Twisted complexes and simplicial homotopies

Abstract

In this paper we consider the dg-category of twisted complexes over simplicial ringed spaces. It is clear that a simplicial map $f:(\mathcal{U},\mathcal{R})\to (\mathcal{V},\mathcal{S})$ between simplicial ringed spaces induces a dg-functor $f^{\ast }:\text{Tw}(\mathcal{V},\mathcal{S})\to \text{Tw}(\mathcal{U},\mathcal{R})$ where $\text{Tw}(\mathcal{U},\mathcal{R})$ denotes the dg-category of twisted complexes on $(\mathcal{U},\mathcal{R})$. In this paper we prove that for simplicial homotopic maps $f$ and $g$, there exists an $A_{\infty }$-natural transformation $\Phi :f^{\ast }\Rightarrow g^{\ast }$ between induced dg-functors. Moreover the $0$th component of $\Phi$ is an objectwise weak equivalence. If we restrict ourselves to the full dg-subcategory of twisted perfect complexes, then we prove that $\Phi$ admits an $A_{\infty }$-quasi-inverse when $(\mathcal{U},\mathcal{R})$ satisfies some additional conditions.

arXiv comment16 pages, minor change, to appear in European Journal of Mathematics

References in corpus (2)

• Descent of dg cohesive modules for open covers on complex manifolds
• A recurrent formula of $A_{\infty }$-quasi inverses of dg-natural transformations between dg-lifts of derived functors