## 60.16 Cosimplicial algebra

This section should be moved somewhere else. A cosimplicial ring is a cosimplicial object in the category of rings. Given a ring $R$, a cosimplicial $R$-algebra is a cosimplicial object in the category of $R$-algebras. A cosimplicial ideal in a cosimplicial ring $A_*$ is given by an ideal $I_ n \subset A_ n$ for all $n$ such that $A(f)(I_ n) \subset I_ m$ for all $f : [n] \to [m]$ in $\Delta$.

Let $A_*$ be a cosimplicial ring. Let $\mathcal{C}$ be the category of pairs $(A, M)$ where $A$ is a ring and $M$ is a module over $A$. A morphism $(A, M) \to (A', M')$ consists of a ring map $A \to A'$ and an $A$-module map $M \to M'$ where $M'$ is viewed as an $A$-module via $A \to A'$ and the $A'$-module structure on $M'$. Having said this we can define a cosimplicial module $M_*$ over $A_*$ as a cosimplicial object $(A_*, M_*)$ of $\mathcal{C}$ whose first entry is equal to $A_*$. A homomorphism $\varphi _* : M_* \to N_*$ of cosimplicial modules over $A_*$ is a morphism $(A_*, M_*) \to (A_*, N_*)$ of cosimplicial objects in $\mathcal{C}$ whose first component is $1_{A_*}$.

A homotopy between homomorphisms $\varphi _*, \psi _* : M_* \to N_*$ of cosimplicial modules over $A_*$ is a homotopy between the associated maps $(A_*, M_*) \to (A_*, N_*)$ whose first component is the trivial homotopy (dual to Simplicial, Example 14.26.3). We spell out what this means. Such a homotopy is a homotopy

$h : M_* \longrightarrow \mathop{\mathrm{Hom}}\nolimits (\Delta [1], N_*)$

between $\varphi _*$ and $\psi _*$ as homomorphisms of cosimplicial abelian groups such that for each $n$ the map $h_ n : M_ n \to \prod _{\alpha \in \Delta [1]_ n} N_ n$ is $A_ n$-linear. The following lemma is a version of Simplicial, Lemma 14.28.4 for cosimplicial modules.

Lemma 60.16.1. Let $A_*$ be a cosimplicial ring. Let $\varphi _*, \psi _* : K_* \to M_*$ be homomorphisms of cosimplicial $A_*$-modules.

1. If $\varphi _*$ and $\psi _*$ are homotopic, then

$\varphi _* \otimes 1, \psi _* \otimes 1 : K_* \otimes _{A_*} L_* \longrightarrow M_* \otimes _{A_*} L_*$

are homotopic for any cosimplicial $A_*$-module $L_*$.

2. If $\varphi _*$ and $\psi _*$ are homotopic, then

$\wedge ^ i(\varphi _*), \wedge ^ i(\psi _*) : \wedge ^ i(K_*) \longrightarrow \wedge ^ i(M_*)$

are homotopic.

3. If $\varphi _*$ and $\psi _*$ are homotopic, and $A_* \to B_*$ is a homomorphism of cosimplicial rings, then

$\varphi _* \otimes 1, \psi _* \otimes 1 : K_* \otimes _{A_*} B_* \longrightarrow M_* \otimes _{A_*} B_*$

are homotopic as homomorphisms of cosimplicial $B_*$-modules.

4. If $I_* \subset A_*$ is a cosimplicial ideal, then the induced maps

$\varphi ^\wedge _*, \psi ^\wedge _* : K_*^\wedge \longrightarrow M_*^\wedge$

between completions are homotopic.

5. Add more here as needed, for example symmetric powers.

Proof. Let $h : M_* \longrightarrow \mathop{\mathrm{Hom}}\nolimits (\Delta [1], N_*)$ be the given homotopy. In degree $n$ we have

$h_ n = (h_{n, \alpha }) : K_ n \longrightarrow \prod \nolimits _{\alpha \in \Delta [1]_ n} K_ n$

see Simplicial, Section 14.28. In order for a collection of $h_{n, \alpha }$ to form a homotopy, it is necessary and sufficient if for every $f : [n] \to [m]$ we have

$h_{m, \alpha } \circ M_*(f) = N_*(f) \circ h_{n, \alpha \circ f}$

see Simplicial, Equation (14.28.1.1). We also should have that $\psi _ n = h_{n, 0 : [n] \to [1]}$ and $\varphi _ n = h_{n, 1 : [n] \to [1]}$.

In each of the cases of the lemma we can produce the corresponding maps. Case (1). We can use the homotopy $h \otimes 1$ defined in degree $n$ by setting

$(h \otimes 1)_{n, \alpha } = h_{n, \alpha } \otimes 1_{L_ n} : K_ n \otimes _{A_ n} L_ n \longrightarrow M_ n \otimes _{A_ n} L_ n.$

Case (2). We can use the homotopy $\wedge ^ ih$ defined in degree $n$ by setting

$\wedge ^ i(h)_{n, \alpha } = \wedge ^ i(h_{n, \alpha }) : \wedge _{A_ n}(K_ n) \longrightarrow \wedge ^ i_{A_ n}(M_ n).$

Case (3). We can use the homotopy $h \otimes 1$ defined in degree $n$ by setting

$(h \otimes 1)_{n, \alpha } = h_{n, \alpha } \otimes 1 : K_ n \otimes _{A_ n} B_ n \longrightarrow M_ n \otimes _{A_ n} B_ n.$

Case (4). We can use the homotopy $h^\wedge$ defined in degree $n$ by setting

$(h^\wedge )_{n, \alpha } = h_{n, \alpha }^\wedge : K_ n^\wedge \longrightarrow M_ n^\wedge .$

This works because each $h_{n, \alpha }$ is $A_ n$-linear. $\square$

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