Let R be a domain with fraction field Q, and let A, C be...

Let $R$ be a domain with fraction field $Q$, and let $A, C$ be $R$-modules. If either $C$ or $A$ is a vector space over $Q$, prove that $\operatorname{Tor}_{n}^{R}(C, A)$ and $\operatorname{Ext}_{R}^{n}(C, A)$ are also vector spaces over $Q$. Hint. Use Exercise $2.38$ on page $97 .$

Key Concepts

Integral Domain and Fraction Field

An integral domain is a commutative ring with no zero divisors, which ensures that its elements can be 'inverted' to form a larger field known as the fraction field. The fraction field Q of a domain R allows one to transition from a more general module theory to the theory of vector spaces, where the existence of multiplicative inverses for nonzero scalars provides stronger algebraic structure and additional tools.

Vector Spaces

A vector space is a module over a field, and having one of the modules (either A or C) be a vector space over Q introduces additional properties such as the ability to multiply by any nonzero element from Q. This additional structure simplifies many algebraic constructions and proofs, as linearity and invertibility of scalars in a field enhance the behavior of derived functors like Tor and Ext.

Derived Functors (Tor and Ext)

Tor and Ext are derived functors in homological algebra that, respectively, measure the failure of flatness and projectivity or injectivity. They arise by resolving modules and then applying tensor product or Hom functor constructions. When one of the modules involved is a vector space over the field Q, these functors inherit a natural Q-module structure, making their outputs vector spaces over Q.

Extension of Scalars and Flatness

Extending scalars from R to its fraction field Q is a process that often preserves and reflects the extra structure supplied by the field. The property of flatness plays a crucial role here because Q, as a localization of R, is a flat R-module. Flatness ensures that tensoring with Q preserves exact sequences, which in turn allows the derived functors Tor and Ext to acquire natural Q-linear (vector space) structures when one of the input modules is already a Q-vector space.

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