The Poincaré series of every finitely generated module over a codimension four almost complete intersection is a rational function
The authors prove the following main results. Assume that \(A\) is a quotient of a regular local ring \((R,m,k)\) which is an almost complete intersection of codimension 4 such that 2 is a unit in \(A\). Then the following statements hold true: (i) The Poincaré series \(P^ M_ A (z) = \sum^ \infty_{i=0} \dim \text{Tor}^ A_ i (k,M) \cdot z^ i\) is a rational function for all finitely generated \(A\)-modules \(M\). (ii) If \(R\) contains \(\mathbb{Q}\) then the cotangent modules \(T_ i (A/R,A)\) vanish for \(i\) large if and only if \(A\) is a complete intersection. (iii) For all finitely generated \(A\)-modules \(M\) whose Betti numbers are bounded, the minimal resolution of \(M\) is eventually periodic of period \(\leq 2\).
scientific article; zbMATH DE number 681808
scientific article