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A reference for this material is Jacobson, Nathan. Basic algebra. I. Second edition. W.
H. Freeman and Company, New York, 1985. xviii+499 pp. ISBN:0-7167-1480-9. More
tailored to smooth manifolds is Conlon Lawrence. Differentiable manifolds, a first
course. Birkhäuser, 1993.
Throughout this section. Let \(A\) be a commutative algebra.
\(A\)-bilinear maps Let \(A\) be a commutative algebra and let \(V,\) \( W,\) \( X\) be \(A\)-modules. An \(A\)-bilinear
function is a distributive function \(f: V \times W \to X\) with
\[ f(av,w) = f(v,aw)= af(v,w) \]
for all \(a \in A\), \(v \in V\), and \(w \in W\).
If \(V = \mathbb {R}^m\) and \(W = \mathbb {R}^n\), then
\[ V \otimes _{\mathbb {R}} W = M_{m \times n} ( \mathbb {R} ) . \]
Tensor products Let \(A\) be a commutative algebra and let \(V\) and \(W\) be \(A\)-modules. Their
tensor product is an \(A\)-module \(V \otimes _A W\) equipped with an \(A\)-bilinear map \(j : V \times W \to V \otimes _A W\) satisfying the following
property:
For any \(A\)-module \(X\) and any \(A\)-bilinear function \(f : V \times W \to X\), there is a unique module
morphism \(\tilde {f} : V \otimes _A W \to X\) with
\[ f = \tilde {f} \circ j . \]
We denote \(j (v,w)\) by \(v \otimes w\).
If no confusion is expected, then \(V \otimes _A W\) is denoted by \(V \otimes W\).
Existence and uniqueness Let \(A\) be a commutative algebra and let \(V\) and \(W\) be \(A\)-modules.
Then the tensor product \(V \otimes W\) exists and is unique up to module isomorphism. It is given
by
\[ V \otimes W = \left \{ \sum _{i = 1}^k a_i (v_i, w_i) \,\Big \vert \, a_i \in A, \, v_i \in V, \, w_i \in W \, \right \} / span_A(S) , \]
where \(S\) is the set consisting of:
For all \(v_1, v_2 \in V\), \(w_1, w_2 \in W\), the linear combinations
Basis of tensor product Let \(V\) and \(W\) be vector spaces. If \(V\) has basis \(\{ e_i \} _{i \in I}\) and \(W\) has basis \(\{ f_j \} _{j \in J}\), then
\(V \otimes W\) has basis
where \(I_{ext} \leq \mathcal {T} (V)\) is
the ideal generated by the set
\[ \{ \, a \otimes b + b \otimes a \, \vert \, a , b \in V \, \} . \]
Symmetric algebra Let \(V\) be a vector space. The symmetric algebra\(S ^{\ast } V\) is given by
\[ S ^{\ast } V : = \mathcal {T}(V) / I_{sym} , \]
where \(I_{sym} \leq \mathcal {T} (V)\)
is the ideal generated by the set
\[ \{ \, a \otimes b - b \otimes a \, \vert \, a, b \in V\, \} . \]
Clifford algebra Let \(V\) be a vector space with an inner product. The Clifford algebra\(Cl( V)\) is
given by
\[ Cl (V) : = \mathcal {T}(V) / I_{Cl} , \]
where \(I_{Cl} \leq \mathcal {T} (V)\) is the ideal generated by the set
\[ \{ \, a \otimes b + b \otimes a + 2 \langle a, b \rangle \, \vert \, a, b \in V \, \} . \]
Duality Let \(A\) be a commutative algebra, and let \(M\) and \(N\) be \(A\)-modules. Consider the
bilinear map
\[ \theta : M^{\ast } \times N \to Hom _A (M , N) \]
given by
\[ \theta ( \lambda , y ) (x ) : = \lambda (x) y . \]
Since \(\theta \) is \(A\)-bilinear, it induces a module morphism
\[ \tilde {\theta } : M ^{\ast } \otimes _A N \to Hom _A( M,N) . \]
If \(A = \mathbb {R}\) and \(M\) and \(N\) are finite-dimensional vector spaces, \(\tilde {\theta }\) is an isomorphism. Later we will
see a more general version of this.
Smooth manifolds