Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2004 • Question 2

Question

Math 6310 • Algebra Fall 2004 • Question 2

Let \(R\) be a commutative ring with identity. Given an \(R\)-module \(M\), recall the \(R\)-module \[ \bigwedge M=\bigoplus _{j\ge 0}\bigwedge ^j M, \] where \(\bigwedge ^j M\) is the \(j\)th exterior power of \(M\). Define a natural operation on morphisms so that the assignment \(M\mapsto \bigwedge M\) becomes a functor and determine whether this functor is exact.

Raw LaTeX

Let \(R\) be a commutative ring with identity. Given an \(R\)-module \(M\), recall the \(R\)-module
\[
\bigwedge M=\bigoplus_{j\ge 0}\bigwedge^j M,
\]
where \(\bigwedge^j M\) is the \(j\)th exterior power of \(M\). Define a natural operation on
morphisms so that the assignment \(M\mapsto\bigwedge M\) becomes a functor and determine whether
this functor is exact.

Topic Tags

Exact Sequences of Modules Ext Functors and Module Extensions Hom Functors and Module Homomorphisms Ideals in Quadratic Integer Rings and Exterior Powers