- #1
bombo1
- 1
- 0
I have this problem that i need to prove and i don't even know where to start. So I have to show that the symmetric algebra ( Sym V ) is isomorphic to free commutative R-algebra on the set {x1, ..., xn}.
Now i know that Sym V could be regarded as +Sym^n V for n>=0. And then we need to show that this is isomorphic to the R-algebra on the set {x1, ..., xn}. What I don't really understand is the +Sym^n V for n>=0
what exactely does the + do? Can i just regard this as the space of homogeneous polynomials of degree n in the variables of e1, e2,... e3, where {ei} is a basis for V.
Every polynomial ring R[x1, ..., xn] is a commutative R-algebra. Do I need to find a map such that it reduces all the polynomials from R to make them homogeneous.. I really don't know how to start this problem.
thnx for everyone who can help me with any ideas:)
Now i know that Sym V could be regarded as +Sym^n V for n>=0. And then we need to show that this is isomorphic to the R-algebra on the set {x1, ..., xn}. What I don't really understand is the +Sym^n V for n>=0
what exactely does the + do? Can i just regard this as the space of homogeneous polynomials of degree n in the variables of e1, e2,... e3, where {ei} is a basis for V.
Every polynomial ring R[x1, ..., xn] is a commutative R-algebra. Do I need to find a map such that it reduces all the polynomials from R to make them homogeneous.. I really don't know how to start this problem.
thnx for everyone who can help me with any ideas:)