Buchsbaum rings with multiplicity 2

Abstract

Let A be a Buchsbaum local ring with the maximal ideal m and let e(A) denote the multiplicity of A. Let Q be a parameter ideal in A and put I = Q : m. Then the equality I 2 = QI holds true, if e(A) = 2 and depth A > 0. The assertion is no longer true, unless e(A) = 2. Counterexamples are given. 1991 Mathematics Subject Classification. Primary 13B22, Secondary 13H10.

Type of blowing-up over a Buchsbaum ring

“…where each map is respectively defined by multiplication with a, b and ab. By 7and (9), all maps are isomorphisms, and hence (7). By (6) this means that mafev must be contained in (a 2 ,b 3 )M. These observations yield mx <= (a 2…”

“…Then as is well-known A has minimal multiplicity and the canonical module K A is Cohen-Macaulay. Hence /A, A {K A ) = 2 by the remark above and Proj R(q) is a locally Gorenstein scheme for every minimal reduction q of m, by(4)(5)(6)(7)(8). (See[7] for the structure theorems and concrete examples of Buchsbaum rings with multiplicity 2.)…”

“…We call this the idealization ofM (over A) and denote it by A xM, cf. [16] Then A is a Buchsbaum ring with maximal embedding dimension, but without minimal multiplicity, e(A) = dimA = 3, depth A = 1, H4(^4) = (0) for i = 0,2 and H, 1 ,,^) = k, and q is a minimal reduction of m and fi A (K A ) = 2; see [23], (3)(4)(5)(6)(7)(8)(9)(10)(11).…”

Buchsbaum homogeneous algebras with minimal multiplicity

“…In [4], the first author classified non-Cohen-Macaulay, Buchsbaum rings (A, m) with e(A) = 2 and proved that such a ring has minimal multiplicity in the sense of Goto [6] [6]. Now let us show that such a ring has minimal multiplicity in our sense.…”

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.…”