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The Ext-algebra for infinitesimal deformations

María Julia Redondo, Lucrecia Román, Fiorela Rossi Bertone

Abstract

Let $f$ be a Hochschild $2$-cocycle and let $A_f$ be an infinitesimal deformation of an associative finite dimensional algebra $A$ over an algebraically closed field $\Bbbk$. We investigate the algebra structure of the Ext-algebra of $A_f$ and, under some conditions on $f$, we describe it in terms of the Ext-algebra of $A$. We achieve this description by getting an explicit construction of minimal projective resolutions in $\mod A_f$.

The Ext-algebra for infinitesimal deformations

Abstract

Let be a Hochschild -cocycle and let be an infinitesimal deformation of an associative finite dimensional algebra over an algebraically closed field . We investigate the algebra structure of the Ext-algebra of and, under some conditions on , we describe it in terms of the Ext-algebra of . We achieve this description by getting an explicit construction of minimal projective resolutions in .
Paper Structure (16 sections, 17 theorems, 125 equations)

This paper contains 16 sections, 17 theorems, 125 equations.

Table of Contents

  1. Preliminaries
  2. Deformations of associative algebras
  3. $A_f$-modules
  4. Projective $A_f$-modules
  5. Projective modules
  6. Simple modules
  7. Projective resolutions
  8. Examples
  9. Projective resolutions in case (A)
  10. Projective resolutions in case (B)
  11. Liftings
  12. Liftings in Case A
  13. Liftings in Case B
  14. Ext-algebras
  15. The product in case (A)
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $P=Ae_i, Q=Ae_j$ be indecomposable projective $A$-modules. Then $(u_0, u_1, u_2) \in \operatorname{Hom}_{A_f}(\hat{P}, \hat{Q})$ if and only if there exist $b, c \in e_iAe_j$ such that for any $a \in A$.

Theorems & Definitions (42)

  • Definition 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Example 3.1
  • Example 3.2
  • ...and 32 more