On the dimension of the space of integrals on coalgebras

TL;DR

This paper investigates the dimension of the space of integrals on coalgebras, providing new bounds and extending previous results, with applications to incidence coalgebras and dual algebra properties.

Contribution

It extends Iovanov's results on integrals, offering new bounds for dimensions of integral spaces on right co-Frobenius coalgebras and exploring dual algebra properties.

Findings

Dimension of left integrals ≤ dimension of the comodule

Dimension of right integrals ≥ dimension of the comodule

Examples of integrals computed for incidence coalgebras

Abstract

We study the injective envelopes of the simple right $C$-comodules, and their duals, where $C$ is a coalgebra. This is used to give a short proof and to extend a result of Iovanov on the dimension of the space of integrals on coalgebras. We show that if $C$ is right co-Frobenius, then the dimension of the space of left $M$-integrals on $C$ is $\leq {\rm dim}M$ for any left $C$-comodule $M$ of finite support, and the dimension of the space of right $N$-integrals on $C$ is $\geq {\rm dim}N$ for any right $C$-comodule $N$ of finite support. If $C$ is a coalgebra, it is discussed how far is the dual algebra $C^*$ from being semiperfect. Some examples of integrals are computed for incidence coalgebras.