Algebra 4 – June 12th – Tensor algebras

Tensor algebras

Definition of the tensor algebra T(M) of an R-module M. Definition of the symmetric (and antisymmetric) algebra S(M) (and $\Lambda(M)$ ).

If M is a cyclic module, then T(M)=S(M).

Let R=k[x,y] and I=(x,y). Show that $\Lambda^2 R=0$ and $\Lambda^2 I\not=0$ . Show that $\Lambda^2$ is not an exact functor. (Hint: The only non-trivial thing is the map $f:I\times I \to R/I$ defined by $f(ax+by,cx+dy)=ad-bc$ .)

References: Dummit and Foote “Abstract Algebra”

Tensor algebras

Lascia un commento