Exterior power "commutes" with direct sum

Let $A$ be a commutative ring. Then $\Lambda(-)$ is a functor from $A$-modules to graded-commutative $A$-algebras which is left adjoint to the functor which takes the degree $1$ part. Because it is left adjoint, it preserves colimits, in particular coproducts. It follows $\Lambda(M \oplus N) \cong \Lambda(M) \otimes \Lambda(N)$. Looking at the $n$th degree part, we obtain $\Lambda^n(M \oplus N) \cong \bigoplus_{p+q=n} \Lambda^p(M) \otimes \Lambda^q(N)$.

For a more direct proof, consider the left hand side as a quotient of $(M \oplus N)^{\otimes n}$ and the right hand side as a quotient of $\bigoplus_{p+q=n} M^{\otimes p} \otimes N^{\otimes q}$. We have the "binomial theorem" $(M \oplus N)^{\otimes n} \cong \bigoplus_{p+q=n} \binom{n}{p} \cdot M^{\otimes p} \otimes N^{\otimes q}$. One easily checks that the quotients agree.

Both arguments work in great generality, $A$ can be any commutative monoid object in a cocomplete linear symmetric monoidal category (this is explained, for example, in my thesis). As usual, there is no need to fiddle around with elements.

It is an isomorphism, where $A$ is a commutative ring, and $V$ and $W$ are $A$-modules. I think we need $V$ and $W$ to be finitely generated and projective (but I'm not sure about this; perhaps someone can opine conerning this).

See Theorem 7 here in Bergman's notes for more details. His proof includes a description of the inverse map. (In his notes, $k$ denotes a commutative ring, not necessarily a field.)

Briefly, consider an element in $\Lambda^n(V \oplus W)$ consisting of $k$ elements from $V$ and $n-k$ elements from $W$, in some order, e.g. $(v_1, w_1, w_2, v_2, w_3, w_4, \dots)$. Let $\sigma$ be the permutation that rearranges that tuple to $(v_1, \dots, v_k, w_1, \dots, w_{n-k})$. Then we map $$\Lambda^n(V \oplus W) \longrightarrow \Lambda^k V \otimes \Lambda^{n-k} W$$ $$(v_1, w_1, w_2, v_2, w_3, w_4, \dots) \longmapsto \operatorname{sgn}(\sigma)(v_1, \dots, v_k)\otimes(w_1, \dots, w_{n-k}).$$

We extend this multilinearly to get an alternating map $$\Lambda^n(V \oplus W) \longrightarrow \bigoplus_{k=0}^n (\Lambda^k V \otimes \Lambda^{n-k} W).$$