0
votes
1answer
43 views

Ideal of an integral domain all of whose exterior powers are nonzero.

I want to find an integral domain $R$ with ideal $I$ (considered as an $R$-module) such that $\bigwedge^k I\neq 0$ for all nonnegative integers $k$. Dummit and Foote gave the example of $R=\mathbb ...
7
votes
1answer
72 views

Direct proof of non-flatness

Consider $k$ a field and the rings $A=k[X^2,X^3]\subset B=k[X]$. How to prove that $B$ is not flat over $A$ by using only the definition of flatness that it maintains exact sequences after making ...
2
votes
2answers
123 views

Eisenbud's proof of right-exactness of the exterior algebra

I'm trying to understand the proof in Eisenbud's Commutative Algebra that, given a right exact sequence $$K \to N \to M \to 0$$ of $R$-modules, we have an exact sequence $$K \otimes \wedge N \to ...
4
votes
1answer
211 views

What is the kernel of the tensor product of two maps?

Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity), then the tensor product $f_1\otimes ...
8
votes
1answer
281 views

Is an ideal generated by multilinear polynomials of different degrees always radical?

Definition. A polynomial $f\in\Bbbk[x_0,\ldots,x_n]$ is called multilinear if $\deg_{x_i}(f)=1$ for each $0\le i \le n$. In other words, $f$ is linear in each variable. If $f$ is homogeneous of ...
4
votes
1answer
67 views

Symmetric and exterior powers of a projective (flat) module are projective (flat)

Assume that $R$ is a commutative ring with unity and $P$ a projective (flat) $R$-module. Why $\mathrm{Sym}^n(P)$ and $\Lambda^n(P)$ are projective (flat) for every $n$?
4
votes
2answers
139 views

Proof that the $d$-th powers generate the $d$-th symmetric power of a vector space

Let $V$ be a $\mathbb{C}$-vector space of finite dimension. Denote its $d$-th symmetric power by $V^{\odot d}$. I am looking for a proof that $V^{\odot d}$ is generated by the elements $v^{\odot d}$ ...
2
votes
2answers
90 views

How can one see that $\operatorname{tr}(f\otimes g)=\operatorname{tr}f\operatorname{ tr }g$?

Suppose you have two free modules $M$ and $N$ of finite rank over a commutative ring $R$. Let's also take some $f\in\operatorname{End}_R(M)$ and $g\in\operatorname{End}_R(N)$, which gives a ...
11
votes
2answers
261 views

Why is it that $\det(\phi-x\text{id})=\sum_{i=0}^n (-1)^ic_ix^i$?

I'm trying to understand a certain formula for the determinant in a more general setting. Say you have a free module $M$ of rank $n$ over a (commutative) ring $R$. Let ...
3
votes
0answers
81 views

Is $M\to M^{\vee\vee}$ injective when $M$ is free?

It's a common theorem that when $M$ is a finite-free $R$-module of rank $n$, there is a natural isomorphism $M\cong M^{\vee\vee}$, where $M^\vee$ denotes the dual. So $M^{\vee\vee}$ is also free of ...