Extension Groups for DG Modules

TL;DR

This paper investigates the relationship between two definitions of Ext-groups for DG modules over a graded algebra, showing they differ generally but coincide under certain conditions.

Contribution

It establishes conditions under which Ext-groups defined via resolutions and extensions are isomorphic, clarifying their relationship in DG module theory.

Findings

Ext-groups via resolutions and extensions are not always isomorphic.

They are isomorphic when the first DG module is semi-projective.

Provides insight into the structure of Ext-groups in DG algebra context.

Abstract

Let $M$ and $N$ be differential graded (DG) modules over a positively graded commutative DG algebra $A$. We show that the Ext-groups $\operatorname{Ext}^i_A(M,N)$ defined in terms of semi-projective resolutions are not in general isomorphic to the Yoneda Ext-groups $\operatorname{YExt}^i_A(M,N)$ given in terms of equivalence classes of extensions. On the other hand, we show that these groups are isomorphic when the first DG module is semi-projective.