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 _!.
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