Tensor product of coaugmented conilpotent coalgebras

Question

Let $\mathbb{K}$ be a field of char. 0.

Let $\mathrm{A}, \mathrm{B}$ be conilpotent cocommutative coaugmented counital dg-coalgebras over $\mathbb{K}$ (i.e. that their corresponding cokernel of their coaugmentation is a conilpotent cocommutative non-counital dg-coalgebra).

Is it always true that their tensorproduct $\mathrm{A} \otimes_{ \mathbb{K} } \mathrm{B}$ is conilpotent as a coaugmented counital dg-coalgebra?

In other words lifts the tensor product $- \otimes_{ \mathbb{K}} -$ of the category of dg-$\mathbb{K}$-vector spaces to the category of cocommutative coaugmented counital dg-coalgebras over $\mathbb{K}$?

From the very definition of conilpotency it follows that the tensorproduct $\mathrm{A} \otimes_{ \mathbb{K} } \mathrm{B}$ of two cocommutative non-counital dg-coalgebras $\mathrm{A}, \mathrm{B}$ over $\mathbb{K}$ is conilpotent as a non-counital dg-coalgebra if $\mathrm{A}$ or $ \mathrm{B}$ are conilpotent as non-counital dg-coalgebras.

But taking the cokernel of the coaugmentation only defines an oplax symmetric monoidal functor from cocommutative coaugmented counital dg-coalgebras to cocommutative non-counital dg-coalgebras.

However it follows from this that if we endow the category of cocommutative coaugmented counital dg-coalgebras with the smash-product $- \wedge_{ \mathbb{K} } -$ over $\mathbb{K}$, then $\mathrm{A} \wedge_{ \mathbb{K} } \mathrm{B}$ is conilpotent as a coaugmented counital dg-coalgebra if $\mathrm{A}$ or $ \mathrm{B}$ are conilpotent as coaugmented counital dg-coalgebras.

But I want to consider the symmetric monoidal structure on the category of cocommutative coaugmented counital dg-coalgebras given by $- \otimes_{ \mathbb{K} } -$ to describe conilpotent cocommutative bialgebras as monoids in the category of conilpotent cocommutative coaugmented counital dg-coalgebras.

Answer 1

Yes, it is true. Moreover, neither cocommutativity nor characteristic $0$ are needed as conditions. The tensor product of two conilpotent coassociative dg-coalgebras over any field $k$ is a conilpotent coassociative dg-coalgebra over $k$.

The definition of a conilpotent dg-coalgebra includes two conditions: the dg-coalgebra $(C,d)$ should be coaugmented as a dg-coalgebra, that is, the coaugmentation $\gamma\colon k\to C$ should be a morphism of dg-coalgebras (which means the equation $d\circ\gamma=0$), and the underlying graded coalgebra should be conilpotent (which means that the cokernel of its coaugmentation $C/\gamma(k)$ should be a conilpotent coalgebra without counit).

Checking that the tensor product of two coaugmented dg-coalgebras is a coaugmented dg-coalgebra is quite easy, so let me skip it. Then the question reduces to showing that the tensor product of two conilpotent coalgebras is a conilpotent coalgebra.

Any (counital or noncounital) coassociative coalgebra is the union of its finite-dimensional subcoalgebras. In particular, a coagumented coalgebra is the union of its finite-dimensional coaugmented subcoalgebras. A finite-dimensional coalgebra is the dual vector space to a finite-dimensional algebra. In particular, a finite-dimensional coaugmented coalgebra is the dual vector space to a finite-dimensional augmented algebra.

A (counital or noncounital) coalgebra is conilpotent if and only if all its finite-dimensional (coaugmented) subcoalgebras are, and if and only if every its element is contained in a conilpotent finite-dimensional subcoalgebra. A finite-dimensional coalgebra $W$ without counit is conilpotent if and only if its dual finite-dimensional algebra $U=W^*$ satisfies $U^n=0$ for $n$ large enough. A finite-dimensional coaugmented coalgebra $C$ is conilpotent if and only if the augmentation ideal $A_+$ of its dual finite-dimensional algebra $A=C^*$ satisfies $A_+^n=0$ for $n$ large enough.

Let $A$ and $B$ be two augmented associative algebras such that $A_+^m=0$ and $B_+^n=0$. The tensor product $A\otimes_k B$ is an augmented associative algebra with the augmentation ideal $$(A\otimes_kB)_+=A_+\otimes B+A\otimes B_+=(A_+\otimes k) \oplus (A_+\otimes B_+) \oplus (k\otimes B_+) \subset A\otimes_kB.$$ Now you can easily see that $(A\otimes_kB)_+^{m+n-1}=0$.

As the duality (anti-equivalence of categories) between finite-dimensional algebras and finite-dimensional coalgebras takes tensor products to tensor products, and as the tensor product of directed unions is the union of tensor products, the desired assertion follows.