thank Michael Wemyss for very helpful comments on the first version, which alerted us to many subtleties in the literature and prompted our revisions. The definitions of NC(C)Rs have been clarified and compared in details to the existing ones. 31 pages
arxiv created 2014/12/02 · arxiv updated 2014/12/04
In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but subsumes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introduce and study the global spectrum of a ring R, that is, the set of all possible finite global dimensions of endomorphism rings of MCM R-modules. Finally, we use a variety of methods to compute global dimension for many endomorphism rings.