The Stacks project

113.17 Variants of cotangent complexes for schemes

This section gives an alternative construction of the cotangent complex of a morphism of schemes. This section is currently in the obsolete chapter as we can get by with the easier version discussed in Cotangent, Section 90.25 for applications.

Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{C}_{X/Y}$ be the category whose objects are commutative diagrams

113.17.0.1
\begin{equation} \label{obsolete-equation-object} \vcenter { \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_ i & A \ar[ld] \\ Y & V \ar[l] } } \end{equation}

of schemes where

  1. $U$ is an open subscheme of $X$,

  2. $V$ is an open subscheme of $Y$, and

  3. there exists an isomorphism $A = V \times \mathop{\mathrm{Spec}}(P)$ over $V$ where $P$ is a polynomial algebra over $\mathbf{Z}$ (on some set of variables).

In other words, $A$ is an (infinite dimensional) affine space over $V$. Morphisms are given by commutative diagrams.

Notation. An object of $\mathcal{C}_{X/Y}$, i.e., a diagram (113.17.0.1), is often denoted $U \to A$ where it is understood that (a) $U$ is an open subscheme of $X$, (b) $U \to A$ is a morphism over $Y$, (c) the image of the structure morphism $A \to Y$ is an open $V \subset Y$, and (d) $A \to V$ is an affine space. We'll write $U \to A/V$ to indicate $V \subset Y$ is the image of $A \to Y$. Recall that $X_{Zar}$ denotes the small Zariski site $X$. There are forgetful functors

\[ \mathcal{C}_{X/Y} \to X_{Zar},\ (U \to A) \mapsto U \quad \text{and}\quad \mathcal{C}_{X/Y} \mapsto Y_{Zar},\ (U \to A/V) \mapsto V. \]

Lemma 113.17.1. Let $X \to Y$ be a morphism of schemes.

  1. The category $\mathcal{C}_{X/Y}$ is fibred over $X_{Zar}$.

  2. The category $\mathcal{C}_{X/Y}$ is fibred over $Y_{Zar}$.

  3. The category $\mathcal{C}_{X/Y}$ is fibred over the category of pairs $(U, V)$ where $U \subset X$, $V \subset Y$ are open and $f(U) \subset V$.

Proof. Ad (1). Given an object $U \to A$ of $\mathcal{C}_{X/Y}$ and a morphism $U' \to U$ of $X_{Zar}$ consider the object $i' : U' \to A$ of $\mathcal{C}_{X/Y}$ where $i'$ is the composition of $i$ and $U' \to U$. The morphism $(U' \to A) \to (U \to A)$ of $\mathcal{C}_{X/Y}$ is strongly cartesian over $X_{Zar}$.

Ad (2). Given an object $U \to A/V$ and $V' \to V$ we can set $U' = U \cap f^{-1}(V')$ and $A' = V' \times _ V A$ to obtain a strongly cartesian morphism $(U' \to A') \to (U \to A)$ over $V' \to V$.

Ad (3). Denote $(X/Y)_{Zar}$ the category in (3). Given $U \to A/V$ and a morphism $(U', V') \to (U, V)$ in $(X/Y)_{Zar}$ we can consider $A' = V' \times _ V A$. Then the morphism $(U' \to A'/V') \to (U \to A/V)$ is strongly cartesian in $\mathcal{C}_{X/Y}$ over $(X/Y)_{Zar}$. $\square$

We obtain a topology $\tau _ X$ on $\mathcal{C}_{X/Y}$ by using the topology inherited from $X_{Zar}$ (see Stacks, Section 8.10). If not otherwise stated this is the topology on $\mathcal{C}_{X/Y}$ we will consider. To be precise, a family of morphisms $\{ (U_ i \to A_ i) \to (U \to A)\} $ is a covering of $\mathcal{C}_{X/Y}$ if and only if

  1. $U = \bigcup U_ i$, and

  2. $A_ i = A$ for all $i$.

We obtain the same collection of sheaves if we allow $A_ i \cong A$ in (2). The functor $u$ defines a morphism of topoi $\pi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}) \to \mathop{\mathit{Sh}}\nolimits (X_{Zar})$.

The site $\mathcal{C}_{X/Y}$ comes with several sheaves of rings.

  1. The sheaf $\mathcal{O}$ given by the rule $(U \to A) \mapsto \mathcal{O}(A)$.

  2. The sheaf $\underline{\mathcal{O}}_ X = \pi ^{-1}\mathcal{O}_ X$ given by the rule $(U \to A) \mapsto \mathcal{O}(U)$.

  3. The sheaf $\underline{\mathcal{O}}_ Y$ given by the rule $(U \to A/V) \mapsto \mathcal{O}(V)$.

We obtain morphisms of ringed topoi

113.17.1.1
\begin{equation} \label{obsolete-equation-pi-schemes} \vcenter { \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}), \underline{\mathcal{O}}_ X) \ar[r]_ i \ar[d]_\pi & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}), \mathcal{O}) \\ (\mathop{\mathit{Sh}}\nolimits (X_{Zar}), \mathcal{O}_ X) } } \end{equation}

The morphism $i$ is the identity on underlying topoi and $i^\sharp : \mathcal{O} \to \underline{\mathcal{O}}_ X$ is the obvious map. The map $\pi $ is a special case of Cohomology on Sites, Situation 21.37.1. An important role will be played in the following by the derived functors $ Li^* : D(\mathcal{O}) \longrightarrow D(\underline{\mathcal{O}}_ X) $ left adjoint to $Ri_* = i_* : D(\underline{\mathcal{O}}_ X) \to D(\mathcal{O})$ and $ L\pi _! : D(\underline{\mathcal{O}}_ X) \longrightarrow D(\mathcal{O}_ X) $ left adjoint to $\pi ^* = \pi ^{-1} : D(\mathcal{O}_ X) \to D(\underline{\mathcal{O}}_ X)$.

Remark 113.17.2. We obtain a second topology $\tau _ Y$ on $\mathcal{C}_{X/Y}$ by taking the topology inherited from $Y_{Zar}$. There is a third topology $\tau _{X \to Y}$ where a family of morphisms $\{ (U_ i \to A_ i) \to (U \to A)\} $ is a covering if and only if $U = \bigcup U_ i$, $V = \bigcup V_ i$ and $A_ i \cong V_ i \times _ V A$. This is the topology inherited from the topology on the site $(X/Y)_{Zar}$ whose underlying category is the category of pairs $(U, V)$ as in Lemma 113.17.1 part (3). The coverings of $(X/Y)_{Zar}$ are families $\{ (U_ i, V_ i) \to (U, V)\} $ such that $U = \bigcup U_ i$ and $V = \bigcup V_ i$. There are morphisms of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _ X) & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _{X \to Y}) \ar[l] \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _ Y) } \]

(recall that $\tau _ X$ is our “default” topology). The pullback functors for these arrows are sheafification and pushforward is the identity on underlying presheaves. The diagram of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (X_{Zar}) \ar[d]^ f & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}) \ar[l]^\pi & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _{X \to Y}) \ar[l] \ar[d] \\ \mathop{\mathit{Sh}}\nolimits (Y_{Zar}) & & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _ Y) \ar[ll] } \]

is not commutative. Namely, the pullback of a nonzero abelian sheaf on $Y$ is a nonzero abelian sheaf on $(\mathcal{C}_{X/Y}, \tau _{X \to Y})$, but we can certainly find examples where such a sheaf pulls back to zero on $X$. Note that any presheaf $\mathcal{F}$ on $Y_{Zar}$ gives a sheaf $\underline{\mathcal{F}}$ on $\mathcal{C}_{Y/X}$ by the rule which assigns to $(U \to A/V)$ the set $\mathcal{F}(V)$. Even if $\mathcal{F}$ happens to be a sheaf it isn't true in general that $\underline{\mathcal{F}} = \pi ^{-1}f^{-1}\mathcal{F}$. This is related to the noncommutativity of the diagram above, as we can describe $\underline{\mathcal{F}}$ as the pushforward of the pullback of $\mathcal{F}$ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _{X \to Y})$ via the lower horizontal and right vertical arrows. An example is the sheaf $\underline{\mathcal{O}}_ Y$. But what is true is that there is a map $\underline{\mathcal{F}} \to \pi ^{-1}f^{-1}\mathcal{F}$ which is transformed (as we shall see later) into an isomorphism after applying $\pi _!$.


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