A criterion for regularity of local rings (Q2495709)

Let \((A, \mathfrak m)\) denote a \(d\)-dimensional local noetherian ring. By Serre' s theorem it follows that \(R\) is regular if and only if the residue field \(k = A/\mathfrak m\) has a finite free resolution. The main result of the present paper is a variation of this classical result: Let \(M : 0 \to M_d \to \ldots \to M_0 \to 0\) denote a complex of free \(A\)-modules with homology modules of finite length. Suppose that \(k\) is a direct summand of \(H_0(M).\) Then \(H_i(M) = 0\) for \(i \geq 1\) and the ring \(A\) is regular, provided \(d \leq 3\) or \(A\) contains a field. The proof uses the balanced big Macaulay modules as they exist in both cases as shown by \textit{M. Hochster} [Topics in the homological theory of modules over commutative rings. Conf. Board Math. Sci. Regional Conf. Ser. Math. No. 24. (1975; Zbl 0302.13003) and J. Algebra 254, 395--408 (2002; Zbl 1078.13506)]. This result is an essential component in the proofs of the McKay correspondence in dimension 3 and of the statement the threefold flops induces equivalences of derived categories, see the first author and \textit{A. Maciocia} [J. Algebr. Geom. 11, 629--657 (2002; Zbl 1066.14047)].