Corollary (« Transitivity Formula for Discriminants »)

Let K be a commutative ring, A a commutative algebra admitting a finite basis $(a_j)_{j=1,\ldots,q}$ over K, and E an algebra over A admitting a finite basis $(e_i)_{i=1,\ldots,n}$. Then $(a_j e_i)$ is a basis of E over K, and we have

Indeed, if we put $\Phi(x, y) = \operatorname{Tr}_{E/A}(xy)$, the K-bilinear form $\Phi'$ of Prop. 6 is $\Phi'(x, y) = \operatorname{Tr}_{E/K}(xy)$ by the transitivity formula for traces (Chap. VIII, § 12, no 2, cor. of Prop. 7).

Exercises. — 1) Let A be an algebra of finite rank over a commutative field K, having a unit element. a) Show that if the radical of A is not zero, the bilinear form $(x, y) \to \operatorname{Tr}_{A/K}(xy)$ on A is degenerate. b) Suppose that K is of characteristic 0. Show that if A is a matrix algebra $M_n(K)$, S the canonical basis of A over K, then $D_{A/K}(S) \neq 0$. c) Deduce from a) and b) that, for an algebra A of finite rank over a field K of characteristic 0 to be absolutely semisimple, it is necessary and sufficient that the bilinear form $(x, y) \to \operatorname{Tr}_{A/K}(xy)$ be non-degenerate (or, what amounts to the same thing, that $D_{A/K}(S) \neq 0$ for every basis S of A over K).

¶ 2) Let B be a ring, A a subring of B containing the unit element of B; B is therefore a (A, A)-bimodule; denote by $^sB$ (resp. $^dB$) the set B considered as a left (resp. right) A-module, by $^sB^*$ (resp. $^dB^*$) the right (resp. left) A-module dual of $^sB$ (resp. $^dB$). For every $x' \in ^sB^*$ and every $b \in B$, $x \to \langle xb, x' \rangle$ is an A-linear form on $^sB$, hence an element of $^sB^*$ which is denoted by $bx'$; the mapping $(b, x') \to bx'$ defines on $^sB^*$ a left B-module structure (cf. chap. III, 2nd ed., App. II, No. 7). a) Let $\varphi$ be a homomorphism of the (A, A)-bimodule B into the (A, A)-bimodule A; in order that the A-bilinear mapping $\Phi : (x, y) \to \varphi(xy)$ from $^sB \times ^dB$ into A be non-degenerate, it is necessary and sufficient that $\varphi(0)$ contain no ideal (left or right) of B distinct from $\{0\}$. One then says that $\varphi$ is a Frobenius homomorphism of B into A. b) Let $\varphi$ be a Frobenius homomorphism of B into A; show that the mapping $d_\varphi$ associated on the right with $\Phi$ is an isomorphism of the left B-module $B_s$ onto a submodule of the left B-module $^sB^*$. Show that $d_\varphi$ is bijective in each of the following two cases: 1° A is a left and right artinian ring satisfying the conditions (N_s) and (N_d) (§ 1, exerc. 11), and $^sB$ and $^dB$ are A-modules of finite length (use exerc. 11 b) of the § 1); 2° A is an artinian commutative and involutive ring (chap. VIII, § 3, exerc. 11), contained in the center of B, and $^sB$ is an A-module of finite length (use exerc. 11 of chap. VIII, § 3).

c) Conversely, show that if $B_s$ and $^sB^*$ are isomorphic, there exists a Frobenius homomorphism of B into A when one is in one of the two cases considered in b) and $^sB$ and $^dB$ have the same length (use exerc. 11 b) of the § 1).

d) With the hypotheses and notations of a), show that the right annihilator (resp. left annihilator) of a left ideal I (resp. of a right ideal r) of B, is the orthogonal $l^0$ (resp. $r^0$) for the form $\Phi$ of the submodule I (resp. r) of $^sB$ (resp. $^dB$).

e) Let $\varphi$ be a Frobenius homomorphism of B into A. Show that if A is an involutive artinian ring (chap. VIII, § 3, exerc. 11) the same is true of B, when one supposes in addition that one or the other of the conditions of b) is satisfied (use b) and d)).

¶ 3) Let A be a commutative field, B a finite rank algebra over A, having a unit element.

a) In order that B be a Frobenius algebra, it is necessary and sufficient that there exist a Frobenius homomorphism of B into A (exerc. 2). (Use exerc. 2 c) and e) above, and exerc. 6 b) of chap. VIII, § 13).

b) Let $\varphi$ be a Frobenius homomorphism of B into A; every A-linear form on B can then be written in one and only one way $x \to \varphi(b'x)$ (resp. $x \to \varphi(xb'')$) where $b'$ and $b''$ belong to B; in order that this form be a Frobenius homomorphism, it is necessary and sufficient that $b'$ (resp. $b''$) be invertible in B.

c) For every $x \in B$, let $x^\sigma$ be the unique element (cf. b)) such that $\varphi(xy) = \varphi(yx^\sigma)$ for every $y \in B$. Show that $x \to x^\sigma$ is an A-automorphism of B. One says that the Frobenius algebra B is symmetric if $\sigma$ is an inner automorphism of B; there is then a Frobenius homomorphism of B into A for which $\sigma$ is the identity (cf. b)). It amounts to the same thing to say that the (B, B)-bimodules B and $^sB^* = ^dB^*$ (which is written B*) are isomorphic (exerc. 2 c)).

d) Let E be a left b-module of finite length, E’ its dual, E’* the dual of E’ considered as vector space over A; E’* is endowed with a structure of left B-module by putting, for $x' \in E'$, $x'' \in {E'}^*$, $b \in B$, $\langle x', bx'' \rangle = \langle x'b, x'' \rangle$ (Chapter III, 2nd ed., App. II, No. 7). For every $x \in E$, let $f_E(x)$ (or simply $f(x)$) be the element of E’* such that $\langle x', f(x) \rangle = \varphi(\langle x, x' \rangle)$ for every $x' \in E'$; now prove that $f$ is a semilinear bijection for the automorphism $\sigma$, of the left B-module E onto the left B-module E’* (use Exercise 10 of Chapter VIII, § 4). For $E = B_s$, one has (with the notations of Exercise 2 b)) $d_\Phi(x^\sigma) = f_{B_s}(x)$ for every $x \in B$.

4) a) Let G be a finite group. Show that the algebra B of the group G over any commutative field A is a symmetric Frobenius algebra (Exercise 3). (Consider the mapping $\varphi$ of B into A which associates, with every element $x = \sum_{s \in G} \xi_s . s$, the element $\varphi(x) = \xi_e$, e denoting the identity element of G).

b) Let E be a finite-dimensional vector space of dimension $n$ over a commutative field A, and B the exterior algebra $\wedge E$. Show that B is a Frobenius algebra. (If $(e_i)_{1 \leq i \leq n}$ is a basis of E, consider the mapping which associates, with every element $x$ of B, the coefficient of $e_1 \wedge e_2 \wedge \ldots \wedge e_n$ in the expression of $x$ using the basis of B corresponding to $(e_i)$). For the Frobenius algebra B to be symmetric, it is necessary and sufficient that $n$ be even or A be of characteristic 2.

¶ 5) a) Show that the tensor product of two Frobenius algebras (resp. Frobenius and symmetric algebras) of finite rank over a commutative field K is a Frobenius algebra (resp. Frobenius and symmetric) (cf. § 1, No. 9, Prop. 9).

b) Let B be an algebra of finite rank over K, L an extension of K of finite degree over K. Show that if the algebra $B_{(L)} = B \otimes_K L$ over L is Frobenius (resp. Frobenius and symmetric), the same is true of B. (Use Exercises 2 c) and 3 c) above and Exercise 2 of Chapter VIII, § 2).

6) Show that every absolutely semisimple algebra B of finite rank over a commutative field A is a symmetric Frobenius algebra. (Reduce to the case where B is simple; use Exercise 5 b) above, as well as Proposition 9 of Chapter VIII, § 12, No. 3).