Let $A \to B$ be a ring map. Consider the category whose objects are $A$-algebra maps $\alpha : P \to B$ where $P$ is a polynomial algebra over $A$ (in some set1 of variables) and whose morphisms $s : (\alpha : P \to B) \to (\alpha ' : P' \to B)$ are $A$-algebra homomorphisms $s : P \to P'$ with $\alpha ' \circ s = \alpha $. Let $\mathcal{C} = \mathcal{C}_{B/A}$ denote the opposite of this category. The reason for taking the opposite is that we want to think of objects $(P, \alpha )$ as corresponding to the diagram of affine schemes
We endow $\mathcal{C}$ with the chaotic topology (Sites, Example 7.6.6), i.e., we endow $\mathcal{C}$ with the structure of a site where coverings are given by identities so that all presheaves are sheaves. Moreover, we endow $\mathcal{C}$ with two sheaves of rings. The first is the sheaf $\mathcal{O}$ which sends to object $(P, \alpha )$ to $P$. Then second is the constant sheaf $B$, which we will denote $\underline{B}$. We obtain the following diagram of morphisms of ringed topoi
The morphism $i$ is the identity on underlying topoi and $i^\sharp : \mathcal{O} \to \underline{B}$ is the obvious map. The map $\pi $ is as in Cohomology on Sites, Example 21.39.1. An important role will be played in the following by the derived functors $ Li^* : D(\mathcal{O}) \longrightarrow D(\underline{B}) $ left adjoint to $Ri_* = i_* : D(\underline{B}) \to D(\mathcal{O})$ and $ L\pi _! : D(\underline{B}) \longrightarrow D(B) $ left adjoint to $\pi ^* = \pi ^{-1} : D(B) \to D(\underline{B})$.
Lemma 92.4.1. With notation as above let $P_\bullet $ be a simplicial $A$-algebra endowed with an augmentation $\epsilon : P_\bullet \to B$. Assume each $P_ n$ is a polynomial algebra over $A$ and $\epsilon $ is a trivial Kan fibration on underlying simplicial sets. Then in $D(\textit{Ab})$, resp. $D(B)$ functorially in $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, resp. $\textit{Mod}(\underline{B})$.
\[ L\pi _!(\mathcal{F}) = \mathcal{F}(P_\bullet , \epsilon ) \]
Proof. We will use the criterion of Cohomology on Sites, Lemma 21.39.7 to prove this. Given an object $U = (Q, \beta )$ of $\mathcal{C}$ we have to show that
is homotopy equivalent to a singleton. Write $Q = A[E]$ for some set $E$ (this is possible by our choice of the category $\mathcal{C}$). We see that
Let $*$ be the constant simplicial set on a singleton. For $b \in B$ let $F_{b, \bullet }$ be the simplicial set defined by the cartesian diagram
With this notation $S_\bullet = \prod _{e \in E} F_{\beta (e), \bullet }$. Since we assumed $\epsilon $ is a trivial Kan fibration we see that $F_{b, \bullet } \to *$ is a trivial Kan fibration (Simplicial, Lemma 14.30.3). Thus $S_\bullet \to *$ is a trivial Kan fibration (Simplicial, Lemma 14.30.6). Therefore $S_\bullet $ is homotopy equivalent to $*$ (Simplicial, Lemma 14.30.8). $\square$
In particular, we can use the standard resolution of $B$ over $A$ to compute derived lower shriek.
Lemma 92.4.2. Let $A \to B$ be a ring map. Let $\epsilon : P_\bullet \to B$ be the standard resolution of $B$ over $A$. Let $\pi $ be as in (92.4.0.1). Then in $D(\textit{Ab})$, resp. $D(B)$ functorially in $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, resp. $\textit{Mod}(\underline{B})$.
\[ L\pi _!(\mathcal{F}) = \mathcal{F}(P_\bullet , \epsilon ) \]
First proof. We will apply Lemma 92.4.1. Since the terms $P_ n$ are polynomial algebras we see the first assumption of that lemma is satisfied. The second assumption is proved as follows. By Simplicial, Lemma 14.34.3 the map $\epsilon $ is a homotopy equivalence of underlying simplicial sets. By Simplicial, Lemma 14.31.9 this implies $\epsilon $ induces a quasi-isomorphism of associated complexes of abelian groups. By Simplicial, Lemma 14.31.8 this implies that $\epsilon $ is a trivial Kan fibration of underlying simplicial sets. $\square$
Second proof. We will use the criterion of Cohomology on Sites, Lemma 21.39.7. Let $U = (Q, \beta )$ be an object of $\mathcal{C}$. We have to show that
is homotopy equivalent to a singleton. Write $Q = A[E]$ for some set $E$ (this is possible by our choice of the category $\mathcal{C}$). Using the notation of Remark 92.3.3 we see that
By Simplicial, Lemma 14.34.3 the map $i(P_\bullet \to B) \to i(B \to B)$ is a homotopy equivalence in $\mathcal{S}$. Hence $S_\bullet $ is homotopy equivalent to
as desired. $\square$
Lemma 92.4.3. Let $A \to B$ be a ring map. Let $\pi $ and $i$ be as in (92.4.0.1). There is a canonical isomorphism in $D(B)$.
\[ L_{B/A} = L\pi _!(Li^*\Omega _{\mathcal{O}/A}) = L\pi _!(i^*\Omega _{\mathcal{O}/A}) = L\pi _!(\Omega _{\mathcal{O}/A} \otimes _\mathcal {O} \underline{B}) \]
Proof. For an object $\alpha : P \to B$ of the category $\mathcal{C}$ the module $\Omega _{P/A}$ is a free $P$-module. Thus $\Omega _{\mathcal{O}/A}$ is a flat $\mathcal{O}$-module. Hence $Li^*\Omega _{\mathcal{O}/A} = i^*\Omega _{\mathcal{O}/A}$ is the sheaf of $\underline{B}$-modules which associates to $\alpha : P \to A$ the $B$-module $\Omega _{P/A} \otimes _{P, \alpha } B$. By Lemma 92.4.2 we see that the right hand side is computed by the value of this sheaf on the standard resolution which is our definition of the left hand side (Definition 92.3.2). $\square$
Lemma 92.4.4. If $A \to B$ is a ring map, then $L\pi _!(\pi ^{-1}M) = M$ with $\pi $ as in (92.4.0.1).
Proof. This follows from Lemma 92.4.1 which tells us $L\pi _!(\pi ^{-1}M)$ is computed by $(\pi ^{-1}M)(P_\bullet , \epsilon )$ which is the constant simplicial object on $M$. $\square$
Lemma 92.4.5. If $A \to B$ is a ring map, then $H^0(L_{B/A}) = \Omega _{B/A}$.
Proof. We will prove this by a direct calculation. We will use the identification of Lemma 92.4.3. There is clearly a map from $\Omega _{\mathcal{O}/A} \otimes \underline{B}$ to the constant sheaf with value $\Omega _{B/A}$. Thus this map induces a map
By choosing an object $P \to B$ of $\mathcal{C}_{B/A}$ with $P \to B$ surjective we see that this map is surjective (by Algebra, Lemma 10.131.6). To show that it is injective, suppose that $P \to B$ is an object of $\mathcal{C}_{B/A}$ and that $\xi \in \Omega _{P/A} \otimes _ P B$ is an element which maps to zero in $\Omega _{B/A}$. We first choose factorization $P \to P' \to B$ such that $P' \to B$ is surjective and $P'$ is a polynomial algebra over $A$. We may replace $P$ by $P'$. If $B = P/I$, then the kernel $\Omega _{P/A} \otimes _ P B \to \Omega _{B/A}$ is the image of $I/I^2$ (Algebra, Lemma 10.131.9). Say $\xi $ is the image of $f \in I$. Then we consider the two maps $a, b : P' = P[x] \to P$, the first of which maps $x$ to $0$ and the second of which maps $x$ to $f$ (in both cases $P[x] \to B$ maps $x$ to zero). We see that $\xi $ and $0$ are the image of $\text{d}x \otimes 1$ in $\Omega _{P'/A} \otimes _{P'} B$. Thus $\xi $ and $0$ have the same image in the colimit (see Cohomology on Sites, Example 21.39.1) $\pi _!(\Omega _{\mathcal{O}/A} \otimes \underline{B})$ as desired. $\square$
Lemma 92.4.6. If $B$ is a polynomial algebra over the ring $A$, then with $\pi $ as in (92.4.0.1) we have that $\pi _!$ is exact and $\pi _!\mathcal{F} = \mathcal{F}(B \to B)$.
Proof. This follows from Lemma 92.4.1 which tells us the constant simplicial algebra on $B$ can be used to compute $L\pi _!$. $\square$
Lemma 92.4.7. If $B$ is a polynomial algebra over the ring $A$, then $L_{B/A}$ is quasi-isomorphic to $\Omega _{B/A}[0]$.
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