Some Excercises
1.12
(Excercise) If \(k\subset F\) is a field extension, there is an additive functor
\[ Cor_k\rightarrow Cor_F \]
sending \(X\) to \(X_F\), and the cycle \(W\in Z(X\times_k Y)\) to \([W\times_kF]\in Z(X_F\times_F Y_F)\).
If \(F\) is finite and separable over \(k\), there is an additive functor \(Cor_F\rightarrow Cor_k\) sendinf \(U/F\) to \(U/k\) (separatedness for smoothness). These are adjoint: if \(U\) is smooth over \(F\) and \(X\) is smooth over \(k\), there is a canonical identification
\[ Cor_F(U,X_F)=Cor_k(U,X). \]
This is kinda obvious since \(U\times_F(X\times_kF)\simeq U\times_k X\).
1.11
- Let \(x\) be a closed point of \(X\), considered as a correspondence from \(k\) to \(X\). Then the composition \(k\rightarrow X\rightarrow k\) is multiplcation by \([\kappa(X):k]\) and \(X\rightarrow k\rightarrow X\) is \(X\times x\subset X\times X\).
This is easy.
- Let \(L/k\) b a finite Galois extension with Galois group \(G\). Prove that \(Cor_k(L,L)\simeq\mathbb{Z}[G]\) (just count the points) and \(L\rightarrow k\rightarrow L\) is \(\sum_{g\in G}g\in\mathbb{Z}[G]\). Then show that \(Cor_k(k,Y)\simeq Cor_k(L,Y)^G\) for every \(Y\).
For the last statement use 1.12 above we see that \(Cor_k(L,Y)\simeq Cor_L(L,Y_L)\), so the question reforms to \(Z_0(Y)=Z_0(Y_L)^G\). Note that this also causes
\[ Cors_k(k,Y)\rightarrow Cors_k(L,Y)\rightrightarrows Cors_k(L\otimes_kL,Y) \]
being an equilizer diagram.
1.13
(Excercise)
- Let \(F\) be a field extension of \(k\) and \(X\) and \(Y\) two smooth schemes over \(k\). Write \(X_F\) for \(X\times_kF\) and so on, show that \(Cor_F(X_F,Y_F)\) is the limit of \(Cor_E(X_E,Y_E)\) as \(E\) ranges over all finitely generated field extensions of \(k\) in \(F\).
If \(F/k\) is finite this is obvious. Say if a scheme \(X/k\) is of finite type (this is the assumption we take at the starting), an ideal sheaf \(\mathcal{I}\) on \(X_F\) is locally determined by a quotient
\[ \mathcal{O}_U^{\oplus n}\xrightarrow{some\ sections}\mathcal{O}_U\rightarrow\mathcal{I}\rightarrow 0 \]
with \(n\) finite (Noetherian). Due to the assumption that \(X\) is quasi-compact, there are in total finitely many local sections determining \(\mathcal{I}\), so take \(E/k\) to assure that those sections are all in \(X_E\). Furthermore, every primary cycle lies in an affine (hence quasi-compact) local so we can push this to locally-of-fintie-type case. Now take \(X\) to be \(X\times_kY\).
- Let \(X/k\) be smooth morphism with \(X\) connected. Denote by \(F\) the function field of \(X\). Show that \(Cor_F(F,Y\times_kF)=colim_{U\subset X} Cor_k(U,Y)\) where \(Y/k\) is smooth.
We need two points here, firstly every (elementary) coprrespondence between \(F\) and \(Y_F\) should go back to some open \(U\times Y\), secondly if two correspondences from \(U\) to \(Y\) coincides on the fibre \(F\), then they shold coincide on some \(V\subset U\).
An elementary correspondence from \(F\) to \(Y_F\) is associated to a closed point on \(Y\). Since \(Y_F\) is smooth over \(Y\), the closed point is locally determined by a regular sequence of length equal to dimension of \(Y\). Additionally our general setting requires that \(Y\) is Noetherian, or at least that closed point generalizes to finitely many general points, so we may assume that \(Y\) is affine. Then the ring of global sections of \(Y_F\) is the colimit of that of \(Y\times_kU\). Since there is always a finite covering, we take the intersection of opens around \(F\).
Say two elementary correspondences from respectively \(U\) and \(U'\) to \(Y\) coincides on the generic point of \(X\). Taking intersection we may assume \(U\) coincides with \(U'\), then note that tensors commutes with colimits of algebras.
Etale sheaf with transfers
6.2
(Lemma) For any scheme \(T/k\), \(\mathbb{Z}_{tr}(T)\) is an etale sheaf.
proof. Let \(X\) be connected and smooth over \(k\) with connected etale cover \(U\rightarrow X\). As \(U\times T\rightarrow X\times T\) is flat and surjective, the pullback of cycles is well-defined and is an injection. The statement of being a sheaf requires the exactness of
\[ 0\rightarrow Cor_k(X,T)\rightarrow Cor_k(U,T)\rightarrow Cor_k(U\times_XU,T). \]
Now we need only to verify the exactness at \(Cor_k(U,T)\). Take \(Z_U\) in \(Cor_k(U,T)\) whose image in \(Cor_k(U\times_XU,T)\) vanish. Denote by \(F\) and \(L\) the function fields of \(X\) and \(U\) respectively. Using 1.11 we know that
\[ Cor_F(F,T_F)\rightarrow Cor_F(L,T_F)\rightrightarrows Cor_F(L\otimes_FL,T_F) \]
is an equilizer diagram. Now \(Z_L\), the fibre of \(Z_U\) at \(L\), vanishes in \(Cor_F(L\otimes_FL,T_F)\) and hence there is a Zariski open \(V\subset X\) and a cycle \(Z_V\) in \(Cor_k(V,T)\) agreeing with \(Z_U\) in \(Cor_k(U\times_XV,T)\) by 1.13. We can further assume that \(Z_V\) is primary (irreducible).
Let \(Z\) be the closure of \(Z_V\) in \(X\times T\). It is irredcible and dominant over \(X\) since \(Z\times_XV\) is, and the lifting of which to \(U\times T\) is \(Z_U\). Then \(Z_U\) being finite over \(U\) implies that \(Z\) is finite over \(X\), then \(Z\) is a finite correspondence in \(Cor_k(X,T)\).
6.12
(Psoposition) Let \(p:U\rightarrow X\) be an etale covering of a scheme \(X\). Then \(\mathbb{Z}_{tr}(\check{U})\) is an etale resolution of the sheaf \(\mathbb{Z}_{tr}(X)\), i.e., the following complex is exact as a complex of etale sheaves
\[ \cdots\xrightarrow{\partial}\mathbb{Z}_{tr}(U\times_XU)\xrightarrow{\partial}\mathbb{Z}_{tr}(U)\xrightarrow{p}\mathbb{Z}_{tr}(X)\rightarrow 0. \]
proof. Problematic!! The key point of this is we need the equation
\[ lim Cor_k(S_i,T)=Cor(S,T) \]
holds somehow where \(S_i\)s are etale neighborhoods of a geometric point of a scheme, and \(S\) is the corresponding strict henselian ring. This is way harder than the case in Zariski topology.
Homotopy (pre)sheaves
2.15
(Definition) A presheaf \(F\) is homotopy invariant if for every \(X\) the map \(p^*:F(X)\rightarrow F(X\times\mathbb{A}^1)\) is an isomorphism. As \(p:X\times\mathbb{A}^1\rightarrow X\) has a section \(p^*\) is always split injective. Thus homotopy invariance of \(F\) is equivalent to \(p^*\) being onto.
Let \(i_\alpha:X\hookrightarrow X\times\mathbb{A}^1\) be the inclusion \(x\mapsto(x,\alpha)\), and \(i_\alpha^*\) the corresponding lifted morphism.
2.16
(Lemma) \(F\) is homotopy invariant iff
\[ i_0^*=i_1^*:F(X\times\mathbb{A}^1)\rightarrow F(X) \]
for all \(X\). (This is the intuition of being homotopy invariant.)
proof. One direction is clear, so we prove the sufficency. Given \(i_0^*=i_1^*\) for all \(X\), let \(m:\mathbb{A}^1\times\mathbb{A}^1\) be the multiplcation map sending \((x,y)\) to \(xy\). Then we have the diagram
For a sheaf \(F\) we denote by \(C_*F\) the sheaf sending an object \(X\) to the chain complex
\[ \cdots\rightarrow F(X\times\Delta^2)\rightarrow F(X\times\Delta^1)\rightarrow F(X\times\Delta^0=X)\rightarrow 0. \]
2.18
(Lemma) Let \(F\) be a presheaf, then the maps \(i_0^*,i_1^*:C_*F(X\times\mathbb{A}^1)\rightarrow C_*F(X)\) are chain homotopic.
proof. Really dull work, the same case as in topology. Want to verify?
2.19
(Corollary) If \(F\) is a presheaf then the homology presheaves
\[ H_nC_*F:X\mapsto H_nC_*F(X) \]
are homotopy invariant for all \(n\).
2.22
(Lemma) Let \(F\) be a presheaf of abelian groups. Suppose that for every smooth scheme \(X\) there is a natural homomorphism \(h_X:F(X)\rightarrow F(X\times\mathbb{A}^1)\) which fits into
then the complex \(C_*F\) is chain contractible.
Proof. By naturality, \(h_X\) induces a map \(C_*h:C_*F(X)\rightarrow C_*F(X\times\mathbb{A}^1)\). Applying \(C_*\) to the diagram above we see that the identity map is then chain homotopic to \(0\).
The prototype for this is the sheaf of global functions \(\mathcal{O}\). Note that \(\mathcal{O}(X\times\mathbb{A}^1)=\mathcal{O}(X)[t]\), then set \(h_X(f)=tf\).
2.24
(Very important corollary) For any \(X\) there is a canonical projection \(p:X\times\mathbb{A}^1\rightarrow X\), which induces a map \(\mathbb{Z}_{tr}(X\times\mathbb{A}^1)\rightarrow\mathbb{Z}_{tr}(X)\). Then \(C_*\mathbb{Z}_{tr}(X\times\mathbb{A}^1)\rightarrow C_*\mathbb{Z}_{tr}(X)\) is a chain homotopy equivalance.
proof. We shall break \(C_*\mathbb{Z}_{tr}(X\times\mathbb{A}^1)\) into \(C_*\mathbb{Z}_{tr}(X)\oplus C_*F\) where \(C_*F\) is chain contractible. Denote by \(F\) the cokernel of (or fit into)
\[ 0\rightarrow\mathbb{Z}_{tr}(X)\xrightarrow{\mathbb{Z}_{tr}(i_0)}\mathbb{Z}_{tr}(X\times\mathbb{A}^1)\rightarrow F\rightarrow 0. \]
Note that this is a short exact seq. for presheaves. Then denote by \(H_U\) the composition
\[ Cor(U,X\times\mathbb{A}^1)\xrightarrow{lifting} Cor(U\times\mathbb{A}^1,(X\times\mathbb{A}^1)\times\mathbb{A}^1) \\ \xrightarrow{multiplcation} Cor(U\times\mathbb{A}^1,X\times\mathbb{A}^1) \]
where the multiplcation is \(\mathbb{A}^2\rightarrow\mathbb{A}^1\). Simple observation shows that \(H_U\) sends \(Cor(U,X\times\{0\})\) to \(Cor(U\times\mathbb{A}^1,X\times\{0\})\), hence it induces a natural map \(h_U:F(U)\rightarrow F(U\times\mathbb{A}^1)\). For \(U=X\times\mathbb{A}^1\) it’s easy to see that the composition of \(H_U\) with \(i_0,i_1:U\rightarrow U\times\mathbb{A}^1\) sends \(id\in Cor(U=X\times\mathbb{A}^1,X\times\mathbb{A}^1)\) to the projection \(i_0p:X\times\mathbb{A}^1\rightarrow X\rightarrow X\times\mathbb{A}^1\) and \(id\) respectively. Therefore \(F(i_0)h_U(id)=0\) and \(F(i_1)h_U(id)=id\) for \(U=X\times\mathbb{A}^1\).
For any other \(U\), every element \(\bar{f}\in F(U)\) is the image of \(id\)
Ouch.
Sheafication remembers transfers
Etale sheafication and Nisnevich sheafications remembers if a presheaf admits transfers structure. Denote by \(\phi\) the forgetful functor getting rid of the transfer structure.
6.17
(Theorem) Let \(F\) be a