On 2-Buchsbaum complexes
Mitsuhiro Miyazaki · Kyoto journal of mathematics · 1990
IntroductionLet K be a field, fixed throughout this p a p e r .By using the Stanley-Reisner ring over K the concept of com m utative algebra such as Cohen-M acaulay or B u c h sb a u m a r e im m e d ia te ly tr a n s f e r r e d to th e c o n c e p t o f s im p lic ia l com plexes.Som e o f th e m su c h as C ohen-M acaulay or Buchsbaum a r e well behaving c o n c e p t a n d it i s k n o w n f o r exam ple Cohen-M acaulayness and B uchsbaum ness are topological properties.(i.e.i f A , a n d 4 2 are sim plicial complexes whose geometric realizations are homeomorphic, th e n A , is Cohen-Macaulay (or Buchsbaum) if and only if A 2 is Cohen-Macaulay (or Buchsbaum resp.)) a n d characterized by th e reduced oriented homology groups o f A .(See [10], [11] and [12].)A nd by the result of Reisner [11] (see Theorem 3.2 of this paper) w e know that if a simplicial complex A is Cohen-Macaulay of dim d > 1 then A is c o n n e c te d .S o one can consider the Cohen-M acaulay property as a specialization o f t h e c o n n e c te d n e s s .B a c la w s k i [1 ] c a lle d t h e Cohen-M acaulayness a s C o h e n -M a c a u la y connectivity a n d d e fin e d the k-C ohen-Macaulayness b y th e sim ilar w ay as the k-connectivity.(See § 1 fo r definition.)Then the 2-Cohen-Macaulayness is a well behaving concept and the following facts are known.(i) 2-Cohen-Macaulayness is a topological property.([17]) (ii) If A is a Cohen-Macaulay complex of dimension r then (r -1)-skeleton of A is 2-Cohen-Macaulay.([6])It is natural to ask if the similar results are valid for Buchsbaum ness.Since a simplicial complex A is Buchsbaum if and only if A is pure and every non-trivial link of A is Cohen-Macaulay (see Theorem 4.1 of this paper), it is also natural to ask if the results similar to (i) and (ii) above are valid for p u re n e ss.The purpose of this paper is to give affirmative answers to these questions.i.e.(i) 2-Buchsbaumness (or 2-pureness) is a topological property.(See Theorems 4.3 and 5.3.)(ii) If A is a Buchsbaum (or pure) complex of dimension r then the (r -1)- skeleton of A is 2-Buchsbaum (or 2-pure resp.).(See Theorems 7.4 and 7.3.)The author w ould like to express his hearty thanks to his advisor Jun-ichi Nishimura for intimate a n d useful conversations.