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Cohomologies and deformations of differential algebra morphisms

Lei Du, Yanhong Bao

Abstract

This paper studies the formal deformations of differential algebra morphisms. As a consequence, we develop a cohomology theory of differential algebra morphisms to interpret the lower degree cohomology groups as formal deformations. Then, we prove the Cohomology Comparison Theorem of differential algebra morphisms, i.e., the cohomology of a morphism of differential algebras is isomorphic to the cohomology of an auxiliary differential algebra. Finally, we can give a minimal model for morphism of differential algebras with weight=0.

Cohomologies and deformations of differential algebra morphisms

Abstract

This paper studies the formal deformations of differential algebra morphisms. As a consequence, we develop a cohomology theory of differential algebra morphisms to interpret the lower degree cohomology groups as formal deformations. Then, we prove the Cohomology Comparison Theorem of differential algebra morphisms, i.e., the cohomology of a morphism of differential algebras is isomorphic to the cohomology of an auxiliary differential algebra. Finally, we can give a minimal model for morphism of differential algebras with weight=0.
Paper Structure (5 sections, 9 theorems, 40 equations)

This paper contains 5 sections, 9 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Cohomology of differential algebra morphisms
  3. Deformations of differential morphisms
  4. CCT theorem of differentail algebra morphisms
  5. The case of $\lambda =0$

Key Result

Theorem 2.1

Let $\psi:(M,d_M)\rightarrow(N,d_N)$ be a morphism of $(A, d_{A})$-bimodules, then $\psi:({}_{\rhd}M_\lhd,d_M) \rightarrow({}_\rhd N_\lhd,d_N)$ is a morphism of $(A, d_{A})$-bimodules.

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.1
  • ...and 22 more