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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2006 • Question 4

Question

Math 6310 • Algebra Fall 2006 • Question 4

Let \(G\) be a finite group and let \(\mathbb {C}[G]\) denote the complex group algebra of \(G\). Show that the invariants functor which (on the level of objects) takes a \(\mathbb {C}[G]\) module \(M\) to \[ M^G:=\{m\in M\mid f\cdot m=m\text { for all }f\in \mathbb {C}[G]\} \] is an exact functor.

Raw LaTeX

Let \(G\) be a finite group and let \(\mathbb{C}[G]\) denote the complex group algebra of \(G\).
Show that the invariants functor which (on the level of objects) takes a \(\mathbb{C}[G]\) module
\(M\) to
\[
M^G:=\{m\in M\mid f\cdot m=m\text{ for all }f\in\mathbb{C}[G]\}
\]
is an exact functor.

Topic Tags

Exact Sequences of Modules Ext Functors and Module Extensions Finite Group Theory Hom Functors and Module Homomorphisms