Quasi-Gorenstein morphisms of commutative local dg-algebras

Technical Summary: Quasi-Gorenstein Morphisms of Commutative Local DG-Algebras

Problem and Context
This paper addresses the extension of the theory of quasi-Gorenstein morphisms from the realm of commutative noetherian local rings to the more general setting of commutative local differential graded (dg) algebras. In the classical setting, Avramov and Foxby introduced quasi-Gorenstein homomorphisms to capture the Gorenstein property: a local ring $R$ is Gorenstein if and only if the map $R \to k$ (to its residue field) is quasi-Gorenstein. Furthermore, for a local homomorphism $\phi: R \to S$, $R$ is Gorenstein if and only if $S$ is Gorenstein, provided $\phi$ is quasi-Gorenstein.

The authors aim to generalize these results to dg-algebras, specifically those where $H_0(R)$ is a noetherian local ring and the homology $H(R)$ is finitely generated over $H_0(R)$. A central challenge in this extension is characterizing these morphisms using homological properties intrinsic to the dg-setting, such as Gorenstein dimension and the "virtually small" property, and applying these characterizations to understand exact sequences in local rings via Koszul complexes.

Methodology
The authors work within the derived category $D(R)$ of dg $R$-modules, focusing on the subcategory $D^f_b(R)$ of modules with bounded and degreewise finitely generated homology. The methodology relies on:

1. Definitions and Background: Establishing the definitions of Gorenstein dimension ($G\text{dim}$), dualizing dg-modules, and the "virtually G small" property (a Gorenstein analogue of the "virtually small" property introduced by Dwyer, Greenlees, and Iyengar).
2. Homological Characterizations: Proving dg-analogs of classical results regarding quasi-Gorenstein morphisms. This involves manipulating derived functors like $\text{RHom}$ and utilizing properties of thick subcategories.
3. Koszul Complex Analysis: Investigating the canonical augmentation $R \to H_0(R)$ where $R$ is a Koszul complex on a sequence of elements. The authors analyze the conditions under which this map is quasi-Gorenstein, linking it to the concept of "exact sequences" (sequences consisting of regular elements or exact zero divisors).
4. Construction of Examples: Utilizing trivial extensions ($R \ltimes M$) and tensor products of resolutions to construct specific dg-algebras and morphisms to test the boundaries of the theory.

Key Contributions and Results

• Definition of Quasi-Gorenstein DG-Morphisms: The paper defines a morphism $\phi: R \to S$ of commutative local dg-algebras (with $G\text{dim}_R S < \infty$) as quasi-Gorenstein if $\text{RHom}_R(S, R)$ is isomorphic in $D(S)$ to a shift of $S$. This definition aligns with the classical ring-theoretic definition when restricted to module-finite local homomorphisms of noetherian local rings.

• Characterization via Virtually G Small Modules (Theorem A):
  The paper establishes a new characterization of quasi-Gorenstein morphisms. Let $\phi: R \to S$ be a morphism with $S_0$ a noetherian local ring and $G\text{dim}_R S < \infty$. The following are equivalent:

  1. $\phi$ is quasi-Gorenstein.
  2. Every dg $S$-module $M$ with bounded, degreewise finite length homology that has finite Gorenstein dimension over $R$ is virtually G small over $S$.
     The authors note that this result is new even in the case where both $R$ and $S$ are ordinary rings, paralleling results by Briggs et al. and Dwyer et al. regarding locally complete intersection morphisms.

• Characterization of Exact Sequences (Theorem B):
  The authors provide a characterization of exact sequences in a noetherian local ring $A$ in terms of quasi-Gorenstein morphisms involving Koszul complexes. Let $x_1, \dots, x_n$ be a sequence in $A$ and $R$ the Koszul complex on this sequence. The sequence is exact if and only if:

  1. The map $R \to H_0(R)$ is quasi-Gorenstein.
  2. The Gorenstein dimension $G\text{dim}_{A/(x_1, \dots, x_{i-1})} A/(x_1, \dots, x_i)$ is finite for all $1 \le i \le n$.
     This result connects the algebraic property of exactness (regular or exact zero divisors) directly to the homological property of the associated Koszul complex.

• Examples and Non-Examples:

  • The paper demonstrates that a dg-algebra $R$ can be Gorenstein while $H_0(R)$ is not, and vice versa, implying that the canonical augmentation is not always quasi-Gorenstein.
  • It provides conditions under which the canonical augmentation of a Koszul complex is quasi-Gorenstein, specifically when the sequence consists of exact zero divisors.
  • The authors construct examples of non-free Gorenstein projective dg-modules over Koszul complexes on sequences of exact zero divisors.

• Dualizing Modules and Trivial Extensions:
  The paper adapts Jørgensen's results to show that a dg-algebra $R$ has a dualizing dg-module if and only if it is a homomorphic image of a Gorenstein dg-algebra. Furthermore, it proves that for a dualizing dg-module $D$ with $\inf D > 0$, the map $R \to R \ltimes D$ is quasi-Gorenstein if and only if $R$ is Gorenstein.

Significance and Claims
The paper claims that the introduction of quasi-Gorenstein morphisms for commutative local dg-algebras successfully extends the foundational work of Avramov and Foxby to the dg-setting. The primary significance lies in the characterization of these morphisms using the "virtually G small" property, a result the authors state is novel even for classical ring homomorphisms.

Additionally, the paper offers a new perspective on exact sequences in local rings, characterizing them via the quasi-Gorenstein nature of the augmentation of their Koszul complexes. This provides a sufficient condition for ideals generated by exact sequences to be quasi-complete intersections. Finally, the work yields concrete constructions of non-free maximal local-Cohen-Macaulay dg-modules over local-Cohen-Macaulay dg-algebras, expanding the known landscape of Gorenstein projective modules in the dg-context.