math.AC — Commutative Algebra
Nonexistence of lim Cohen–Macaulay modules following [Che26] and [Ang26]
Using the inequality stated in [ Che26, Main Claim 1.1], we prove nonexistence of lim Cohen–Macaulay sequences for the completed section rings of smooth projective surfaces over C with an ample globally generated divisor H satisfying KY ≡ 4H and 15H2/8 > χ(OY ). Such a pair is exhibited in [Ang26, Corollary 4.3], with H2 = 1080 and χ(OY ) = 1926. For a complete local domain (R, m) with ℓRHj m(R) < ∞ for 0 ≤ j < dim R, we prove that a lim Cohen–Macaulay sequence in the sense of Bhatt, Hochster, and Ma [ BHM26 ] exists if and only if a lim Γm-Cohen–Macaulay sequence exists (Definition 1.1). These may be different sequences. Chen’s geometric inequality [Che26, Main Claim 1.1] obstructs lim Γm-Cohen–Macaulay sequences over the completed section rings above. The comparison therefore rules out lim Cohen–Macaulay sequences over those rings.