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3
votes
0answers
125 views

Symmetric power of an algebra

Given an algebra $A$ over $k$ with characteristic zero and a positive integer $n$, the subspace of $A^{\otimes n}$ consisting of all tensors invariant under the action of all permutations ...
0
votes
2answers
251 views

Tensor powers of an algebra all isomorphic

Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism. EDIT: Assume ...
4
votes
2answers
405 views

Characterizing $\mathbb{Q}[X]$ via a property of its tensor powers

Let $\varphi: \mathbb{Q}[X] \longrightarrow R$ an inclusion of commutative rings. Suppose that the map $$- \circ \varphi: \operatorname{Hom}_{\mathbb{Q}\operatorname{-alg}}(R, R^{\otimes_{\mathbb{Q}} ...
2
votes
1answer
87 views

Iterated Reduced Tensor Power of Graded Vector spaces

This might be inappropriate for the MO-level. If so I'll delete it... Suppose $V$ is a $\mathbb{Z}$-graded vector space and $\overline{T}(V):=V \oplus V\otimes V \oplus \otimes^3 V \ldots$ is the ...
6
votes
2answers
316 views

Alternating multilinear invariants of GL(n) on End (k^n)

Introduction. Let $k$ be a field of characteristic $0$, and let $n\in\mathbb N$. Let $V=k^n$. The group $\mathrm{GL}_n\left(k\right)=\mathrm{GL} V$ acts on $\mathrm{End} V$ by conjugation, and thus ...
6
votes
3answers
584 views

Torsion-free tensor powers

Let $R$ be an integral domain. If $M$ is an $R$-module such that every tensor power of $M$ over $R$ is $R$-torsion-free, then is $M$ necessarily flat as an $R$-module? If not, then does this ...
1
vote
0answers
231 views

Decomposition of product of exterior products

Suppose $V$ is a finite dimensional vector space of dimension n. What is the kernel of the map $$\bigwedge^p V \otimes \bigwedge^q V ----> \bigwedge^{p+q} V$$ ? (here $p+q< n$) Thanks.. ...
4
votes
1answer
389 views

Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?

Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$. For any $k$-vector space $V$, consider the canonical ...
1
vote
1answer
401 views

Restricted universal enveloping algebra of Abelian p-Lie algebra

Question: Let $p$ be a prime. Let $k$ be a commutative ring such that $p=0$ in $k$. Let $\mathfrak g$ be an abelian $p$-restricted Lie algebra over $k$. In other words, let $\mathfrak g$ be a ...
13
votes
5answers
1k views

What do gerbes and complex powers of line bundles have to do with each other?

We all know how to take integer tensor powers of line bundles. I claim that one should be able to also take fractional or even complex powers of line bundles. These might not be line bundles, but ...