The Stacks project

Example 14.34.8. Going back to the example discussed in Example 14.34.5 our Lemma 14.34.3 signifies that for any ring map $A \to B$ the map of simplicial rings

\[ \xymatrix{ A[A[A[B]]] \ar[d] \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & A[A[B]] \ar[d] \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & A[B] \ar[d] \ar@<0ex>[l] \\ B \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & B \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & B \ar@<0ex>[l] } \]

is a homotopy equivalence on underlying simplicial sets. Moreover, the inverse map constructed in Lemma 14.34.3 is in degree $n$ given by

\[ b \longmapsto [\ldots [b]\ldots ] \]

with obvious notation. In the other direction the lemma tells us that for every set $E$ there is a homotopy equivalence

\[ \xymatrix{ A[A[A[A[E]]]] \ar[d] \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & A[A[A[E]]] \ar[d] \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & A[A[E]] \ar[d] \ar@<0ex>[l] \\ A[E] \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & A[E] \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & A[E] \ar@<0ex>[l] } \]

of rings. The inverse map constructed in the lemma is in degree $n$ given by the ring map

\[ \sum a_{e_1,\ldots ,e_ p}[e_1][e_2] \ldots [e_ p] \longmapsto \sum a_{e_1,\ldots ,e_ p}[\ldots [e_1]\ldots ][\ldots [e_2]\ldots ] \ldots [\ldots [e_ p]\ldots ] \]

(with obvious notation).

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