Appendix C. The exterior algebra
FC13 — Matsumura, Commutative Ring Theory.
This chapter contains 4 definitions, 5 theorems, 1 example, 2 constructions and 1 remark.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: alternating multilinear map
An r-multilinear map M^r→N is alternating if it vanishes whenever two arguments coincide; swapping two arguments changes the sign.
Definition/construction: exterior power
The rth exterior power Λ^rM represents alternating r-multilinear maps. It is the quotient of M^{⊗r} by tensors with repeated factors; x_1∧⋯∧x_r denotes the universal alternating image.
Theorem: exterior powers of finite free modules
If M is free of rank n with basis e_1,…,e_n, then Λ^rM=0 for r>n and for r≤n it is free of rank binom(n,r) with basis e_{i_1}∧⋯∧e_{i_r}, i_1<⋯<i_r.
Example: Λ² of an ideal can be nonzero
Although Λ²A=0, Λ²I need not vanish for an ideal I⊆A; for A=k[x,y] and I=(x,y), an alternating pairing I×I→A/I with value 1 on (x,y) shows Λ²I≠0.
Theorem: exterior powers commute with scalar extension
For an A-algebra B and A-module M, (Λ^r_A M)⊗_A B≅Λ^r_B(M⊗_A B).
Theorem: Theorem C1
Let A be a domain, I_1,…,I_r ideals and M=I_1⊕⋯⊕I_r. If T is the torsion of Λ^rM, then (Λ^rM)/T≅I_1⋯I_r. Hence an isomorphism ⊕I_i≅⊕J_i implies I_1⋯I_r≅J_1⋯J_r.
Remark: direct sums of ideals over Dedekind rings
For a Dedekind ring, the source records I_1⊕⋯⊕I_r≅A^{r−1}⊕I_1⋯I_r.
Theorem: Theorem C2
For A-modules M,N, Λ^r(M⊕N)≅⊕_{s+t=r}(Λ^sM)⊗_A(Λ^tN).
Definition: skew-commutative graded algebra
A graded A-algebra E is skew-commutative when E_pE_q⊆E_{p+q}, xy=(−1)^{pq}yx for homogeneous x,y, and every odd-degree homogeneous element squares to zero.
Definition: skew derivation
A skew derivation d on a skew-commutative graded algebra lowers degree by one and satisfies d(xy)=(dx)y+(−1)^p x(dy) for x of degree p.
Construction: exterior algebra
The wedge maps ΛpM×ΛqM→Λ^{p+q}M define an associative multiplication on ΛM=⊕_{n≥0}Λ^nM, making it a skew-commutative graded A-algebra.
Theorem: skew derivation generated by a linear functional
Every A-linear map α:M→A extends uniquely to a skew derivation d of ΛM, with d(x_1∧⋯∧x_p)=Σ_i(−1)^{i−1}α(x_i)x_1∧⋯∧x̂_i∧⋯∧x_p.
Construction: Koszul complex as an exterior algebra
For free M=Ae_1⊕⋯⊕Ae_n and scalars c_i, the skew derivation with d(e_i)=c_i turns ΛM into the Koszul complex K(c_1,…,c_n).