Tensor Products

Abstract

Let K be a commutative ring, and let V ₁, V ₂, …, V _n and W be K-modules. A map

$$\displaystyle{ \varphi: V _{1} \times V _{2} \times \,\cdots \, \times V _{n} \rightarrow W }$$

(1.1)

is called multilinear or n - linear if φ is linear in each argument while all the other arguments are fixed, i.e.,

$$\displaystyle{ \varphi (\,\ldots,\,\lambda v' +\mu v'',\,\ldots \,) =\lambda \,\varphi (\,\ldots,\,v',\,\ldots \,) +\mu \,\varphi (\,\ldots,\,v'',\,\ldots \,) }$$

for all λ, μ ∈ K, v′, v″ ∈ V _i, $1\leqslant i\leqslant n$. For example, the 1-linear maps V → V are the ordinary linear endomorphisms of V, and the 2-linear maps V × V → K are the bilinear forms on V. The multilinear maps (1.1) form a K- module with the usual addition and multiplication by constants defined for maps taking values in a K-module. We denote the K- module of multilinear maps (1.1) by $\mathop{\mathrm{Hom}}\nolimits (V _{1},V _{2},\ldots,V _{n};W)$, or by $\mathop{\mathrm{Hom}}\nolimits _{K}(V _{1},V _{2},\ldots,V _{n};W)$ when explicit reference to the ground ring is required.