Resolutions by mapping cones (Q1605642)

The authors construct (minimal) free resolutions from iterated mapping cones as follows. Let \(R\) be a ring, \(f_1, \ldots, f_n \in R\), \(I_j = (f_1, \ldots, f_j)\). There are exact sequences \[ 0 \to R/(I_j : f_{j+1}) \to R/I_j \to R/I_{j+1} \to 0. \] If free resolutions \(F\) and \(G\) of \(R/(I_j : f_{j+1})\) and \(R/I_j\), respectively, are known, then the mapping cone of a map \(\varphi : F \to G\) extending \(R/(I_j : f_{j+1}) \to R/I_j\) gives a resolution of \(R/I_{j+1}\). The authors consider the cases when \(R\) is a polynomial ring over a field and \(I_j : f_{j+1}\) is generated by a subset of the variables. Special cases are stable ideals [see \textit{S. Eliahou} and \textit{M. Kervaire}, J. Algebra 129, 1-25 (1990; Zbl 0701.13006)], square-free stable ideals [see \textit{A. Aramova, J. Herzog} and \textit{T. Hibi}, Math. Z. 228, 353-378 (1998; Zbl 0914.13007)] and matroidal ideals. In fact, the authors generalize the Eliahou-Kervaire construction (loc. cit.) of free resolutions of stable ideals to monomial ideals which have a ``regular decomposition function'': Then the map \(\varphi\) is constructible. In section 2 the authors prove some properties of mapping cones of two-sided DG-modules over DG-algebras, and in section~3 they apply these to DG-algebra resolutions of quotients of polynomial rings. Special cases are Koszul-type resolutions, which include resolutions of regular and monomial sequences.