The Stacks project

Lemma 50.19.2. There exists a unique rule that to every finite syntomic morphism of schemes $f : Y \to X$ assigns $\mathcal{O}_ X$-module maps

\[ \Theta ^ p_{Y/X} : f_*\Omega ^ p_{Y/\mathbf{Z}} \longrightarrow \Omega ^ p_{X/\mathbf{Z}} \]

satisfying the following properties

  1. the composition with $\Omega ^ p_{X/\mathbf{Z}} \otimes _{\mathcal{O}_ X} f_*\mathcal{O}_ Y \to f_*\Omega ^ p_{Y/\mathbf{Z}}$ is equal to $\text{id} \otimes \text{Trace}_ f$ where $\text{Trace}_ f : f_*\mathcal{O}_ Y \to \mathcal{O}_ X$ is the map from Discriminants, Section 49.3,

  2. the rule is compatible with base change.

Proof. First, assume that $X$ is locally Noetherian. By Lemma 50.18.3 we have a canonical map

\[ c^ p_{Y/X} : \Omega _{Y/S}^ p \longrightarrow f^*\Omega _{X/S}^ p \otimes _{\mathcal{O}_ Y} \det (\mathop{N\! L}\nolimits _{Y/X}) \]

By Discriminants, Proposition 49.13.2 we have a canonical isomorphism

\[ c_{Y/X} : \det (\mathop{N\! L}\nolimits _{Y/X}) \to \omega _{Y/X} \]

mapping $\delta (\mathop{N\! L}\nolimits _{Y/X})$ to $\tau _{Y/X}$. Combined these maps give

\[ c^ p_{Y/X} \otimes c_{Y/X} : \Omega _{Y/S}^ p \longrightarrow f^*\Omega _{X/S}^ p \otimes _{\mathcal{O}_ Y} \omega _{Y/X} \]

By Discriminants, Section 49.5 this is the same thing as a map

\[ \Theta _{Y/X}^ p : f_*\Omega _{Y/S}^ p \longrightarrow \Omega _{X/S}^ p \]

Recall that the relationship between $c^ p_{Y/X} \otimes c_{Y/X}$ and $\Theta _{Y/X}^ p$ uses the evaluation map $f_*\omega _{Y/X} \to \mathcal{O}_ X$ which sends $\tau _{Y/X}$ to $\text{Trace}_ f(1)$, see Discriminants, Section 49.5. Hence property (1) holds. Property (2) holds for base changes by $X' \to X$ with $X'$ locally Noetherian because both $c^ p_{Y/X}$ and $c_{Y/X}$ are compatible with such base changes. For $f : Y \to X$ finite syntomic and $X$ locally Noetherian, we will continue to denote $\Theta ^ p_{Y/X}$ the solution we've just found.

Uniqueness. Suppose that we have a finite syntomic morphism $f: Y \to X$ such that $X$ is smooth over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and $f$ is étale over a dense open of $X$. We claim that in this case $\Theta ^ p_{Y/X}$ is uniquely determined by property (1). Namely, consider the maps

\[ \Omega ^ p_{X/\mathbf{Z}} \otimes _{\mathcal{O}_ X} f_*\mathcal{O}_ Y \to f_*\Omega ^ p_{Y/\mathbf{Z}} \to \Omega ^ p_{X/\mathbf{Z}} \]

The sheaf $\Omega ^ p_{X/\mathbf{Z}}$ is torsion free (by the assumed smoothness), hence it suffices to check that the restriction of $\Theta ^ p_{Y/X}$ is uniquely determined over the dense open over which $f$ is étale, i.e., we may assume $f$ is étale. However, if $f$ is étale, then $f^*\Omega _{X/\mathbf{Z}} = \Omega _{Y/\mathbf{Z}}$ hence the first arrow in the displayed equation is an isomorphism. Since we've pinned down the composition, this guarantees uniqueness.

Let $f : Y \to X$ be a finite syntomic morphism of locally Noetherian schemes. Let $x \in X$. By Discriminants, Lemma 49.11.7 we can find $d \geq 1$ and a commutative diagram

\[ \xymatrix{ Y \ar[d] & V \ar[d] \ar[l] \ar[r] & V_ d \ar[d] \\ X & U \ar[l] \ar[r] & U_ d } \]

such that $x \in U \subset X$ is open, $V = f^{-1}(U)$ and $V = U \times _{U_ d} V_ d$. Thus $\Theta ^ p_{Y/X}|_ V$ is the pullback of the map $\Theta ^ p_{V_ d/U_ d}$. However, by the discussion on uniqueness above and Discriminants, Lemmas 49.11.4 and 49.11.5 the map $\Theta ^ p_{V_ d/U_ d}$ is uniquely determined by the requirement (1). Hence uniqueness holds.

At this point we know that we have existence and uniqueness for all finite syntomic morphisms $Y \to X$ with $X$ locally Noetherian. We could now give an argument similar to the proof of Lemma 50.18.3 to extend to general $X$. However, instead it possible to directly use absolute Noetherian approximation to finish the proof. Namely, to construct $\Theta ^ p_{Y/X}$ it suffices to do so Zariski locally on $X$ (provided we also show the uniqueness). Hence we may assume $X$ is affine (small detail omitted). Then we can write $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ as the limit over a directed set $I$ of Noetherian affine schemes. By Algebra, Lemma 10.127.8 we can find $0 \in I$ and a finitely presented morphism of affines $f_0 : Y_0 \to X_0$ whose base change to $X$ is $Y \to X$. After increasing $0$ we may assume $Y_0 \to X_0$ is finite and syntomic, see Algebra, Lemma 10.168.9 and 10.168.3. For $i \geq 0$ also the base change $f_ i : Y_ i = Y_0 \times _{X_0} X_ i \to X_ i$ is finite syntomic. Then

\[ \Gamma (X, f_*\Omega ^ p_{Y/\mathbf{Z}}) = \Gamma (Y, \Omega ^ p_{Y/\mathbf{Z}}) = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \Gamma (Y_ i, \Omega ^ p_{Y_ i/\mathbf{Z}}) = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \Gamma (X_ i, f_{i, *}\Omega ^ p_{Y_ i/\mathbf{Z}}) \]

Hence we can (and are forced to) define $\Theta ^ p_{Y/X}$ as the colimit of the maps $\Theta ^ p_{Y_ i/X_ i}$. This map is compatible with any cartesian diagram

\[ \xymatrix{ Y' \ar[r] \ar[d] & Y \ar[d] \\ X' \ar[r] & X } \]

with $X'$ affine as we know this for the case of Noetherian affine schemes by the arguments given above (small detail omitted; hint: if we also write $X' = \mathop{\mathrm{lim}}\nolimits _{j \in J} X'_ j$ then for every $i \in I$ there is a $j \in J$ and a morphism $X'_ j \to X_ i$ compatible with the morphism $X' \to X$). This finishes the proof. $\square$

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