Semi-fiber products of algebras and lifting of complexes

TL;DR

This paper introduces semi-fiber products of commutative algebras, explores their properties, and characterizes the liftability of the residue field in terms of these structures and algebraic decompositions.

Contribution

It defines semi-fiber products of algebras and links their properties to the liftability of the residue field in local algebra settings.

Findings

Semi-fiber products include fiber products of local algebras.

Liftability of the residue field relates to semi-fiber product decompositions.

Characterization involves retractions, sections, and algebraic conditions.

Abstract

Let $k$ be a field. In this paper, we define the notion of semi-fiber products of commutative $k$-algebras and show that the class of such rings contains several classes of commutative rings, including that of the fiber products of local $k$-algebras over their common residue field $k$. For a noetherian local $k$-algebra $R$ and an ideal $I$ of $R$, under certain conditions, we characterize the liftability of $k$ along the natural surjection $R\twoheadrightarrow R/I$ in terms of retractions, sections, and the existence of semi-fiber product decompositions of $R$.