Maximal Cohen-Macaulay DG-complexes

Let $R$ be a commutative noetherian local differential graded (DG) ring. In this paper we propose a definition of a maximal Cohen-Macaulay DG-complex over $R$ that naturally generalizes a maximal Cohen-Macaulay complex over a noetherian local ring, as studied by Iyengar, Ma, Schwede, and Walker. Our proposed definition extends the work of Shaul on Cohen-Macaulay DG-rings and DG-modules, as any maximal Cohen-Macaulay DG-module is a maximal Cohen-Macaulay DG-complex. After proving necessary lemmas in derived commutative algebra, we establish the existence of a maximal Cohen-Macaulay DG-complex for every DG-ring with constant amplitude that admits a dualizing DG-module. We then use the existence of these DG-complexes to establish a derived Improved New Intersection Theorem for all DG-rings with constant amplitude.

💡 Research Summary

The paper introduces a notion of maximal Cohen‑Macaulay (MCM) differential graded (DG) complexes that extends the classical definition of maximal Cohen‑Macaulay complexes over Noetherian local rings to the setting of commutative Noetherian local DG‑rings. After reviewing the necessary background on DG‑rings, derived categories, derived Hom and tensor functors, derived torsion and completion, and dualizing DG‑modules, the author focuses on DG‑rings of “constant amplitude”, i.e., those for which localization at any prime does not reduce the amplitude of the ring. This condition guarantees that depth and dimension calculations behave uniformly across the spectrum.

A maximal Cohen‑Macaulay DG‑complex (M) is defined for an object (M\in D^{b}{f}(R)) by two requirements: (1) the derived local cohomology (R\Gamma{\mathfrak m}(M)) has amplitude zero (so depth equals the lowest non‑vanishing cohomology), and (2) the natural map (H^{0}(M)\to H^{0}(k\otimes^{\mathbf L}_{R}M)) is non‑zero. These conditions mirror those of Iyengar‑Ma‑Schwede‑Walker for ordinary rings but allow the DG‑complex to have larger amplitude than the underlying DG‑ring.

The first main result (Theorem A) shows that if (R) is a Noetherian local DG‑ring of constant amplitude (n) and (D) is a right‑normalized dualizing DG‑module (i.e., (\inf D=0)), then the DG‑module \