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 92.25 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
of schemes where
$U$ is an open subscheme of $X$,
$V$ is an open subscheme of $Y$, and
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 (115.19.0.1), 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
Lemma 115.19.1. Let $X \to Y$ be a morphism of schemes.
The category $\mathcal{C}_{X/Y}$ is fibred over $X_{Zar}$.
The category $\mathcal{C}_{X/Y}$ is fibred over $Y_{Zar}$.
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
$U = \bigcup U_ i$, and
$A_ i = A$ 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.
The sheaf $\mathcal{O}$ given by the rule $(U \to A) \mapsto \mathcal{O}(A)$.
The sheaf $\underline{\mathcal{O}}_ X = \pi ^{-1}\mathcal{O}_ X$ given by the rule $(U \to A) \mapsto \mathcal{O}(U)$.
The sheaf $\underline{\mathcal{O}}_ Y$ given by the rule $(U \to A/V) \mapsto \mathcal{O}(V)$.
We obtain morphisms of ringed topoi
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 21.38.1. 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 115.19.2. 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 115.19.1 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 (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 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 $\pi _!$.
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _ X) & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _{X \to Y}) \ar[l] \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _ Y) } \]
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (X_{Zar}) \ar[d]^ f & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}) \ar[l]^\pi & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _{X \to Y}) \ar[l] \ar[d] \\ \mathop{\mathit{Sh}}\nolimits (Y_{Zar}) & & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _ Y) \ar[ll] } \]
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)