Dg-separable dg-extensions

TL;DR

The paper defines and studies dg-separable dg-extensions $(A,d_A)\to(B,d_B)$ of differential graded algebras, characterized by a splitting of the multiplication map $\mu:(B,d_B)\otimes_A(B,d_B)\to(B,d_B)$ as dg-bimodules. It provides a complete characterization in the graded-commutative dg-field setting: if $(A,d_A)$ is acyclic and the induced extension on cycles $ker(d_A)\to ker(d_B)$ is graded-separable, then the dg-extension is dg-separable, with the converse holding when the characteristic is different from $2$. The analysis uses explicit models $A \simeq ker(d_A)[T;D_A]/(T^2 - y_A^2)$ and $B \simeq ker(d_B)[T;D_B]/(T^2 - y_B^2)$ to reduce to the cycles, and demonstrates that the restriction of separability to cycles controls the dg-separability of the extension. Furthermore, dg-separability implies that the restriction functor $dg-(B,d_B)\text{-mod}\to dg-(A,d_A)\text{-mod}$ is separable, ensuring that short exact sequences of dg-modules split upon restriction and yielding semisimplicity phenomena in the acyclic setting.

Abstract

We define and characterise completely dg-separable dg-extensions $\varphi:(A,d_A)\rightarrow (B,d_B)$. We completely characterise the case of graded commutative dg-division algebras in characteristic different from $2$. We prove that for a dg-separable extension a short exact sequence of dg-modules over $(B,d_B)$ splits if and only if the restriction to $(A,d_A)$ splits.

Theorems & Definitions (23)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 1.4
• Corollary 1.5
• Proposition 2.1
• Definition 2.2
• Proposition 2.3
• Lemma 2.4
• Proposition 3.1