Projective Localization, The Exterior Power of a Projective Module

Exterior Power and Base Change

Let M be an R module and let S be an R algebra, so that M×S is an S module. Let U be the tensor product of j copies of M, and let V be the tensor product of j copies of M×S. View V as an S module. Now the j components of S can all be folded together into one element c from S. At the same time, U×S is built from tuples in M cross M cross M … j times, cross something in S. This establishes a canonical isomorphism between U×S and V.

Let E be the exterior power of M, hence E = U/K. Let F be the exterior power of M×S, hence F = V/L. Carry K×S into V, using the aforementioned isomorphism. Let a tuple equal x in the third and fifth components of U. When this is crossed with 1 in S, and mapped to V, the result is the same as placing 1 along side each entry in the tuple. Since x,1 equals x,1, the image lies in L. Multiply the tuple by c in S, and by linearity, the image still lies in L. This is because K×S and L are both S modules. Therefore all of K×S maps into L, and US/KS maps into F.

Since tensor and quotient commute, the aforementioned map carries E tensor S into F.

Since U×S maps onto V, E maps onto F.

Now reverse the isomorphism, going from V back to U×S. Let a tuple in L set two components equal. These are equal entities in M tensor S. Pass the action of R between M and S as necessary, so that both components are x,c for some x in M and some c in S. Within V, gather all the elements of S together, including the two copies of c, into one element d. Pull this across to U tensor S. The result is a tuple in U, with two components set to x, crossed with d. This is an element of K tensor S. The submodule L maps into KS, and F maps into E.

We have a map from E onto F, and a map from F onto E. Both maps are derived from the original isomorphism. The composition of the two maps, in either order, gives the identity map. Therefore E and F are isomorphic, as S modules. Exterior power commutes with base change.

Finitely Presented

If M is finitely generated, with n generators, then the tensor product of j copies of M is finitely generated, with n^j generators. The exterior power E is a quotient of the tensor product, hence it too is finitely generated.

Assume R is noetherian, whence every finitely generated R module is also finitely presented. Therefore M and E are finitely presented.

The Exterior Power of a Projective Module

Assume R is noetherian and M is a finitely generated R module. Let E be the j^th exterior power of M. By the above, M and E are finitely presented. Apply an earlier theorem, and M is projective iff its localizations are free. The same holds for E. Localization is a base change, which commutes with exterior power. By the previous theorem, the exterior power of a free module is free. Therefore the localizations of E are free, and E is projective. If R is noetherian, the exterior power of a finite flat module is finite flat.

Notice that a module of constant rank produces another module of constant rank. If M has constant rank l, then E has constant rank (l:j).