CancellationTheorem

The ultimate goal of this note is to read Cancellation Theorem by V.Voevodsky, 2010.

Relative cycles

Preliminaries

Some definitions

Let \(X\) be Noetherian. Denoty by \(Cycl(X)\) (resp. \(Cycl^{eff}(X)\)) the free abelian group (resp. monoid) of cycles on \(X\). For a cycle \(\mathcal{Z}\in Cycl(X)\) denote by \(supp(\mathcal{Z})\) the support of such a cycle, considered as a reduced closed subscheme of \(X\). Define by \(cycl_X(\bullet)\) the function sending a closed subscheme \(Z\) of \(X\) to the corresponding element in \(Cycl^{eff}(X)\), in the obvious way. There is a natually defined flat pullback satisfying \(p^*cycl_S(Z)=cycl_X(p^{-1}Z)\) for \(p:X\rightarrow S\) flat, which is then injective provided by \(p\) being surjective.

2.3.2

(Lemma) For \(X/k\) of finite type and a finite normal field extension \(k'/k\) with Galois group \(G\), we can descend a cycle \(\mathcal{Z}'\in Cycl(X_{k'})^G\) to a cycle (with potentially rational coefficients) \((1/[k':k]_{insep})\mathcal{Z}\) such that \([k':k]_{insep}\mathcal{Z}'=\mathcal{Z}\otimes_k k'\).

Proof. We may just assume that \(X\) is a finitly generated (maybe transcedental) field over \(k\). We shall now investigate the tensor product \(X\times_k k'\). By elementary field theory we know that

  • there is an intermediate Galois extension \(k'/k_0/k\),
  • and we can decompose the purely inseparable extension \(k'/k_0\) into a tower with every two adjacent items be like

\[ (k_i[x]/(x^p-\alpha))/k_i \]

where \(\alpha\in k_i\) has no \(p\)th roots.

Now we need only the statement hold for Galois extensions and simple purely inseparable extensions. The case for a Galois extension is kinda obvious. Consider the tensor product \(k'\otimes_k K\) where \(k'\) is the field get by adding the \(p\)th root of \(\alpha\in k\). We claim that

  • if there is a \(p\)th root in \(K\), namely \(\beta\), then the tensor product is isomorphic to \(K[x]/(x-\beta)^p\),
  • otherwise the tensor product still has only one point in the meaning of being a scheme.

The first statement is straightforward, and to see the second one just add such a root to \(K\) to get \(K_0\). We know then the tensor product of \(k'\times_k K\) with \(K_0\) has only one point, which means that the former term cannot has more than two points. Moreover this tensor product must be a field instead of an Artinian algebra with nilpotents.

2.3.3

(Corollary) In the preceding assumptions and notations the homomorphism

\[ Cycl(X)[1/p]\rightarrow (Cycl(X_{k'})[1/p])^G \]

is an isomorphism.

Proper definition of a non-flat pullback

Relative Cycles

We know that being flat over a DVR (even an arbitary valuation ring) is equivalent to being torsion-free. So we may extend the notion of a flat supported cycle, which behaves quite well in doing pullpacks, to the notion of a relative cycle, which is defined by testing using DVRs.

3.1.1

(Definition) Let \(S\) be Noetherian, and \(x:k\rightarrow S\) be a \(k\)-point. A fat point over \(x\) is a triple \((x_0,x_1,R)\) where \(R\) is a DVR and … well just see the illustration

3.1.2

(Lemma) Let \(S\) be Noetherian and \(X/S\), with \(Z\) being a closed subscheme in \(X\). Further let \(R\) be a DVR and \(f:R\rightarrow S\) be a morphism. Thenre there is a unique closed subscheme \(\phi_f(Z)\) in \(Z\times_S R\) s.t.

  • \(\phi_f(Z)\rightarrow Z\times_S R\) is an isomorphism over the generic point of \(R\),
  • \(\phi_f(Z)\) is flat over \(R\).

Just mod out the torsion subgroup will do the trick.

Let \(X/S\) be a scheme of finite type over a Noetherian base, and \(Z\) a closed subscheme of \(X\). For any fat point \((x_0,x_1)\) over a \(k\)-point \(x\) of \(S\) we denote by \((x_0,x_1)^*(Z/S)\) the cycle on \(X\times_kS\) associated with the closed subscheme \(\phi_{x_1}(Z)\times_R k\).

Basically this is a process of specialization.

3.1.3

(Definition) Let \(S\) be a Noetherian scheme and \(X/S\) be of finite type. A relative cycle on \(X/S\) is a cycle \(\mathcal{Z}=\sim m_iz_i\) on \(X\) satisfying

  • the points \(z_i\) lie over the generic points of \(S\),
  • and for any \(k\)-point \(x\) of \(S\), the pullback along a fat point lies over \(x\) should not rely on the selection of that fat point.

We say such a cycle is a relative cycle of dimension \(r\) if each point \(z_i\) has dimension \(r\) in it’s fibre over \(S\). The corresponding abelian group is denoted by \(Cycl(X/S,r)\).

We say such a cycle is an equidimensional relative cycle on dimension \(r\) if \(supp(\mathcal{Z})\) is equidimensional of dimension \(r\) over \(S\). The corresponding abelian group is denoted by \(Cycl_{equi}(X/S,r)\).

We say such a cycle is a proper relative cycle is \(supp(\mathcal{Z})\) is proper over \(S\). \(PropCycl(X/S,r)\) and \(PropCycl_{equi}(X/S,r)\).

For the related abelian monoid we will still use the notation \(\bullet^{eff}\).

3.1.4

(Lemma) We can construct enough fat points. To be precise, Let \(S\) be Noetherian with a generic point denoted by \(\eta\) and a point \(s\) generalize to \(\eta\). Let further \(L\) be an extension of finite type of \(\kappa(\eta)\). Then there is a DVR \(R\) and a morphism \(f:R\rightarrow S\) such that

  • \(f\) maps the generic point of \(R\) to \(\eta\) and \(K(R)\) is isomorphisc to \(L\),
  • \(f\) maps the closed point of \(R\) to \(s\).

Proof. EGA2, 7.1.7.

[Something Omitted… Turning back later]

Now we turn to the so called flat cycles. Again use over generic situation \(p:X/S\), and denote by \(Hilb(X/S,r)\) (resp. \(PropHilb(X/S,r)\)) the set of closed subschemes \(Z\) of \(X\) which are flat (resp. flat and proper) and equidimensional of dimension \(r\) over \(S\). The corresponding freely generated abelian monoids (resp. groups) are denoted by \(N(\bullet)\) (resp. \(Z(\bullet)\)).

A good property is that those flat cycles have well-defined pullbacks. The assignment \(S'/S\rightarrow N(Hilb(X\times S'/S',r))\) defines a presheaf of abelian monoids on the category of Noetherian schemes over \(S\). The same way works for the \(Z(\bullet)\) version.

3.2.2

(Proposition) Let \(X/S\) be as our generic settings and \(S'/S\) a Noetherian scheme over \(S\). Let further \(\mathcal{Z}=\sum n_iZ_i\) be an element of \(Z(Hilb(X/S,r))\). Then \(cycl_{X\times S'}(\mathcal{Z}\times S')=0\) provided that \(cycl_X(\mathcal{Z})=0\).

In fact, there is a morphism \(Z(Hilb(X/S,r))\rightarrow Cycl(X)\) with which image lies in \(Cycl_{equi}(X/S,r)\).

Now we head into the theory of Chow presheaves. This is exciting!

3.3.1

(Theorem) Let \(X/S\) be as our generic settings and \(f:T\rightarrow S\) be a Noetherian base change. For a \(\mathcal{Z}\in Cycl(X/S,r)\) there is a unique element \(\mathcal{Z}_T\) in \(Cycl(X\times T/T,r)\otimes\mathbb{Q}\) such that for any commutative diagram

of fat points we have

\[ (y_0,y_1)^*(\mathcal{Z}_T)=(x_0,x_1)^*(\mathcal{Z}). \]