# The Stacks Project

## Tag 08X5

Definition 34.4.19. Define the functor $f_*$ $: DD_{S/R} \to \text{Mod}_R$ by taking $f_*(M, \theta)$ to be the $R$-submodule of $M$ for which the diagram $$\tag{34.4.19.1} \xymatrix@C=8pc{f_*(M,\theta) \ar[r] & M \ar@<1ex>^{\theta \circ (1_M \otimes \delta_0^1)}[r] \ar@<-1ex>_{1_M \otimes \delta_1^1}[r] & M \otimes_{S, \delta_1^1} S_2 }$$ is an equalizer.

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 1182–1194 (see updates for more information).

\begin{definition}
\label{definition-pushforward}
Define the functor {\it $f_*$} $: DD_{S/R} \to \text{Mod}_R$ by taking
$f_*(M, \theta)$ to be the $R$-submodule of $M$ for which the diagram

\label{equation-equalizer-f}
\xymatrix@C=8pc{f_*(M,\theta) \ar[r] & M \ar@<1ex>^{\theta \circ (1_M \otimes
\delta_0^1)}[r] \ar@<-1ex>_{1_M \otimes \delta_1^1}[r] &
M \otimes_{S, \delta_1^1} S_2
}

is an equalizer.
\end{definition}

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