2019/03/03 by Michael Hellus, Hellus, Michael
Mathematics · #13C40 (Secondary) #13D45 (Primary) 14M10 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1903.00874
openalex publication_date 2019/03/03 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
It is a well-known result that, in projective space over a field, every\nset-theoretical complete intersection of positive dimension in connected in\ncodimension one (Hartshorne [H1,3.4.6] or [H2, Theorem 1.3]). Another important\nconnectedness result is that a local ring with disconnected punctured sprectrum\nhas depth at most 1 ([H1, Proposition 2.1]). The two results are related,\nHartshorne calls the latter "the keystone to the proof" of the former (loc.\ncit).\n In this short note we show that the latter result generalizes smoothly from\nset-theoretical to cohomologically complete intersections, i. e. to ideals for\nwhich there is in terms of local cohomology no obstruction to be a complete\nintersection ([HeSc1], [HeSc2]). The proof is based on the fact that, for\ncohomologically complete intersections over a complete local ring, the\nendomorphism ring of the (only) local cohomology cohomology module is the ring\nitself ([HeSt, Theorem 2.2 (iii)]) and hence indecomposable as a module.\n