Mathematics Lecture Notes

Theorem2.15Tensor-Hom Adjunction

For $R$ -modules $A$ , $B$ , and $C$ , there is a natural isomorphism: $\text{Hom}_R(A \otimes_R B, C) \cong \text{Hom}_R(A, \text{Hom}_R(B, C))$

This makes $- \otimes_R B$ left adjoint to $\text{Hom}_R(B, -)$ .

Proof

Define $\Phi: \text{Hom}(A \otimes B, C) \to \text{Hom}(A, \text{Hom}(B, C))$ by: $\Phi(f)(a)(b) = f(a \otimes b)$

And $\Psi: \text{Hom}(A, \text{Hom}(B, C)) \to \text{Hom}(A \otimes B, C)$ by: $\Psi(g)(a \otimes b) = g(a)(b)$

These are mutually inverse isomorphisms, and naturality follows from the universal property of tensor products.

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Remark

Adjunctions are fundamental in category theory. They preserve limits/colimits and provide natural transformations between functors. The tensor-Hom adjunction explains why tensoring is right exact while Hom is left exact.

Theorem2.16Projection Formula

Let $f: R \to S$ be a ring homomorphism, $M$ an $R$ -module, and $N$ an $S$ -module. Then: $M \otimes_R N \cong M \otimes_R (S \otimes_S N)$

as $S$ -modules. More generally, for finitely generated projective modules, projection formulas relate direct and inverse image functors.

ExampleChange of Rings

Given a ring homomorphism $\phi: R \to S$ , any $S$ -module $N$ can be viewed as an $R$ -module via restriction of scalars. The induced functors:

Extension of scalars: $- \otimes_R S: \text{Mod}_R \to \text{Mod}_S$
Restriction of scalars: Forgetting $S$ -action

form an adjoint pair. This is essential in representation theory and algebraic geometry.

Theorem2.17Exactness of Direct Limits

Direct limits (filtered colimits) are exact: if $\{A_i \to B_i \to C_i\}$ is a directed system of exact sequences, then: $\varinjlim A_i \to \varinjlim B_i \to \varinjlim C_i$ is exact.

Remark

The exactness of direct limits is crucial in algebraic geometry (sheaf cohomology) and commutative algebra (localization). In contrast, inverse limits are generally only left exact, with $\varprojlim^1$ measuring the failure of right exactness.

ExampleLocalization is Exact

Localization at a multiplicative set $S \subset R$ can be expressed as a direct limit: $S^{-1}M = \varinjlim_{s \in S} M$

where the directed system consists of multiplication maps $M \xrightarrow{\cdot s} M$ . Since direct limits are exact, localization is an exact functor.

Theorem2.18Nakayama's Lemma

Let $R$ be a commutative ring, $\mathfrak{m}$ a maximal ideal, and $M$ a finitely generated $R$ -module. If $\mathfrak{m}M = M$ , then $M = 0$ .

More generally, if $N \subseteq M$ and $M = N + \mathfrak{m}M$ , then $M = N$ .

Remark

Nakayama's Lemma is indispensable in commutative algebra. It allows us to deduce global properties from local information modulo the maximal ideal, making it fundamental for studying local rings and completing proofs by reduction to residue fields.

Math Notes

Short and Long Exact Sequences - Applications