I'm having a little trouble with the definition of the contraction operation for tensors via the universal property of the tensor product. The construction I've found (for instance, suggested here and outlined here) goes as follows: suppose we have a commutative ring $R$ (for example, the smooth real valued functions on a manifold) and an $R$-module $E$ which is finitely generated and projective (for example, the module of smooth sections of the tangent bundle of a smooth manifold), with dual $E^{*}$. Consider the tensor product
$$
\bigotimes_{i=1}^{r}E \otimes \bigotimes_{i=1}^{s}E^{*}
$$
and call $\otimes^r_s$ the corresponding canonical multilinear mapping between the cartesian product to the tensor product. To define the contraction $C^\ell_{k}$ between the $\ell$-th $E$ factor and the $k$-th $E^{*}$ factor, one defines the (suppossedly) multilinear mapping:
$$
f:\prod_{i=1}^{r}E \times \prod_{i=1}^{s}E^{*}\longrightarrow \prod_{i=1}^{r-1}E \times \prod_{i=1}^{s-1}E^{*}
$$
$$
f:(U_1,\ldots,U_r,\omega^1,\ldots,\omega^s)\mapsto \omega^k(U_\ell)(U_1,\ldots,\widehat{U_\ell},\ldots,U_r,\omega_1,\ldots,\widehat{\omega^k},\ldots,\omega_s)
$$
where hats mean ommitted arguments, and where (I presume) the product spaces are understood to be $R$-modules through the usual construction as direct sums/products. Then by the universal property of the tensor product, there exists a unique module homomorphism
$$
h:\bigotimes_{i=1}^{r}E \otimes \bigotimes_{i=1}^{s}E^{*}\longrightarrow \prod_{i=1}^{r-1}E \times \prod_{i=1}^{s-1}E^{*}
$$
such that $f = h\circ\otimes^r_s$; then the contraction would suppossedly be defined as $C^\ell_k:= \otimes^{r-1}_{s-1}\circ h$, and the universal property should yield:
$$
C^\ell_k(U_1\otimes\cdots\otimes U_r\otimes\omega^1\otimes\cdots\otimes\omega^s) = \omega^k(U_\ell)U_1\otimes\cdots,\otimes\widehat{U_\ell}\otimes\cdots\otimes U_r\otimes\omega_1\otimes\cdots\otimes\widehat{\omega^k}\otimes\cdots\otimes\omega_s
$$
I just can't get this to work. My questions/problems are:
- How exactly is $f$ suppossed to be multilinear? If addition and multiplication by elements of $R$ on $\prod_{i=1}^{r-1}E \times \prod_{i=1}^{s-1}E^{*}$ are defined component-wise, then it seems that $f$ would only be multilinear in the two arguments that are being contracted.
- Even if we admit that $h$ exists and is unique, if we understand that multiplication by elements of $R$ acts componentwise, then we should have:
$$
\otimes^{r-1}_{s-1}\left(\omega^k(U_\ell)(U_1,\ldots,\widehat{U_\ell},\ldots,U_r,\omega_1,\ldots,\widehat{\omega^k},\ldots,\omega_s)\right) = \omega^k(U_\ell)^{(r-1)(s-1)}U_1\otimes\cdots,\otimes\widehat{U_\ell}\otimes\cdots\otimes U_r\otimes\omega_1\otimes\cdots\otimes\widehat{\omega^k}\otimes\cdots\otimes\omega_s
$$
since the factor $\omega^k(U_\ell)$ multiplies all the components and thus appears $(r-1)(s-1)$ times in the tensor product.
What am I missing here?
Best Answer
Your $f$ should go directly to the tensor product space.
How do you multiply an element of the direct product by a scalar (element of $R$)? Well, we can do that: multiply each coordinate - that's how your question 2. possibly arose, but that's not what we want here: the scalar $\omega^k(U_\ell)$ is meant to be a single factor of the single tensor $$U_1\otimes\cdots\otimes \widehat{U_\ell}\otimes\cdots\otimes U_r\,\otimes\,\omega_1\otimes\cdots \otimes\widehat{\omega_k}\otimes \cdots\otimes \omega_s\,.$$ As you observed, the part $\omega^k(U_\ell)$ is $R$-linear in both variables, and using this tensor, you can prove $R$-linearity of the other variables as well.