Buchsbaum homogeneous algebras with minimal multiplicity
In the present paper, the authors introduce a new class of graded algebras and study its properties. Let \(A\) be a homogeneous graded algebra over a field and \(\mathfrak m\) the irrelevant ideal of \(A\). The first author [J. Algebra 76, 383--399 (1982; Zbl 0482.13012)] showed that \[ e(A) \geq \sum_{i=1}^{d-1} \ell(H_{\mathfrak m}^i(A)) \tag{1} \] if \(A\) is a Buchsbaum ring where \(e(A)\) is the multiplicity of \(A\), \(d\) the dimension of \(A\) and \(H_{\mathfrak m}^i(M)\) the local cohomology module of \(A\) with respect to \(\mathfrak m\). Furthermore he showed that \(A\) has good properties, for example, \(A\) has a \(2\)-linear free resolution, if the equality of (1) holds. In the paper under review, the authors gave a fine bound of \(e(A)\). Assume that \(A\) is a Buchsbaum ring. Let \(c\) be the codimension of \(A\) and \(q\) the initial degree of \(A\). Then they show that \[ e(A) \geq \binom{c+q-2}{q-2} + \sum_{i=1}^{d-1} \ell(H_{\mathfrak m}^i(A)). \tag{2} \] They also find good properties of \(A\) when the equality of (2) holds.
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