Let R be a k-algebra, where k is a commutative ring, which...

Let $R$ be a $k$-algebra, where $k$ is a commutative ring, which is flat as a $k$-module. Prove that if $B$ is an $R$-module (and hence a $k$-module), then $$ R \otimes_{k} \operatorname{Tor}_{n}^{k}(B, C) \cong \operatorname{Tor}_{n}^{R}\left(B, R \otimes_{k} C\right) $$ for all $k$-modules $C$ and all $n \geq 0$.

Let $R$ be a $k$-algebra, where $k$ is a commutative ring, which is flat as a $k$-module. Prove that if $B$ is an $R$-module (and hence a $k$-module), then
$$
R \otimes_{k} \operatorname{Tor}_{n}^{k}(B, C) \cong \operatorname{Tor}_{n}^{R}\left(B, R \otimes_{k} C\right)
$$
for all $k$-modules $C$ and all $n \geq 0$.

Key Concepts

k-Algebra

A k-algebra is a ring that also carries the structure of a module over a commutative ring k, with the ring multiplication being k-bilinear. This dual structure allows one to study algebraic objects with extra module-theoretic properties, and it plays a key role in connecting ring theory with module theory and homological algebra.

Flat Module

A module is called flat if the tensor product with any exact sequence of modules preserves the exactness. This property is critical when dealing with tensor products in homological algebra because it ensures that taking tensor products does not introduce new homological complications, making flat modules a central concept in studying derived functors like Tor.

Tensor Product

The tensor product is a fundamental construction in module theory, used to 'multiply' two modules over a ring to obtain another module, encapsulating bilinear interactions. It is particularly important in homological algebra when defining and computing derived functors, as it allows for extending properties from simple modules to more complex constructions.

Tor Functor

Tor is a derived functor that measures the failure of flatness in modules; it computes, in a homological sense, how much the tensor product functor fails to be exact for non-flat modules. The n-th Tor group, Tor_n, arises as the homology of a projective resolution when tensored with a module, making it a key tool in both homological algebra and the study of module extensions.

Derived Functors

Derived functors provide a systematic way to extend and measure the behavior of non-exact functors by capturing their 'higher-order' information. In homological algebra, they are used to study and compute invariants like Tor and Ext, which encapsulate essential structural and extension properties of modules over rings.

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