10.2 Flat modules and flatness criteria

Flat modules are a key concept in commutative algebra, generalizing free modules. They preserve exactness when tensored with other modules, making them crucial for understanding algebraic structures and their relationships.

Flatness criteria help determine when a module is flat, including the tensor product and local flatness criteria. Flat modules are always torsion-free over integral domains, and in some cases, like PIDs, the concepts are equivalent.

Flat Modules and Their Properties

Flat modules in commutative algebra

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• Flat modules defined as modules $M$ over ring $R$ where functor $- \otimes_R M$ preserves exactness of sequences
• Importance stems from preserving short exact sequences generalizing free modules crucial in algebraic geometry
• Characterized by tensor product of exact sequences remaining exact preserving injective homomorphisms
• Free modules and localizations of modules exemplify flat modules (Z, Q as Z-modules)

Criteria for flatness

• Tensor product criterion states $M$ flat iff $I \otimes_R M \to R \otimes_R M$ injective for all ideals $I$ of $R$
• Local flatness criterion asserts $M$ flat over $R$ iff $M_\mathfrak{p}$ flat over $R_\mathfrak{p}$ for all prime ideals $\mathfrak{p}$ of $R$
• Faithfully flat modules defined as $M$ where $M \otimes_R N = 0$ implies $N = 0$ for all $R$-modules $N$
• Faithfully flat criterion requires $M$ flat and $M \otimes_R (R/\mathfrak{m}) \neq 0$ for all maximal ideals $\mathfrak{m}$ of $R$

Relationships and Applications of Flat Modules

Flat vs torsion-free modules

• Torsion-free modules defined as $R$-modules $M$ where $rm = 0$ implies $r = 0$ or $m = 0$ for $r \in R, m \in M$
• Flat modules always torsion-free over integral domains converse not always true
• Dedekind domains equate flat modules to torsion-free modules
• Principal ideal domains (PIDs) define flat modules precisely as torsion-free modules

Examples of flat modules

• Projective modules direct limits of flat modules and tensor products of flat modules exemplify flat modules
• Non-flat modules include torsion modules over integral domains and quotient modules $R/I$ where $I$ not flat ideal
• Flatness determination techniques: 1 Apply tensor product criterion 2 Use local flatness criterion 3 Check preservation of exactness for specific sequences
• $\mathbb{Z}/n\mathbb{Z}$ as $\mathbb{Z}$-module non-flat for $n > 1$ demonstrates non-flatness by losing exactness after tensoring short exact sequence