(i) Find an abelian group B for which Extℤ^1(ℚ, B) ≠{0}. (ii) Prove that ℚ ⊗ℤ Extℤ^1(ℚ, B) ≠{0}

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(i) Find an abelian group B for which Extℤ^1(ℚ, B) ≠{0}....

(i) Find an abelian group $B$ for which $\operatorname{Ext}_{\mathbb{Z}}^{1}(\mathbb{Q}, B) \neq\{0\}$. (ii) Prove that $\mathbb{Q} \otimes_{\mathbb{Z}} \operatorname{Ext}_{\mathbb{Z}}^{1}(\mathbb{Q}, B) \neq\{0\}$ for the group $B$ in (i). (iii) Prove that Proposition $7.39$ may be false when $A$ is not finitely generated, even when $R=\mathbb{Z}$.

(i) Find an abelian group $B$ for which $\operatorname{Ext}_{\mathbb{Z}}^{1}(\mathbb{Q}, B) \neq\{0\}$.
(ii) Prove that $\mathbb{Q} \otimes_{\mathbb{Z}} \operatorname{Ext}_{\mathbb{Z}}^{1}(\mathbb{Q}, B) \neq\{0\}$ for the group $B$ in (i).
(iii) Prove that Proposition $7.39$ may be false when $A$ is not finitely generated, even when $R=\mathbb{Z}$.

Key Concepts

Homological Algebra Techniques

Techniques of homological algebra, such as the use of long exact sequences, projective and injective resolutions, and the manipulation of extension groups, are central to solving problems like those posed. These methods allow one to compute and analyze Ext groups and to understand how operations like tensoring interact with them. The interplay of these techniques is particularly important when dealing with complex cases such as non-finitely generated modules or when extensions do not split.

Finitely Generated vs Non-Finitely Generated Modules

Many classical results in homological algebra, including propositions concerning the behavior of Ext and tensoring, often rely on the modules being finitely generated. Finitely generated modules have a well-behaved structure that allows for robust decomposition and classification theorems. However, when dealing with non-finitely generated modules, these results may fail. This distinction is crucial to understanding counterexamples where expected properties (like those stated in Proposition 7.39) might not hold, emphasizing that assumptions about finite generation can be essential.

Ext Functor

The Ext functor, particularly Ext¹, is a central construction in homological algebra that measures the extent to which a short exact sequence 0 ? B ? E ? A ? 0 fails to split. Essentially, Ext¹_Z(?, B) classifies equivalence classes of extensions of a module A by B. In many contexts, such as when dealing with abelian groups, the computation of these groups reveals structural properties about the modules involved, and non-trivial Ext groups can indicate the existence of interesting or non-split extensions.

Divisible Groups

A divisible group is an abelian group in which every element can be divided by any nonzero integer; that is, for every element a and every nonzero integer n, there exists an element b such that nb = a. The rational numbers Q are a prototypical example of a divisible abelian group. Divisibility of Q plays a key role in extension problems, as it can influence the vanishing or non-vanishing of Ext groups by affecting the splitting of exact sequences.

Tensor Product over ?

The tensor product is a fundamental bilinear operation in module theory, creating a new module from two given modules over a ring, in this case over ?. It is especially useful for 'localizing' problems or isolating torsion phenomena. When applied to extension groups like Ext¹, the tensor product operation can reveal intricate relations between the modules involved and can sometimes show that certain extension phenomena persist even after tensoring with fields such as ?.

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