Resolutions over strict complete intersections

Let \((Q, \mathfrak {n} )\) be a regular local ring and let \(f_1, \ldots , f_c \in \mathfrak {n} ^2\) be a Q-regular sequence. Set \((A, \mathfrak {m} ) = (Q/(\textbf{f} ), \mathfrak {n} /(\textbf{f} ))\). Further assume that the initial forms \(f_1^*, \ldots , f_c^*\) form a \(G(Q) = \bigoplus _{n \ge 0}\mathfrak {n} ^i/\mathfrak {n} ^{i+1}\)-regular sequence. Without loss of any generality, assume \(\operatorname {ord}_Q(f_1) \ge \operatorname {ord}_Q(f_2) \ge \cdots \ge \operatorname {ord}_Q(f_c)\). Let M be a finitely generated A-module and let \((\mathbb {F} , \partial )\) be a minimal free resolution of M. Then we prove that \(\operatorname {ord}(\partial _i) \le \operatorname {ord}_Q(f_1) - 1\) for all \(i \gg 0\). We also construct an MCM A-module M such that \(\operatorname {ord}(\partial _{2i+1}) = \operatorname {ord}_Q(f_1) - 1\) for all \(i \ge 0\). We also give a considerably simpler proof regarding the periodicity of ideals of minors of maps in a minimal free resolution of modules over arbitrary complete intersection rings (not necessarily strict).