Cohomology of presheaves with oriented weak transfers

Let $(f,\alpha):X\hookrightarrow Y\times\mathbb{A}^{n}$ and $(f,\beta):X\hookrightarrow Y\times\mathbb{A}^{m}$ be $N$-trivial embeddings. Consider the closed immersion $\iota:\mathbb{A}^{1}\times X\to\mathbb{A}^{1}\times Y\times\mathbb{A}^{n}\times\mathbb{A}^{m}$ defined by $\iota(t,x)=(t,f(x),t\alpha,(1-t)\beta)$. Since both $\mathbb{A}^{1}\times X$ and $\mathbb{A}^{1}\times Y$ have trivial sheaf of Kähler differentials, Lemma A.1 implies that the product of $\iota$ with a constant morphism $\mathbb{A}^{1}\times X\to\mathbb{A}^{\dim Y+1}$ is an $N$-trivial embedding. We denote the product morphism by $(\iota,c)$.

Homotopy invariance implies the maps $i_{0}^{*},i_{1}^{*}:\cal{F}(\mathbb{A}^{1}\times X)\to\cal{F}(X)$ are both inverse to the isomorphism induced by the projection, hence are equal. Now the compatibility of the weak transfers with the inclusions $0\times Y,1\times Y\hookrightarrow\mathbb{A}^{1}\times Y$ implies $(1\times f,\iota)_{*}=(f,\iota|_{0\times X})_{*}=(f,\iota|_{1\times X})_{*}$.

We have $\iota(0,x)=(0,f(x),0,\beta,c)$. Since the weak transfers are compatible with the addition of irrelevant summands, we have $(f,\iota|_{0\times X})_{*}=(f,\beta)_{*}$. Similarly, since $\iota(1,x)=(1,f(x),\alpha,0,c)$ we have $(f,\iota|_{1\times X})_{*}=(f,\alpha)_{*}$. ∎

The compatibility with the transverse squares determined by the embeddings $0\times S,1\times S\hookrightarrow\mathbb{A}^{1}\times S$ implies that $i_{0}^{\ast}\circ f_{\ast}={f_{0}}_{\ast}\circ{g_{0}}^{\ast}:\cal{F}(X)\to\cal{F}(S)$ (and similarly for the fiber at $1\times S$). Since $i_{0}^{*}=i_{1}^{*}$, the result follows. ∎

One finds an element $\alpha\in I\subset A$ with suitable vanishing properties. The main point is that for almost all $r\in k^{\times}$, the element $s:=\alpha-rt^{N}$ (here $N$ is large compared to the degrees of the coefficients $p_{i}(t)$ of an equation expressing the integral dependence of $\alpha$ on $R[t]$; see the proof in [18] for further explanation of the notation) satisfies the conclusions. Then for almost all $r\in k^{\times}$, the element $s$ will have the required smoothness and étaleness properties.

The ideal $I$ is locally principal since it is a section to smooth morphism of relative dimension 1. Locally on $\operatorname{Spec}A$, we can write $\alpha=a\alpha^{\prime},t=a^{l}t^{\prime}$, with $(a)=I$ and $\alpha^{\prime},t^{\prime}\notin I$. By factoring out a suitable power of $a$ we obtain a local defining equation for the residual part $\operatorname{Spec}(A/J)$ of $V(s)$. By Bertini’s theorem, $V(s)$ is regular for most choices of $r\in k^{\times}$. Since there is a finite cover of $\operatorname{Spec}A$ on which $I$ is principal, we can find an $r$ which works uniformly. Similarly, most choices of $r$ will produce a regular fiber at $s=1$. The purity of the branch locus, together with the assumed $R$-smoothness along $A/I$, implies the resulting finite extensions are generically étale. ∎

We can find a neighborhood of $S$ such that $\Omega^{1}_{X/k}\cong\cal{O}^{d}_{X}$. ∎

The presheaves $\ker\varphi$ and $\operatorname{coker}\varphi$ inherit structures of homotopy invariant presheaves with weak transfers for affine varieties, so Theorem 3.2 applies to them. Since sheafification is exact it suffices to show the Zariski sheafification of the kernel and cokernel both vanish. By hypothesis we know they vanish on fields, hence their stalks vanish by Theorem 3.2. ∎

By Quillen’s trick [23], we can find a finite surjective morphism $p:X\to\mathbb{A}^{d}$ such that composing with the projection away from the (say) first factor gives a morphism $X\to\mathbb{A}^{d-1}$ of relative dimension 1, smooth at $S$, and so that $Z$ is finite over $\mathbb{A}^{d-1}$. For any open $U\subset X$ (which we will assume contains $S$), by applying $-\times_{\mathbb{A}^{d-1}}U$, we obtain the following diagram:

We have an exact sequence of sheaves $0\to p^{\ast}(\Omega^{1}_{\mathbb{A}^{d-1}/k})\to\Omega^{1}_{X/k}\to\Omega^{1}_{X/\mathbb{A}^{d-1}}\to 0$, all of which are locally free near $S$. Since $\Omega^{1}_{\mathbb{A}^{d-1}/k}\cong{\cal{O}}^{d-1}_{\mathbb{A}^{d-1}}$, taking determinants shows $\Omega^{1}_{X/\mathbb{A}^{d-1}}\cong\cal{O}_{X}$. As $X$ is affine, the sequence also gives $\Omega^{1}_{X/k}\cong{\cal{O}}^{d}_{X}$. (Hence also $\Omega^{1}_{U/k}\cong{\cal{O}}^{d}_{U}$.)

By the geometric presentation lemma (Lemma 3.1), there is a morphism $X\times_{\mathbb{A}^{d-1}}U\to\mathbb{A}^{1}$ such that the induced morphism $\pi:X\times_{\mathbb{A}^{d-1}}U\to U\times\mathbb{A}^{1}$ of $U$-schemes has the following properties:

1. (1)

   $\pi$ is finite (hence flat, since source and target are regular of the same dimension);

2. (2)

   $\pi^{-1}(U\times 0)=\Delta(U)\coprod R_{0}$;

3. (3)

   $\delta:\Delta(U)\to U$ étale;

4. (4)

   $R_{0}$ is regular (hence smooth), $R_{0}\subset(X\setminus Z)\times_{\mathbb{A}^{d-1}}U$, and $r_{0}:R_{0}\to U$ is generically étale; and

5. (5)

   $\pi^{-1}(U\times 1)=:F_{1}$ is regular (hence smooth), $F_{1}\subset(X\setminus Z)\times_{\mathbb{A}^{d-1}}U$, and $f_{1}:F_{1}\to U$ generically étale.

We claim the morphism $\pi$ admits an $N$-trivial embedding. By Lemma A.2 it suffices to show $X\times_{\mathbb{A}^{d-1}}U\to U$ admits an $N$-trivial embedding. Since $U\to\mathbb{A}^{d-1}$ is smooth, by Lemma A.3, it suffices to show $X\to\mathbb{A}^{d-1}$ admits an $N$-trivial embedding. This is a consequence of Lemma A.1. We choose such an embedding. The geometric situation is summarized in the following diagram.

Hence we have a commutative diagram in which the paired arrows are inverse isomorphisms, and the unlabeled arrows pointing down are induced by the base change $0\hookrightarrow\mathbb{A}^{1}$:

We omit the choices of $N$-trivial embedding and trivializations in the notation, with the understanding that the embeddings are induced from an $N$-trivial embedding of $\pi$, and whichever trivialization is used to define $\pi_{*}$ is also used to define $\delta_{*},{r_{0}}_{*},{f_{1}}_{*}$. Note that $\delta_{\ast}\circ{pr}_{\cal{F}(\Delta)}(i_{0}^{*})\circ p^{*}_{1}=j_{U}^{*}:\cal{F}(X)\to\cal{F}(U)$.

We can consider the similar diagram induced by $1\hookrightarrow\mathbb{A}^{1}$, replacing $\cal{F}(\Delta)\oplus\cal{F}(R_{0})$ with $\cal{F}(F_{1})$. Then we conclude $(\delta_{\ast}+{r_{0}}_{\ast})\circ i_{0}^{*}\circ p^{*}_{1}={f_{1}}_{\ast}\circ i_{1}^{*}\circ p^{*}_{1}:\cal{F}(X)\to\cal{F}(U)$, as both are equal to ${p^{U}_{1}}^{*}*\circ\pi_{\ast}\circ i^{*}\circ p^{*}_{13}\circ{p^{X}_{1}}^{*}$.

Since $R_{0}$ and $F_{1}$ avoid $Z$, we have a commutative diagram (for $C=R_{0}$ or $F_{1}$, $c=r_{0},f_{1}$):

We conclude the map $({f_{1}}_{\ast}\circ i_{1}^{*}\circ p^{*}_{1})-({r_{0}}_{\ast}\circ{pr}_{\cal{F}(R_{0})}(i_{0}^{*})\circ p^{*}_{1}:\cal{F}(X)\to\cal{F}(U)$ factors through $j_{X\setminus Z}^{*}$. By our previous calculations, this map is exactly $j_{U}^{*}$.

Observe we only used transfers along the finite generically étale morphism $\pi:X\times_{\mathbb{A}^{d-1}}U\to U\times\mathbb{A}^{1}$ and its base changes along $0,1\hookrightarrow\mathbb{A}^{1}$, which are again generically étale by purity of the branch locus. ∎

We use the standard coordinates $X_{0},X_{1}$ on $\mathbb{P}^{1}$, so that $\infty\hookrightarrow\mathbb{P}^{1}$ is defined by the vanishing of $X_{0}$, and $X_{1}\over X_{0}$ is the canonical coordinate on $\mathbb{A}^{1}_{X_{0}\neq 0}\subset\mathbb{P}^{1}$. Let $u$ denote the canonical coordinate on $U$. Let $F$ denote a homogeneous form generating the ideal of $\mathbb{A}^{1}-U\hookrightarrow\mathbb{P}^{1}$ with the reduced scheme structure, i.e., $F\in\Gamma(\mathbb{P}^{1},\cal{O}_{\mathbb{P}^{1}}(\mathbb{A}^{1}-U))$. Let $G$ denote a form generating the ideal of $U-V\hookrightarrow\mathbb{P}^{1}$ with the reduced structure, and let $s_{\Delta}=X_{1}-uX_{0}$ denote the canonical section of the diagonal bundle $\cal{O}(\Delta_{U})$ on $U\times\mathbb{P}^{1}$. By rescaling we may assume $F(X_{0},X_{1})=X_{1}^{\deg F}+\ldots$ and $G(X_{0},X_{1})=X_{1}^{\deg G}+\ldots$. We let $f,g$ denote the dehomogenizations of $F,G$.

Let $N=\deg(FG)+1$. The zero scheme of the section $s_{\Delta}F(X)G(X)+X_{0}^{N}\in H^{0}(U\times\mathbb{P}^{1},\cal{O}(N))$ has the following properties: it is disjoint from $\Delta_{U}$; it is disjoint from $U\times(\mathbb{P}^{1}-V)$; it is the graph of a function $V\to\mathbb{A}^{1}$, hence it is smooth; and it is $U$-finite.

We set $s_{0}:=s_{\Delta}(s_{\Delta}FG+X_{0}^{N})\in H^{0}(U\times\mathbb{P}^{1},L)$. Here $L=\cal{O}(\Delta_{U})\otimes\cal{O}(N)\cong\cal{O}(N+1)$. With this choice of $s_{0}$, properties (1) and (2) are satisfied. We write $s_{R_{0}}:=s_{\Delta}FG+X_{0}^{N}$.

Now we find sections $s_{1}\in H^{0}(U\times\mathbb{P}^{1},L)$ satisfying (3)-(5). The method used here is explicit but does not readily generalize. There exist $a(x),b(x)\in k[x]$ such that $a(x)f(x)+b(x)g(x)=1$ and with $\deg(b)<\deg(f)$ and $\deg(a)<\deg(g)$. Homogenizing we have (for some $r>0$):

Now for any $q(x)$ of degree $\deg(g)+2$ such that $f(x)q(x)=x^{N}+\ldots$ (i.e., the leading coefficient is $1$), the section

has the following properties:

• •

  along $X_{0}=0$, we have $s_{1}(Q)=X_{1}^{N}(=s_{0}|_{U\times\infty})$;

• •

  along $F=0$, we have $s_{1}(Q)=s_{\Delta}X_{0}^{N}(=s_{0}|_{U\times(\mathbb{A}^{1}-U)})$; and

• •

  along $G=0$, we have $s_{1}(Q)=F(X)Q(X)$.

The first two properties imply $s_{1}(Q)=s_{0}$ along $U\times(\mathbb{P}^{1}-U)$. The third implies $s_{1}(Q)$ generates along $U\times(U-V)$ for $q$ generic. Now we show that having fixed $B,G,F$, we can choose $Q$ appropriately.

The first item shows $Z(s_{1}(Q))\cap(U\times\infty)=\emptyset$ for any $Q$. Choosing $Q$ so that $Z(Q)$ is disjoint from $Z(GB)$, we see that $Z(s_{1}(Q))\cap(U\times Z(GB))\subset U\times(Z(F)\cap Z(GB))$. But the relation $af+bg=1$ implies that $Z(F)\cap Z(GB)=\emptyset$, hence $Z(s_{1}(Q))\cap(U\times Z(GB))=\emptyset$ and $Z(s_{1}(Q))$ is the graph of the function $u={f(x)q(x)\over b(x)g(x)}+x$. Hence for general $Q(X)$, the zero scheme $Z(s_{1}(Q))$ is smooth and disjoint from $U\times Z(G)=U\times(U-V)$.

We need to check $Q$ can be chosen so the intersections $Z(s_{1}(Q))\cap\Delta_{U}$ and $Z(s_{1}(Q))\cap R_{0}$, i.e.,