[Paper Review] Homologically optimal categories of sequences lead to N-complexes

Djalal Mirmohades|arXiv (Cornell University)|May 15, 2014

Homotopy and Cohomology in Algebraic Topology8 references3 citations

TL;DR

This paper introduces a generalized homology functor for $
\mathbb{Z}$-indexed sequences over an abelian category, parameterized by integers $a$ and $b$, and shows that the category of $N$-complexes with $N = a + b$ is the homologically optimal subcategory: it supports projective resolutions and induces an equivalent derived category to any larger category under this homology. The homology reduces to Kapranov's $
\ker d^a / \operatorname{im} d^b$ in this case.

ABSTRACT

We study the category of $\mathbb{Z}$-indexed sequences over an abelian category and certain generalized homology functors for this category of sequences which are indexed by positive integers $a$ and $b$. By looking at the corresponding derived category, we show that there is an "optimal" subcategory of sequences for every choice of our generalized homology functors, namely, the category of $N$-complexes (sequences for which the differential $d$ satisfies $d^N = 0$) where $N = a + b$. In this optimal case we show that our homology functors reduce to Kapranov's homology functors $\operatorname{ker} d^a / \operatorname{im} d^b$.

Motivation & Objective

• To identify the minimal subcategory of sequences over an abelian category that supports projective resolutions under a generalized homology functor $\mathrm{H}^{(a,b)} = \ker d^a / \operatorname{im} d^b$.
• To determine the conditions under which the derived category with respect to $\mathrm{H}^{(a,b)}$ inherits the property of having enough projectives.
• To establish that the category $\mathrm{Com}_N(\mathcal{A})$ of $N$-complexes with $N = a + b$ is optimal in the sense of derived category equivalence.
• To show that Kapranov's homology $\ker d^a / \operatorname{im} d^b$ arises naturally as the restriction of the generalized homology to $N$-complexes.

Proposed method

• Define the category $\mathrm{Seq}(\mathcal{A})$ of $\mathbb{Z}$-indexed sequences with differential $d$ satisfying $d^N = 0$ for $N$-complexes.
• Introduce a generalized homology functor $\mathrm{H}^{(a,b)} = \ker d^a / \left(\ker d^a \cap \operatorname{im} d^b\right)$ for positive integers $a, b$.
• Construct a functor $\mathrm{R}_N$ from chain complexes to $N$-complexes that preserves quasi-isomorphisms under the generalized homology.
• Prove that the derived category $\mathrm{D}_\mathrm{H}(\mathcal{B})$ of any category $\mathcal{B}$ containing $\mathrm{Com}_N(\mathcal{A})$ is equivalent to $\mathrm{D}_\mathrm{H}(\mathrm{Com}_N(\mathcal{A}))$.
• Use the inclusion $\operatorname{im} d^b \subset \ker d^a$ in $N$-complexes (since $d^N = 0$) to simplify $\mathrm{H}^{(a,b)}$ to Kapranov's homology.
• Establish $\mathrm{H}^{(a,b)}$-projective resolutions in $\mathrm{Com}_N(\mathcal{A})$ by lifting classical projective resolutions via $\mathrm{R}_N$.

Experimental results

Research questions

• RQ1 Which subcategory of sequences over an abelian category is minimal yet supports projective resolutions under the generalized homology $\mathrm{H}^{(a,b)}$?
• RQ2 Under what conditions does the derived category $\mathrm{D}_\mathrm{H}(\mathcal{B})$ of a subcategory $\mathcal{B}$ of sequences inherit the property of having enough projectives?
• RQ3 Why is the category $\mathrm{Com}_N(\mathcal{A})$ with $N = a + b$ optimal in the sense of derived category equivalence?
• RQ4 How does the generalized homology $\mathrm{H}^{(a,b)}$ reduce to Kapranov's homology $\ker d^a / \operatorname{im} d^b$ in the $N$-complex setting?
• RQ5 What is the minimal $N$ such that a non-projective object in $\mathcal{A}$ admits an $\mathrm{H}^{(a,b)}$-projective resolution in $\mathrm{Com}_N(\mathcal{A})$?

Key findings

• The category $\mathrm{Com}_N(\mathcal{A})$ with $N = a + b$ is homologically optimal: any larger category $\mathcal{B}$ containing it yields a derived category $\mathrm{D}_\mathrm{H}(\mathcal{B})$ equivalent to $\mathrm{D}_\mathrm{H}(\mathrm{Com}_N(\mathcal{A}))$.
• The generalized homology $\mathrm{H}^{(a,b)}$ reduces to Kapranov's homology $\ker d^a / \operatorname{im} d^b$ in $\mathrm{Com}_N(\mathcal{A})$ due to the relation $d^N = 0$ implying $\operatorname{im} d^b \subset \ker d^a$.
• If $\mathcal{A}$ has enough projectives, then every object in $\mathcal{A}$ admits an $\mathrm{H}^{(a,b)}$-projective resolution in $\mathrm{Com}_N(\mathcal{A})$ via the functor $\mathrm{R}_N$ applied to classical resolutions.
• For a non-projective object $X \in \mathcal{A}$, any $\mathrm{H}^{(a,b)}$-projective resolution in $\mathrm{Com}_N(\mathcal{A})$ cannot lie in $\mathrm{Com}_{N-1}(\mathcal{A})$, proving minimality of $N = a + b$.
• The functor $\mathrm{R}_N$ preserves quasi-isomorphisms: it maps ordinary quasi-isomorphisms in $\mathrm{Com}_2(\mathcal{A})$ to $\mathrm{H}^{(a,b)}$-quasi-isomorphisms in $\mathrm{Com}_N(\mathcal{A})$.
• The derived category of $\mathrm{Com}_N(\mathcal{A})$ is the minimal and maximal such category in the sense of derived equivalence for the given generalized homology.