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Semi-fiber products of algebras and lifting of complexes

Saeed Nasseh, Maiko Ono, Yuji Yoshino

TL;DR

The paper introduces semi-fiber products of $k$-algebras and demonstrates that they encompass fiber products, trivial extensions, and tensor algebras. It develops a lifting theory for modules and complexes, linking liftability of the residue field $k$ along a surjection $R\twoheadrightarrow R/I$ to the existence of retractions or semi-fiber product decompositions, under conditions like flatness and $\mathfrak{m}_RI=(0)$. A central result shows the equivalence of liftability, the presence of a section, and a semi-fiber product decomposition, with a concrete criterion that $I$ be generated by part of a minimal generating set of $\mathfrak{m}_R$. The work provides practical tools for understanding deformations and liftings in commutative algebra and connects homological invariants such as Poincaré series to these structural decompositions, highlighting the interplay between algebraic and homological properties in lifting problems.

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 $R$.

Semi-fiber products of algebras and lifting of complexes

TL;DR

The paper introduces semi-fiber products of -algebras and demonstrates that they encompass fiber products, trivial extensions, and tensor algebras. It develops a lifting theory for modules and complexes, linking liftability of the residue field along a surjection to the existence of retractions or semi-fiber product decompositions, under conditions like flatness and . A central result shows the equivalence of liftability, the presence of a section, and a semi-fiber product decomposition, with a concrete criterion that be generated by part of a minimal generating set of . The work provides practical tools for understanding deformations and liftings in commutative algebra and connects homological invariants such as Poincaré series to these structural decompositions, highlighting the interplay between algebraic and homological properties in lifting problems.

Abstract

Let be a field. In this paper, we define the notion of semi-fiber products of commutative -algebras and show that the class of such rings contains several classes of commutative rings, including that of the fiber products of local -algebras over their common residue field . For a noetherian local -algebra and an ideal of , under certain conditions, we characterize the liftability of along the natural surjection in terms of retractions, sections, and the existence of semi-fiber product decompositions of .
Paper Structure (4 sections, 11 theorems, 24 equations)

This paper contains 4 sections, 11 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Semi-fiber products of algebras
  3. Lifting of complexes
  4. Proof of Main Theorem

Key Result

Proposition 2.10

Let $(R, \mathfrak{m}_R)$ and $(S, \mathfrak{m}_S)$ be commutative noetherian local $k$-algebras. If the $R$-module structure on $\mathfrak{m}_S$ is induced through a $k$-algebra homomorphism $f\colon R \to S$, then the map $\psi\colon R \ltimes _k S \to R \times _k S$ defined by the formula with $\ell \in k$, $x \in \mathfrak{m}_R$, and $y \in \mathfrak{m}_S$, is an isomorphism of $k$-algebras.

Theorems & Definitions (33)

  • Definition 2.3
  • Definition 2.4
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Proposition 2.10
  • proof
  • Proposition 2.11
  • proof
  • ...and 23 more