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Tuesday, July 30, 2013

Exterior Algebras and Fraction Fields (11.5.8c)

Dummit and Foote Abstract Algebra, section 11.5, exercise 8(c):

MathJax TeX Test Page (c) Give an example of an integral domain R with fraction field F and ideal IF considered as an R-module such that nI0 for all n.

Proof: Let R=Z[x1,...] and I=(x1,...). It suffices to find an alternating n-multilinear map φn:I×...×IZ (n factors) such that φn(x1,...,xn)=1 for all n. To that end, define φn as follows,φn:I×...×IZφn(a1,ixi,...,an,ixi)=det (aij)1i,jnwhere aij is the constant term of aij. Note that aixi=bixi implies that ai=bi for all i, so that (aij)1i,jn is uniquely determined and φn is well defined. This map is multilinear alternating on components of I×...×I just as det is multilinear alternating on matrix rows. Here (x1,...,xn) represents the identity matrix, and as such φn(x1,...,xn)=1.

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