90.4 Simplicial resolutions and derived lower shriek

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

$\xymatrix{ \mathop{\mathrm{Spec}}(B) \ar[d] \ar[r] & \mathop{\mathrm{Spec}}(P) \ar[ld] \\ \mathop{\mathrm{Spec}}(A) }$

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

90.4.0.1
$$\label{cotangent-equation-pi} \vcenter { \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \underline{B}) \ar[r]_ i \ar[d]_\pi & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \\ (\mathop{\mathit{Sh}}\nolimits (*), B) } }$$

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 21.38.1. 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 90.4.1. 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

$L\pi _!(\mathcal{F}) = \mathcal{F}(P_\bullet , \epsilon )$

in $D(\textit{Ab})$, resp. $D(B)$ functorially in $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, resp. $\textit{Mod}(\underline{B})$.

Proof. We will use the criterion of Cohomology on Sites, Lemma 21.38.7 to prove this. Given an object $U = (Q, \beta )$ of $\mathcal{C}$ we have to show that

$S_\bullet = \mathop{Mor}\nolimits _\mathcal {C}((Q, \beta ), (P_\bullet , \epsilon ))$

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

$S_\bullet = \mathop{Mor}\nolimits _{\textit{Sets}}((E, \beta |_ E), (P_\bullet , \epsilon ))$

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

$\xymatrix{ F_{b, \bullet } \ar[r] \ar[d] & P_\bullet \ar[d]_\epsilon \\ {*} \ar[r]^ b & B }$

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 90.4.2. 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 (90.4.0.1). Then

$L\pi _!(\mathcal{F}) = \mathcal{F}(P_\bullet , \epsilon )$

in $D(\textit{Ab})$, resp. $D(B)$ functorially in $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, resp. $\textit{Mod}(\underline{B})$.

First proof. We will apply Lemma 90.4.1. 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 14.33.5 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 21.38.7. Let $U = (Q, \beta )$ be an object of $\mathcal{C}$. We have to show that

$S_\bullet = \mathop{Mor}\nolimits _\mathcal {C}((Q, \beta ), (P_\bullet , \epsilon ))$

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 90.3.3 we see that

$S_\bullet = \mathop{Mor}\nolimits _\mathcal {S}((E \to B), i(P_\bullet \to B))$

By Simplicial, Lemma 14.33.5 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

$\mathop{Mor}\nolimits _\mathcal {S}((E \to B), (B \to B)) = \{ *\}$

as desired. $\square$

Lemma 90.4.3. Let $A \to B$ be a ring map. Let $\pi$ and $i$ be as in (90.4.0.1). There is a canonical isomorphism

$L_{B/A} = L\pi _!(Li^*\Omega _{\mathcal{O}/A}) = L\pi _!(i^*\Omega _{\mathcal{O}/A}) = L\pi _!(\Omega _{\mathcal{O}/A} \otimes _\mathcal {O} \underline{B})$

in $D(B)$.

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 90.4.2 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 90.3.2). $\square$

Proof. This follows from Lemma 90.4.1 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 90.4.5. 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 90.4.3. 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

$H^0(L_{B/A}) = H^0(L\pi _!(\Omega _{\mathcal{O}/A} \otimes \underline{B})) = \pi _!(\Omega _{\mathcal{O}/A} \otimes \underline{B}) \to \Omega _{B/A}$

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 10.130.6). 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 10.130.9). 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 21.38.1) $\pi _!(\Omega _{\mathcal{O}/A} \otimes \underline{B})$ as desired. $\square$

Lemma 90.4.6. If $B$ is a polynomial algebra over the ring $A$, then with $\pi$ as in (90.4.0.1) we have that $\pi _!$ is exact and $\pi _!\mathcal{F} = \mathcal{F}(B \to B)$.

Proof. This follows from Lemma 90.4.1 which tells us the constant simplicial algebra on $B$ can be used to compute $L\pi _!$. $\square$

Lemma 90.4.7. If $B$ is a polynomial algebra over the ring $A$, then $L_{B/A}$ is quasi-isomorphic to $\Omega _{B/A}[0]$.

[1] It suffices to consider sets of cardinality at most the cardinality of $B$.

Comment #4567 by Anonymous on

In the definition of the category $\mathcal{C}_{B/A}$, i suppose that a morphisms between two objects $P\rightarrow B$, $P'\rightarrow B$ is just an $A$-map $P\rightarrow P'$. Is it suppsoed to commute with the structure maps?

Comment #4758 by on

Well, if you have a category then you also have a category of arrows, so I think it was clear. But since this category is used a lot it makes sense to be very precise, so I've elucidated the definition. See changes here.

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