(i) Prove that ℚ is a flat ℤ-module that is not faithfully flat. (ii) Prove that an abelian grou

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(i) Prove that ℚ is a flat ℤ-module that is not faithfully...

(i) Prove that $\mathbb{Q}$ is a flat $\mathbb{Z}$-module that is not faithfully flat.
(ii) Prove that an abelian group $G$ is a faithfully flat $\mathbb{Z}$-module if and only if it is torsion-free and $p G \neq G$ for all primes $p$.

Key Concepts

Flat Module

A flat module over a ring is one for which the tensor product with any exact sequence (of modules over that ring) preserves exactness. This means that when you tensor an exact sequence with a flat module, the resulting sequence remains exact, thereby reflecting a certain 'niceness' in the behavior of tensor operations. Flatness is closely related to the idea of not introducing new torsion or collapsing existing structure when extending scalars.

Faithfully Flat Module

A module is faithfully flat if it is flat and, in addition, the functor given by tensoring with that module is faithful; that is, it reflects nonzero modules. In other words, a module M is faithfully flat if whenever N is a nonzero module, M ? N is also nonzero. This property is essential in descent theory and change of scalars because it ensures that properties of modules can be detected at the level of the tensor product.

Localization

Localization is a process of inverting a multiplicative set in a ring, often used to ‘zoom in’ on certain parts of the spectrum of the ring. When a module is formed as the localization of a ring, it is automatically flat over the original ring. For example, Q, viewed as the localization of Z by inverting all nonzero integers, is flat over Z. However, such localizations may fail to be faithfully flat if the localization annuls nonzero modules.

Torsion-Free Abelian Groups

An abelian group (or Z-module) is torsion-free if no nonzero element has finite order, meaning that multiplying any nonzero element by any nonzero integer never gives zero. Torsion-freeness is a necessary condition for a module to be faithfully flat over Z since the presence of torsion can lead to the collapse of information under tensoring, thereby failing the faithful part of the flatness condition.

p-Divisibility Condition in Abelian Groups

The condition that pG ? G for every prime p ensures that the abelian group does not have an excessive amount of divisibility by any particular prime, which would effectively make parts of the structure 'invisible' under tensoring with Z/pZ. This condition, together with torsion-freeness, characterizes those abelian groups that are faithfully flat as Z-modules by guaranteeing that the module detects the presence of nonzero elements in any Z-module upon tensoring.

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