Fix a field $\mathbf{k}$ and an $\mathbb{N}$-graded commutative $\mathbf{k}$-algebra $A = \bigoplus\limits_{n = 0}^{\infty} A_n$ of finite type. ("Finite type" means that each $A_n$ is a finite-dimensional $\mathbf{k}$-vector space. All algebras are understood to contain $1$.) Let $B = \bigoplus\limits_{n = 0}^{\infty} B_n$ be an $\mathbb{N}$-graded $\mathbf{k}$-subalgebra of $A$ (so that $B_n \subseteq A_n$ for all $n$). Assume that $A$ is free as a $B$-module.

 

Question 1. Does it follow that the $B$-module $A$ has a basis consisting of homogeneous elements?

 

Question 2. If yes: Can we omit the requirement that $A$ is of finite type?

 

Question 3. If yes: Does this still hold if $\mathbf{k}$ is a commutative ring rather than a field?

 

Question 4. If no, does it help to assume that $A$ is a graded subalgebra of a polynomial ring over $\mathbf{k}$?

 

Question 5. Does it help if $A_0 \cong \mathbf{k}$ and the $\mathbf{k}$-algebra $A$ is generated by $A_1$ ?

Sorry for the onslaught of questions -- I am hoping that an answer to one will likely solve most of the others, which is why I prefer not to split them across several topics.

The question cloud originates from working with Vic Reiner on cyclic quasisymmetric functions, but I find it more fundamental. I originally thought these would be easy exercises in graded linear algebra (using projection maps, linear combinations and recurrent constructions), but I see no obvious point of vantage for such tactics.

Fix a field $\mathbf{k}$ and an $\mathbb{N}$-graded commutative $\mathbf{k}$-algebra $A = \bigoplus\limits_{n = 0}^{\infty} A_n$ of finite type. ("Finite type" means that each $A_n$ is a finite-dimensional $\mathbf{k}$-vector space. All algebras are understood to contain $1$.) Let $B = \bigoplus\limits_{n = 0}^{\infty} B_n$ be an $\mathbb{N}$-graded $\mathbf{k}$-subalgebra of $A$ (so that $B_n \subseteq A_n$ for all $n$). Assume that $A$ is free as a $B$-module.

Question 1. Does it follow that the $B$-module $A$ has a basis consisting of homogeneous elements?

Question 2. If yes: Can we omit the requirement that $A$ is of finite type?

Question 3. If yes: Does this still hold if $\mathbf{k}$ is a commutative ring rather than a field?

Question 4. If no, does it help to assume that $A$ is a graded subalgebra of a polynomial ring over $\mathbf{k}$?

Question 5. Does it help if $A_0 \cong \mathbf{k}$ and the $\mathbf{k}$-algebra $A$ is generated by $A_1$ ?

Sorry for the onslaught of questions -- I am hoping that an answer to one will likely solve most of the others, which is why I prefer not to split them across several topics.

The question cloud originates from working with Vic Reiner on cyclic quasisymmetric functions, but I find it more fundamental. I originally thought these would be easy exercises in graded linear algebra (using projection maps, linear combinations and recurrent constructions), but I see no obvious point of vantage for such tactics.