Example

(1) The canonical isomorphism $A \to A \otimes_A A$ (II, § 3, no. 5) defines an A-cogebra structure on A.

(2) Let E be a cogebra, $c$ its coproduct and $\sigma$ the canonical automorphism of the A-module $E \otimes_A E$ such that $\sigma(x \otimes y) = y \otimes x$ for $x \in E, y \in E$; the A-linear mapping $\sigma \circ c$ defines a new cogebra structure on E. With this structure E is called the opposite cogebra to the given cogebra E.

(3) Let B be an A-*algebra* and let $m : B \otimes_A B \to B$ be the A-linear mapping defining multiplication on B (§ 1, no. 3). The transpose $^t m$ is then an A-linear mapping of the dual $B^*$ of the A-module B into the dual $(B \otimes_A B)^*$ of the A-module $B \otimes_A B$. If also B is a finitely generated projective A-module, the canonical mapping $\mu : B^* \otimes_A B^* \to (B \otimes_A B)^*$ is an A-module isomorphism (II, § 4, no. 4); the mapping $c = \mu^{-1} \circ ^t m$ is then a coproduct defining a cogebra structure on the dual $B^*$ of the A-module B.

(4) Let X be a set, $A^{(X)}$ the A-module of formal linear combinations of elements of X with coefficients in A (II, § 1, no. 11) and $(e_x)_{x \in X}$ the canonical basis of $A^{(X)}$. An A-linear mapping $c : A^{(X)} \to A^{(X)} \otimes_A A^{(X)}$ is defined by the condition $c(e_x) = e_x \otimes e_x$ and a canonical cogebra structure is thus obtained on $A^{(X)}$.

(5) Let M be an A-module and $T(M)$ the tensor algebra of M (§ 5, no. 1); by II, § 3, no. 9 there exists one and only one A-linear mapping $c$ of the A-module $T(M)$ into the A-module $T(M) \otimes_A T(M)$ such that, for all $n \geq 0$,

for all $x_i \in M$ ($x_1 x_2 \ldots x_n$ denotes the product in the algebra $T(M)$). Thus $T(M)$ is given a cogebra structure.

(6) Let M be an A-module and $S(M)$ the symmetric algebra of M (§ 6, no. 1); the diagonal mapping $\Delta : x \mapsto (x, x)$ of M into $M \times M$ is an A-linear mapping to which there therefore corresponds a homomorphism $S(\Delta)$ of the A-algebra $S(M)$ into the A-algebra $S(M \times M)$ (§ 6, no. 2, Proposition 3). On the other hand, in § 6, no. 6 we defined a canonical graded algebra isomorphism $h : S(M \times M) \to S(M) \otimes_A S(M)$; by composition we therefore obtain an A-algebra homomorphism

thus defining on $S(M)$ a cogebra structure. For all $x \in M$, by definition $S(\Delta)(x) = (x, x)$ and the definition of $h$ given in § 6, no. 6 shows that

It follows that $c$ is the unique algebra homomorphism such that, for all $x \in M$,

(4)

As $c$ is an algebra homomorphism, it follows that, for every sequence $(x_i)_{1 \leq i \leq n}$ of $n$ elements of $M$,

(5)

the summation on the right hand side of (5) being taken over all ordered pairs of strictly increasing sequences (in some cases empty)

of elements of $\{1, n\}$, whose sets of elements are complementary. The element $c(x_1 x_2 \ldots x_n)$ is an element of total degree $n$ in $S(M) \otimes_A S(M)$ and its component of bidegree $(p, n-p)$ is

(6)

where the summation is taken over all permutations $\sigma \in S_n$ which are increasing in each of the intervals $\{1, p\}$ and $\{p+1, n\}$.

(7) Let $M$ be an $A$-module and proceed with the exterior algebra $\Lambda(M)$ as with $S(M)$ in Example 6; the diagonal mapping $\Delta : M \to M \times M$ this time defines a homomorphism $\Lambda(\Delta)$ of the $A$-algebra $\Lambda(M)$ into the $A$-algebra $\Lambda(M \times M)$ (§ 7, no. 2, Proposition 2); on the other hand there is a canonical graded algebra isomorphism

(§ 7, no. 7, Proposition 10), whence by composition there is an algebra homomorphism $c = h \circ \Lambda(\Delta) : \Lambda(M) \to \Lambda(M)^g \otimes_A \Lambda(M)$, which can be considered as an $A$-module homomorphism $\Lambda(M) \to \Lambda(M) \otimes_A \Lambda(M)$ and which therefore defines on $\Lambda(M)$ a cogebra structure. It can be proved as in Example 6 that $c$ is the unique algebra homomorphism such that, for all $x \in M$,

(7)

whence, for every sequence $(x_i)_{1 \leq i \leq n}$ of elements of $M$,

where the product on the right hand side is taken in the algebra to calculate this product, consider, for every ordered pair of strictly increasing sequences $i_1 < i_2 < \cdots < i_p,\ j_1 < j_2 < \cdots < j_{n-p}$ of elements of $[1, n]$, whose sets of elements are complementary, the product $y_1 y_2 \ldots y_n$, where $y_{i_h} = x_{i_h} \otimes 1$ ($1 \leq h \leq p$) and $y_{j_k} = 1 \otimes x_{j_k}$ ($1 \leq k \leq n - p$) and the sum is taken over all these products. As the graded algebra $\bigwedge(M)^{\mathfrak{g}} \otimes_A \bigwedge(M)$ is anticommutative and the elements $x_i \otimes 1$ and $1 \otimes x_i$ are of total degree 1, by § 4, no. 6, Lemma 3 and Lemma 1, $v$ being the number of ordered pairs $(h, k)$ such that $j_k < i_h$ and the summation being taken over the same set as in (5). The element $c(x_1 \wedge \cdots \wedge x_n)$ is of total degree $n$ in $\bigwedge(M)^{\mathfrak{g}} \otimes_A \bigwedge(M)$ and its homogeneous component of bi-degree $(p, n - p)$ is equal to the summation being taken over permutations $\sigma \in S_n$ which are increasing in each of the intervals $[1, p]$ and $[p + 1, n]$.

When in future we speak of $A^{(x)}, T(M), S(M)$ or $\bigwedge(M)$ as cogebras, we shall mean, unless otherwise mentioned, with the cogebra structures defined in Examples 4, 5, 6 and 7 respectively.

(8) Let $E, F$ be two $A$-cogebras and $c, c'$ their respective coproducts. Let $\tau : (E \otimes_A E) \otimes_A (F \otimes_A F) \to (E \otimes_A F) \otimes_A (E \otimes_A F)$ denote the associativity isomorphism such that $\tau((x \otimes x') \otimes (y \otimes y')) = (x \otimes y) \otimes (x' \otimes y')$ for $x, x'$ in $E$ and $y, y'$ in $F$. Then the composite linear mapping

defines a cogebra structure on the $A$-module $E \otimes_A F$, called the tensor product of the cogebras $E$ and $F$.

Let $E$ be a cogebra and $\Delta$ a commutative monoid. A graduation $(E_\lambda)_{\lambda \in \Delta}$ on the $A$-module $E$ is said to be compatible with the coproduct $c$ of $E$ if $c$ is a graded homomorphism of degree $O$ of the graded $A$-module $E$ into the graded $A$-module (of type $\Delta$) $E \otimes_A E$, in other words (II, § 11, no. 5) if

In what follows, we shall most often limit our attention to graduations of type $\mathbf{N}$ compatible with the coproduct; a cogebrà with such a graduation will also be called a graded cogebrà. If F is another graded cogebrà, a graded cogebrà morphism $\phi : E \to F$ is by definition a cogebrà morphism (Definition 2) which is also a graded homomorphism of degree 0 of graded A-modules.