[Paper Review] Dg analogues of the Zuckerman functors and the dual Zuckerman functors II

Takuma Hayashi|arXiv (Cornell University)|Jun 14, 2016

Homotopy and Cohomology in Algebraic Topology28 references4 citations

TL;DR

This paper constructs derived functors for dg analogues of Zuckerman functors and their duals using model category theory, establishing combinatorial model structures on categories of modules over dg pairs and weak pairs. The key contribution is the construction of unbounded derived functors via left Quillen functors, with equivalence results between injective and projective model structures under characteristic zero and reductive group assumptions.

ABSTRACT

In the first part of this series of papers we constructed dg analogues of the Zuckerman functors over commutative rings and the dual Zuckerman functors over the field of complex numbers. In this paper we construct their derived functors in view of the theory of model categories.

Motivation & Objective

• To define and construct derived functors for dg analogues of Zuckerman functors and their duals in the context of unbounded derived categories.
• To establish combinatorial model structures on categories of modules over dg weak pairs and pairs, suitable for deriving functors.
• To prove that the projective and injective model structures on these module categories are Quillen equivalent under characteristic zero and reductive group assumptions.
• To extend previous results on dg analogues of Zuckerman functors by incorporating homotopical algebra via model categories.

Proposed method

• Utilizes the theory of model categories, particularly Quillen's framework, to construct derived functors for dg Zuckerman functors.
• Applies the transfer theorem (Theorem 2.5.1) to lift the projective model structure from $K extrm{-} extrm{mod}$ to $( ilde{A},K) extrm{-} extrm{mod}_{w}$ and $( ilde{A},K) extrm{-} extrm{mod}$ via adjoint functors.
• Constructs path objects using the $[-1]$-shift of the mapping cone of $M igoplus M \to M$ to define right homotopies.
• Establishes that right homotopies correspond precisely to chain homotopies of $(\mathcal{A},K)$-modules.
• Applies the Quillen equivalence criterion to show that the identity functor induces a Quillen equivalence between injective and projective model structures.
• Relies on the assumption that $k$ is a field of characteristic zero and $K$ is reductive to ensure the equivalence of model structures.

Experimental results

Research questions

• RQ1 How can derived functors be constructed for dg analogues of Zuckerman functors using model category theory?
• RQ2 Under what conditions do the injective and projective model structures on categories of $(\mathcal{A},K)$-modules become Quillen equivalent?
• RQ3 What is the relationship between chain homotopies and right homotopies in the context of dg modules over weak pairs?
• RQ4 How do the functors $P^{\mathcal{B},L}_{\mathcal{A},K,w}$ and $P^{\mathcal{B},L}_{\mathcal{A},K}$ behave as left Quillen functors under the projective model structure?
• RQ5 What conditions ensure that the forgetful functors between module categories admit left adjoints that are compatible with derived functors?

Key findings

• A combinatorial model structure is constructed on $(\mathcal{A},K)\textrm{-}\mathrm{mod}_{w}$ and $(\mathcal{A},K)\textrm{-}\mathrm{mod}$ using transfer from $K\textrm{-}\mathrm{mod}$, with fibrations as surjective maps and weak equivalences as quasi-isomorphisms.
• The functors $P^{\mathcal{B},L}_{\mathcal{A},K,w}$ and $P^{\mathcal{B},L}_{\mathcal{A},K}$ are shown to be left Quillen functors with respect to the projective model structures when $K$ and $L$ are reductive.
• The unbounded derived functors $\mathbb{L}P^{\mathcal{B},L}_{\mathcal{A},K,w}$ and $\mathbb{L}P^{\mathcal{B},L}_{\mathcal{A},K}$ are well-defined via the left Quillen property.
• Right homotopies with respect to the path object construction are equivalent to chain homotopies of $(\mathcal{A},K)$-modules.
• When $k$ is a field of characteristic zero and $K$ is reductive, the identity functor induces a Quillen equivalence between the injective and projective model structures on $(\mathcal{A},K)\textrm{-}\mathrm{mod}_{w}$ and $(\mathcal{A},K)\textrm{-}\mathrm{mod}$.
• The model structures are proper, and the weak equivalences satisfy the two-out-of-three property and are closed under retracts.