Skip to main content
MathOverflow

Return to Question

Became Hot Network Question
deleted 154 characters in body
Source Link
Tim Campion
Tim Campion
  • 63.9k
  • 13
  • 143
  • 384

A finite complex over a field $k$ is pretty simple: it's the direct sum of its homology with a split-exact complex. How complicated can a finite double complex be? Does it make a difference if $k$ is algebraically closed?

It seems as though quiver representation theory may be relevant here, but since a double complex is a representation of a quiver with relations, I'm not really sure where to start looking. So my question is:

Question: Let $k$ be a field. How complicated is the category of finite-(total) dimensional representations of the category $\mathbb Z \times \mathbb Z$ (where $\mathbb Z$ is considered as an ordered set), i.e. the category of finite-dimensional double complexes over $k$?

A finite complex over a field $k$ is pretty simple: it's the direct sum of its homology with a split-exact complex. How complicated can a finite double complex be? Does it make a difference if $k$ is algebraically closed?

It seems as though quiver representation theory may be relevant here, but since a double complex is a representation of a quiver with relations, I'm not really sure where to start looking. So my question is:

Question: Let $k$ be a field. How complicated is the category of finite-(total) dimensional representations of the category $\mathbb Z \times \mathbb Z$ (where $\mathbb Z$ is considered as an ordered set), i.e. the category of finite-dimensional double complexes over $k$?

A finite complex over a field $k$ is pretty simple: it's the direct sum of its homology with a split-exact complex. How complicated can a finite double complex be? Does it make a difference if $k$ is algebraically closed?

It seems as though quiver representation theory may be relevant here, but since a double complex is a representation of a quiver with relations, I'm not really sure where to start looking. So my question is:

Question: Let $k$ be a field. How complicated is the category of finite-(total) dimensional double complexes over $k$?

Source Link
Tim Campion
Tim Campion
  • 63.9k
  • 13
  • 143
  • 384

How complicated can a finite double complex over a field be?

A finite complex over a field $k$ is pretty simple: it's the direct sum of its homology with a split-exact complex. How complicated can a finite double complex be? Does it make a difference if $k$ is algebraically closed?

It seems as though quiver representation theory may be relevant here, but since a double complex is a representation of a quiver with relations, I'm not really sure where to start looking. So my question is:

Question: Let $k$ be a field. How complicated is the category of finite-(total) dimensional representations of the category $\mathbb Z \times \mathbb Z$ (where $\mathbb Z$ is considered as an ordered set), i.e. the category of finite-dimensional double complexes over $k$?