Bourbaki

Lemma 1

Tag 0681, permanent. Cite it as [Bourbaki, Tag 0681].

Algebra, chapter III, § 11. Cogebras, products of multilinear forms, inner products and duality

(i) In the associative algebra E∗=HomA(E,A)E^* = \mathrm{Hom}_A(E, A), the product of nn elements u1,u2,…,unu_1, u_2, \ldots, u_n of degree 1 is given by

u1u2…un=mn∘(u1⊗u2⊗⋯⊗un)∘cn. u_1 u_2 \ldots u_n = m_n \circ (u_1 \otimes u_2 \otimes \cdots \otimes u_n) \circ c_n.
(ii) Suppose also that the cogebra EE is graded. Then, in the graded associative algebra E∗gr=HomgrA(E,A)E^{*gr} = \mathrm{Homgr}_A(E, A), the product of nn elements u1,u2,…,unu_1, u_2, \ldots, u_n of degree 1 is given by
u1u2…un=mn∘(u1⊗u2⊗⋯⊗un)∘δn u_1 u_2 \ldots u_n = m_n \circ (u_1 \otimes u_2 \otimes \cdots \otimes u_n) \circ \delta_n
where δn:E→E⊗n\delta_n : E \to E^{\otimes n} is the linear mapping which maps each x∈Ex \in E to the component of cn(x)c_n(x) of multidegree (1,1,…,1)(1, 1, \ldots, 1).

Formula (26) is just the definition of the product in E∗E^* for n=2n = 2; to prove it by induction on nn, observe that

u1u2…un u_1 u_2 \ldots u_n
=m∘((u1u2…un−1)⊗un)∘c = m \circ ((u_1 u_2 \ldots u_{n-1}) \otimes u_n) \circ c
=m∘((mn−1∘(u1⊗u2⊗⋯⊗un−1)∘cn−1)⊗un)∘c = m \circ ((m_{n-1} \circ (u_1 \otimes u_2 \otimes \cdots \otimes u_{n-1}) \circ c_{n-1}) \otimes u_n) \circ c
=m∘(mn−1⊗1A)∘(u1⊗u2⊗⋯⊗un−1⊗un)∘(cn−1⊗1E)∘c = m \circ (m_{n-1} \otimes 1_A) \circ (u_1 \otimes u_2 \otimes \cdots \otimes u_{n-1} \otimes u_n) \circ (c_{n-1} \otimes 1_E) \circ c
=mn∘(u1⊗u2⊗⋯⊗un)∘cn = m_n \circ (u_1 \otimes u_2 \otimes \cdots \otimes u_n) \circ c_n
by virtue of (24), (25), II, § 3, no. 3, formula (5) and the relation
un=1A∘un∘1E∗. u_n = 1_A \circ u_n \circ 1_{E^*}.

When E is graded and the elements ui∈E∗gru_i \in E^{*gr} homogeneous of degree 1, then by definition for homogeneous elements xi∈Ex_i \in E,

(u1⊗u2⊗⋯⊗un)(x1⊗x2⊗⋯⊗xn)=0 (u_1 \otimes u_2 \otimes \cdots \otimes u_n)(x_1 \otimes x_2 \otimes \cdots \otimes x_n) = 0

unless all the xix_i are of degree 1, whence formula (27).

It follows from formulae (3), (5) and (7) of no. 1 and formula (24) that when E is taken to be one of the three graded cogebras T(M)\mathbf{T}(M), S(M)\mathbf{S}(M) and Λ(M)\mathbf{\Lambda}(M), we obtain respectively by induction on nn (using the fact that the coproduct is a graded homomorphism of degree 0), for x1,x2,…,xnx_1, x_2, \ldots, x_n in M:

when E=T(M)E = \mathbf{T}(M),

δn(x1x2⋯xn)=x1⊗x2⊗⋯⊗xn \delta_n(x_1 x_2 \cdots x_n) = x_1 \otimes x_2 \otimes \cdots \otimes x_n

when E=S(M)E = \mathbf{S}(M),

δn(x1x2⋯xn)=∑σ∈Snxσ(1)⊗xσ(2)⊗⋯⊗xσ(n) \delta_n(x_1 x_2 \cdots x_n) = \sum_{\sigma \in \mathfrak{S}_n} x_{\sigma(1)} \otimes x_{\sigma(2)} \otimes \cdots \otimes x_{\sigma(n)}

when E=Λ(M)E = \mathbf{\Lambda}(M),

δn(x1x2⋯xn)=∑σ∈Snεσ⋅xσ(1)⊗xσ(2)⊗⋯⊗xσ(n). \delta_n(x_1 x_2 \cdots x_n) = \sum_{\sigma \in \mathfrak{S}_n} \varepsilon_\sigma \cdot x_{\sigma(1)} \otimes x_{\sigma(2)} \otimes \cdots \otimes x_{\sigma(n)}.

It suffices to note, for example when E=Λ(M)E = \mathbf{\Lambda}(M), that in the expression

cn(x1x2⋯xn)=(cn−1⊗1E)(∑(−1)v(xi1…xip)⊗(xj1…xjn−p)) c_n(x_1 x_2 \cdots x_n) = (c_{n-1} \otimes 1_E) \left( \sum (-1)^v (x_{i_1} \ldots x_{i_p}) \otimes (x_{j_1} \ldots x_{j_{n-p}}) \right)
arising from formula (8) of no. 1, the only terms which can give a term of multidegree (1,1,…,1)(1, 1, \ldots, 1) are those for which n−p=1n - p = 1 and hence
δn(x1x2⋯xn) \delta_n(x_1 x_2 \cdots x_n)
is the term of multidegree (1,1,…,1)(1, 1, \ldots, 1) in the sum
∑i=1n(−1)n−icn−1(x1…xi−1xi+1…xn)⊗xi \sum_{i=1}^n (-1)^{n-i} c_{n-1}(x_1 \ldots x_{i-1} x_{i+1} \ldots x_n) \otimes x_i
and this term is necessarily equal to
∑i=1n(−1)n−iδn−1(x1…xi−1xi+1…xn)⊗xi, \sum_{i=1}^n (-1)^{n-i} \delta_{n-1}(x_1 \ldots x_{i-1} x_{i+1} \ldots x_n) \otimes x_i,
whence the result by the induction hypothesis.

Using Lemma 1, the product in T(M)∗gr\mathbf{T}(M)^{*gr} of nn linear forms x1∗,x2∗,…,xn∗x_1^*, x_2^*, \ldots, x_n^* of M∗M^* is given by

⟨x1∗x2∗…xn∗,x1x2…xn⟩=∏i=1n⟨xi∗,xi⟩ \langle x_1^* x_2^* \ldots x_n^*, x_1 x_2 \ldots x_n \rangle = \prod_{i=1}^n \langle x_i^*, x_i \rangle
for xi∈Mx_i \in M (1≤i≤n1 \leq i \leq n); the product of these nn forms in S(M)∗gr\mathbf{S}(M)^{*gr} is given by

⟨x1∗x2∗…xn∗,x1x2…xn⟩=∑σ∈Sn(∏i=1n⟨xσ(i)∗,xi⟩); \langle x_1^* x_2^* \ldots x_n^*, x_1 x_2 \ldots x_n \rangle = \sum_{\sigma \in \mathfrak{S}_n} \left( \prod_{i=1}^n \langle x_{\sigma(i)}^*, x_i \rangle \right);

finally, the product of these forms in ⋀(M)∗gr\bigwedge(M)^{*gr} is given by

⟨x1∗x1∗…xn∗,x1x2…xn⟩=det⁡(⟨xi∗,xj⟩). \langle x_1^* x_1^* \ldots x_n^*, x_1 x_2 \ldots x_n \rangle = \det(\langle x_i^*, x_j \rangle).
In each of the three cases, we have respectively
θT(x1∗⊗x2∗⊗…⊗xn∗)=x1∗x2∗…xn∗ \theta_T(x_1^* \otimes x_2^* \otimes \ldots \otimes x_n^*) = x_1^* x_2^* \ldots x_n^*
θS(x1∗x2∗…xn∗)=x1∗x2∗…xn∗ \theta_S(x_1^* x_2^* \ldots x_n^*) = x_1^* x_2^* \ldots x_n^*
θΛ(x1∗∧x2∗∧⋯∧xn∗)=xn∗xn−1∗…x1∗=(−1)n(n−1)/2x1∗x2∗…xn∗ \theta_\Lambda(x_1^* \wedge x_2^* \wedge \cdots \wedge x_n^*) = x_n^* x_{n-1}^* \ldots x_1^* = (-1)^{n(n-1)/2} x_1^* x_2^* \ldots x_n^*
and hence we deduce from (28), (29) and (30) the relations
(28 bis)⟨θT(x1∗⊗x2∗⊗⋯⊗xn∗),x1⊗x2⊗⋯⊗xn⟩=∏i=1n⟨xi∗,xi⟩(28\ \mathrm{bis})\quad \langle \theta_T(x_1^* \otimes x_2^* \otimes \cdots \otimes x_n^*), x_1 \otimes x_2 \otimes \cdots \otimes x_n \rangle = \prod_{i=1}^n \langle x_i^*, x_i \rangle
(in other words θT\theta_T restricted to T2(M∗)T^2(M^*) is just the canonical homomorphism of II, § 4, no. 4)
(29 bis)⟨θS(x1∗x2∗…xn∗),x1x2…xn⟩=∑σ∈Sn(∏i=1n⟨xσ(i)∗,xi⟩)(29\ \mathrm{bis})\quad \langle \theta_S(x_1^* x_2^* \ldots x_n^*), x_1 x_2 \ldots x_n \rangle = \sum_{\sigma \in S_n} \left( \prod_{i=1}^n \langle x_{\sigma(i)}^*, x_i \rangle \right)
(30 bis)⟨θΛ(x1∗∧x2∗∧⋯∧xn∗),x1∧x2∧⋯∧xn⟩(30\ \mathrm{bis})\quad \langle \theta_\Lambda(x_1^* \wedge x_2^* \wedge \cdots \wedge x_n^*), x_1 \wedge x_2 \wedge \cdots \wedge x_n \rangle
=(−1)n(n−1)/2det⁡(⟨xi∗,xj⟩). = (-1)^{n(n-1)/2} \det(\langle x_i^*, x_j \rangle).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

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Cites

Transcribed from the printed edition, 1998, Springer. Not translated.