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.
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_*.
If \varphi _* and \psi _* are homotopic, then
\wedge ^ i(\varphi _*), \wedge ^ i(\psi _*) : \wedge ^ i(K_*) \longrightarrow \wedge ^ i(M_*)
are homotopic.
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.
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.
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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