Ring Constructions and Generation of the Unbounded Derived Module Category

Abstract

We consider the smallest triangulated subcategory of the unbounded derived module category of a ring that contains the injective modules and is closed under set indexed coproducts. If this subcategory is the entire derived category, then we say that injectives generate for the ring. In particular, we ask whether, if injectives generate for a collection of rings, do injectives generate for related ring constructions, and vice versa. We provide sufficient conditions for this statement to hold for various constructions including recollements, ring extensions and module category equivalences.

A finite dimensional algebra with infinite delooping level

“…Proof. Since Λ[M (q)] op is a triangular matrix ring, by a result of Cummings it is sufficient to show that injectives generate for Λ op [3,Example 6.11].…”

Left–right symmetry of finite finitistic dimension

“…Proof As $$\tilde{A}$$ is a triangular matrix ring, the statement holds by [6, Example 6.11].$$\Box$$…”

“…Then $$\bar{C} = {C} {\otimes _{k}} {\bar{k}}$$ is a finite‐dimensional algebra over the algebraically closed field $$\bar{k}$$. Moreover, if injectives generate for $$\bar{C}$$, then injectives generate for $$C$$ by the application of [6, Lemma 5.2] to the ring homomorphism $$C \hookrightarrow \bar{C}$$. Every finite‐dimensional algebra is Morita equivalent to a basic finite‐dimensional algebra and injective generation is invariant under Morita equivalence because it is invariant under derived equivalence [17, Theorem 3.4].…”