Change of ring

Suppose $𝜌 : 𝑅 \to 𝑆$ is a ring homomorphism. For any $𝑆$ -module $𝑉$ , we have a (left) $𝑅$ -module $𝜌 * 𝑉$ by restriction of scalars. Explicitly, $𝜌 * 𝑉$ is the $𝑅$ -module with the same underlying abelian group as $𝑉$ and the $𝑅$ -action defined by $𝑥 𝑣 = 𝜌 (𝑥) 𝑣$ . Since any $𝑆$ -morphism $𝑉 \to 𝑊$ becomes an $𝑅$ -morphism of $𝜌 * 𝑉 \to 𝜌 * 𝑊$ , we have a “forgetful” faithful functor $𝜌 * : 𝑆 𝖬 𝗈 𝖽 \to 𝑅 𝖬 𝗈 𝖽$ . In fact, this functor has both left and right adjoints, giving us an adjoint triple

where

$𝜌!$ is Extension of scalars
$𝜌 *$ is Coëxtension of scalars

Note that is some cases $𝜌 * ≅ 𝜌!$ are naturally isomorphic: When $𝜌$ is a Frobenius extension.

Extension of scalars

Let $𝜌 : 𝑅 \to 𝑆$ be a ring homomorphism, and let $𝑆 𝑆 𝑅$ denote $𝑆$ regarded as a $(𝑆, 𝑅)$ -bimodule, where the right action is given by $𝑠 ◃ 𝑟 = 𝑠 𝜌 (𝑟)$ . We define $𝜌!$ from the Tensor product of bimodules as the functor

𝜌! : = 𝑆 𝑆 𝑅 \otimes 𝑅 ? .

It follows for an $𝑅$ -module $𝑉$ , we have the $𝑆$ -action

𝑠 1 (𝑠 2 \otimes 𝑣) = 𝑠 1 𝑠 2 \otimes 𝑣

for $𝑥 \in 𝑆$ , $𝑦 \in 𝑅$ , and $𝑣 \in 𝑉$ .

The unit of the adjunction $𝜌! ⊣ 𝜌 *$ is the (not necessarily injective) “inclusion,” with components

𝜂 𝑉 : 𝑉 \to 𝜌 * 𝜌! (𝑉) 𝑣 \mapsto 1 \otimes 𝑣 .

Proof of adjunction

Suppose $𝑊$ is a $𝑆$ -module and $𝑓 : 𝑉 \to 𝜌 * 𝑊$ is an $𝑅$ -morphism. The right diagram says that for $𝑣 \in 𝑉$ ,

𝑓 (1 \otimes 𝑣) = 𝑓 ♯ 𝜂 𝑉 (𝑣) = 𝑓 (𝑣) .

which by $𝑆$ -linearity uniquely defined $𝑓$ .

Coëxtension of scalars

Let $𝜌 : 𝑅 \to 𝑆$ be a ring homomorphism, and let $𝑅 𝑆$ denote $𝑆$ regarded as a left $𝑅$ -module via the action $𝑥 ▹ 𝑦 = 𝜌 (𝑥) 𝑦$ .

We define the functor $𝜌 * : 𝑅 𝖬 𝗈 𝖽 \to 𝑆 𝖬 𝗈 𝖽$ so that

the image $𝜌 * 𝑉$ of an $𝑅$ -module $𝑉$ is the $𝑆$ -module with underlying abelian group $𝑅 𝖬 𝗈 𝖽 (𝑅 𝑆, 𝑉)$ and $𝑆$ -action $𝑠 1 ▹ 𝑓 = 𝑠 2 \mapsto 𝑓 (𝑠 2 𝑠 1)$ .¹
the image $𝜌 * 𝑓$ of an $𝑅$ -morphism $𝑓 : 𝑉 \to 𝑊$ is the $𝑆$ -morphism with underlying group homomorphism $𝑅 𝖬 𝗈 𝖽 (𝑅 𝑆, 𝑔) : 𝑅 𝖬 𝗈 𝖽 (𝑅 𝑆, 𝑉) \to 𝑅 𝖬 𝗈 𝖽 (𝑅 𝑆, 𝑊) 𝑓 \mapsto 𝑔 \circ 𝑓 .$

Proof of

𝑆

-linearity

To reduce parentheses, the action $▹$ takes precedence over application of functions.

To see that $𝜌 * 𝑉$ has a valid $𝑆$ -action: closure follows from

𝑠 1 ▹ 𝑓 (𝑟 1 ▹ 𝑠 2 + 𝑟 2 ▹ 𝑠 3) = 𝑠 1 ▹ 𝑓 (𝜌 (𝑟 1) 𝑠 2 + 𝜌 (𝑟 2) 𝑠 3) = 𝑓 (𝜌 (𝑟 1) 𝑠 2 𝑠 1 + 𝜌 (𝑟 2) 𝑠 3 𝑠 1) = 𝑟 1 𝑓 (𝑠 2 𝑠 1) + 𝑟 2 𝑓 (𝑠 3 𝑠 1) = 𝑟 1 (𝑠 1 ▹ 𝑓) (𝑠 2) + 𝑟 2 (𝑠 1 ▹ 𝑓) (𝑠 3);

unitality follows immediately; multiplicativity follows from

𝑠 1 𝑠 2 ▹ 𝑓 (𝑠 3) = 𝑓 (𝑠 3 𝑠 1 𝑠 2) = 𝑠 2 ▹ 𝑓 (𝑠 3 𝑠 1) = 𝑠 1 ▹ 𝑠 2 ▹ 𝑓 (𝑠 3);

scalar distributivity follows from

𝑠 1 ▹ 𝑓 1 + 𝑓 2 (𝑠 2) = (𝑓 1 + 𝑓 2) (𝑠 2 𝑠 1) = 𝑓 1 (𝑠 2 𝑠 1) + 𝑓 2 (𝑠 2 𝑠 1) = 𝑠 1 ▹ 𝑓 1 (𝑠 2) + 𝑠 1 ▹ 𝑓 2 (𝑠 2) = (𝑠 1 ▹ 𝑓 1 + 𝑠 1 ▹ 𝑓 2) (𝑠 2);

vector distributivity follows from

(𝑠 1 + 𝑠 2) ▹ 𝑓 (𝑠 3) = 𝑓 (𝑠 3 (𝑠 1 + 𝑠 2)) = 𝑓 (𝑠 2 𝑠 1 + 𝑠 3 𝑠 2) = 𝑓 (𝑠 2 𝑠 1) + 𝑓 (𝑠 3 𝑠 2) = 𝑠 1 ▹ 𝑓 (𝑠 3) + 𝑠 2 ▹ 𝑓 (𝑠 1) .

That $𝜌 * 𝑔$ is well-defined and preserved addition follows from the internal-hom of $𝖠 𝖻$ . To see that $𝜌 * 𝑔$ preserves the $𝑆$ -action,

𝜌 * 𝑔 (𝑠 1 ▹ 𝑓) (𝑠 2) = 𝑔 \circ (𝑠 1 ▹ 𝑓) (𝑠 2) = 𝑔 (𝑓 (𝑠 2 𝑠 1)) = 𝑠 1 ▹ (𝑔 \circ 𝑓) (𝑠 2)

as required.

The coünit of the adjunction $𝜌 * ⊣ 𝜌 *$ is the “projection” with components

𝜖 𝑉 : 𝜌 * 𝜌 * 𝑉 \to 𝑉 𝑓 \mapsto 𝑓 (1)

Proof of

𝑅

-linearity and adjunction

To see that $𝜖 𝑉$ is $𝑅$ -linear, it suffices to note

𝜖 𝑉 (𝜌 (𝑟) ▹ 𝑓 1) = 𝜌 (𝑟) ▹ 𝑓 (1) = 𝑓 (𝜌 (𝑟)) = 𝑟 𝑓 (1) = 𝑟 1 𝜖 𝑉 (𝑓)

by the $𝑅$ -linearity of $𝑓$ .

Next we show that $𝜖 𝑉$ is a coünit of adjunction. Suppose that $𝑊$ is a $𝑆$ -module and $𝑓 : 𝜌 * 𝑊 \to 𝑉$ is an $𝑅$ -morphism. The left diagram says that for $𝑤 \in 𝑊$ we have

𝑓 ♭ (𝑤) (1) = 𝜖 𝑉 𝑓 ♭ (𝑤) = 𝑓 (𝑤) \in 𝜌 * 𝑉 = 𝑅 𝖬 𝗈 𝖽 (𝑅 𝑆, 𝑉)

This determines the $𝑓 ♭ (𝑤)$ completely, as for any $𝑠 \in 𝑆$ we have

𝑓 ♭ (𝑠 𝑤) (1) = 𝑠 ▹ 𝑓 ♭ (𝑤) (1) = 𝑓 ♭ (𝑤) (𝑠)

by $𝑆$ -linearity.

#state/tidy | #lang/en | #SemBr

Crucially, the restricted $𝑅$ -action on $𝜌 * 𝜌 * 𝑉$ will not be the usual $𝑅$ -action on $𝑅 𝖬 𝗈 𝖽 (𝑅 𝑇, 𝑉)$ , but instead $𝑟 ▹ 𝑓 (𝑡) = 𝑟 ▹ 𝑓 (𝑡 𝜌 (𝑟))$ . ↩

Change of ring

Extension of scalars

Coëxtension of scalars

Footnotes