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Piecewise hereditary algebras under field extensions

Jie Li

TL;DR

The paper proves that a finite-dimensional $k$-algebra $A$ is derived equivalent to a hereditary algebra if and only if $A\otimes_kK$ is, for any finite separable extension $K/k$. It develops a framework based on directed objects in hereditary triangulated categories and leverages a Galois-descent approach to descend this property from $A\otimes_kK$ to $A$. It also shows that the same base-change principle holds for canonical and tilted algebras, via strong-global-dimension characterizations and tilting theory. The result clarifies how base-field changes interact with prominent structural classes in representation theory and connects skew-group extensions with simple field extensions through a descent mechanism.

Abstract

Let $A$ be a finite-dimensional $k$-algebra and $K/k$ be a finite separable field extension. We prove that $A$ is derived equivalent to a hereditary algebra if and only if so is $A\otimes_kK$.

Piecewise hereditary algebras under field extensions

TL;DR

The paper proves that a finite-dimensional -algebra is derived equivalent to a hereditary algebra if and only if is, for any finite separable extension . It develops a framework based on directed objects in hereditary triangulated categories and leverages a Galois-descent approach to descend this property from to . It also shows that the same base-change principle holds for canonical and tilted algebras, via strong-global-dimension characterizations and tilting theory. The result clarifies how base-field changes interact with prominent structural classes in representation theory and connects skew-group extensions with simple field extensions through a descent mechanism.

Abstract

Let be a finite-dimensional -algebra and be a finite separable field extension. We prove that is derived equivalent to a hereditary algebra if and only if so is .

Paper Structure

This paper contains 7 sections, 13 theorems, 27 equations.

Table of Contents

  1. introduction
  2. derived categories and Galois extensions
  3. Derived categories and field extensions
  4. Action of the Galois group
  5. piecewise hereditary algebras under field extensions
  6. Piecewise hereditary algebras
  7. Directing objects and Galois extensions

Key Result

Lemma 2.1

Given two objects $X$ and $Y$ in $\mathbf{D}^{\mathrm{b}}(A\hbox{-{\rm mod}})$, we have an isomorphism of vector spaces In particular, we have an isomorphism of $K$-algebras and an isomorphism of vector spaces

Theorems & Definitions (22)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 12 more