On Rees algebras over Buchsbaum rings

Castelnuovo's regularity and multiplicity

“…From these sequences we get sup {i + e(Hi(A/xA)) } = sup {sup {i + e(Hi(A)), i+ e(Hi + I(A)( -1))}} i>=s i>s = sup{i+e(Hi(A)) }, q.e.d.i>sFrom the proof of Lemma 1 we obtain the following corollary which strengthens Lemma 2.6 of[9]. Applying the functor Hip(. )…”

Buchsbaumness in Rees Modules Associated to Ideals of Minimal Multiplicity in the Equi-I-Invariant Case

“…(1) This is shown by an argument similar to that in [GS,Theorem 2.5] (see also the proof of [St,Theorem 13]). First of all, after passing through the completion of A, we may assume without loss of generality that A is a regular local ring.…”

“…The Rees modules (and also the associated graded modules) play very important roles not only in algebraic geometry but also in commutative algebra, particularly in the theory of Buchsbaum modules. Many authors studied in this field, see, e.g., [Br,GS,GY,Sc2,St,SV,T] and also [G5,N,SY,Y3] for recent topics. However, we are here interested in their ring theoretical behaviours, especially whether they obtain the Buchsbaumness.…”

The structure of Sally modules – towards a theory of non-Cohen–Macaulay cases

“…To see this, let (A, m) be a Buchsbaum local ring with dim A = 2, depth A = 1, and h 1 (A) = 1, e.g., look at the ring A = kJ X, Y , Z , W K/(X, Y ) ∩ (Z , W ) in Example 2.8, the ring A = C M in Example 4.6, or the ring A in Theorem 5.3 of Section 5. (The reader may consult also [G1,G2,G3,GS] for the ubiquity of this kind of Buchsbaum local rings.) Let a 1 , a 2 be a system of parameters of A and put U(a i ) = (a i ) : m for i = 1, 2.…”