Cohen-Macaulay differential graded modules and negative Calabi-Yau configurations

The Karoubi envelope and weak idempotent completion of an extriangulated category

“…There are many extriangulated categories which a neither exact nor triangulated. For example, it is shown in [8,Theorem 2.4] that the category of Cohen-Macaulay differential graded modules over certain Gorenstein differential graded algebras is extriangulated. Another is the subcategory K [−1,0] (proj ), which is the subcategory of complexes concetrated in degree -1 and degree 0 in K b (proj ), where is an Artin algebra; see [15,Proposition 4.39].…”

Resolutions and homological dimensions of DG-modules

“…Therefore by Lemma 2.13 pd M j = pd M j−1 − 1 − supM j−1 + supM j for j ≤ i. Thus, we have (2)(3)(4)(5)(6)(7)(8)(9) pd It remains to prove the implication (4) ⇒ (1).…”

“…Let φ : Q 3 → Q ′′ 2 and φ ′ : Q ′′ 2 → Q 3 be the retraction and the section of the left direct sum decomposition of (2)(3)(4)(5)(6)(7)(8). Let ψ : Q 2 → Q ′′ 2 and ψ ′ : Q ′′ 2 → Q 2 be the retraction and the section of of the middle direct sum decomposition (2)(3)(4)(5)(6)(7)(8). Note that we have δ 3 = ψ ′ φ .…”

“…Jin [8] introduced the notion of Gorenstein DG-algebra and studied their representation theory. In [8] a Gorenstein DG-algebra is defined to be a connective DG-algebra which is proper, i.e., dim k ∑ i∈Z H i < ∞ and satisfies the condition (J) perfR = thick R * .…”

“…Jin [8] introduced the notion of Gorenstein DG-algebra and studied their representation theory. In [8] a Gorenstein DG-algebra is defined to be a connective DG-algebra which is proper, i.e., dim k ∑ i∈Z H i < ∞ and satisfies the condition (J) perfR = thick R * . We note that for a locally finite dimensional connective DG-algebra R over a field k, properness follows from the condition (J).…”