A motivating example: calculating the pullback

Here comes the case: say we have three nonsingular separated varieties \(X\), \(Y\) and \(Z\) over a field \(k\). Moreover, we have two correspondences \(\Gamma_g\in Cor_k(X,Y)\) and \([W]\in Cor_k(Y,Z)\) arised from a smooth (even flat) morphism \(g:X\rightarrow Y\) and a integral subscheme \(W\hookrightarrow Y\times Z\) respectively. Now we want to see that whether \[[W]\circ\Gamma_g=(g\times id_Z)^*([X\times W])\in Cor_k(X,Z)\] holds. We hope to do something like this

\[ \begin{aligned}[] [W]\circ\Gamma_g&=p_*(([X\times W])\cdot(\Gamma_g\times Z)) \\ &=p_*(([X\times W])\cdot(g\times id_Z)_*[X\times Z]) \\ &=p_*(g\times id_Z)_*((g\times id_Z)^*([X\times W])\cdot[X\times Z]) \\ &=(g\times id_Z)^*([X\times W]). \end{aligned} \] However this stupid calculation does not make sense in general, since the projection formula in the second equation should not hold in such a case. In fact, we have many versions of stuffs like the projection formula in the intersection theory for cycles, namely Serre’s treatment. Moreover, Voevodsky gives a version suitable for his finite correspondence case in the Lemma 17A.11. However, these descrete lemmas are not so flexible and likely to cause a headache in most times.

From cycles to coherent sheaves

Following Lemma 02S9 of 0FDQ, we have a pair of mutually inverse isomorphisms, namely \[ \begin{aligned}[] Z_k(X)&\rightarrow K_0(Coh_{\leq k}(X)/Coh_{\leq k-1}(X)) \\ [Z]&\mapsto [\mathcal{O}_Z] \end{aligned} \] and \[ [\cdot]_k:K_0(Coh_{\leq k}(X)/Coh_{\leq k-1}(X))\rightarrow Z_k(X) \] where \(k\) is the truncating dimension. We will denote \(K_0(Coh_{\leq k}(X)/Coh_{\leq k-1}(X))\) by \(gr_kK_0'(X)\) in the following sections.

[Lemma 0] For a flat morphism \(f:X\rightarrow Y\) of relative dimension \(r\), the inverse image map \(f^*:Coh(Y)\rightarrow Coh(X)\) sends \(Coh_{\leq k}(Y)\) into \(Coh_{\leq k+r}(X)\); for a proper morphism \(g:X\rightarrow Y\), the pushforward maps \(R^ig_*:Coh(X)\rightarrow Coh(Y)\) send \(Coh_{\leq k}(X)\) into \(Coh_{\leq k}(Y)\).

proof For \(f^*\) and \(R^0g_*\) this is just 02RE and 02R6. For the higher case, note that \(g\) is a closed map sending a closed scheme \(Z\hookrightarrow X\) to \(g(Z)\hookrightarrow Y\)

Then for any coherent sheaf \(M\) supported on \(Z\), we have \(g_*M=g_*(i_X)_*M=(i_Y)_*(g_Z)_*M\). Now take the derived functor on both sides.

In this case, we may say \(f^*\) respects the filtration with shift \(r\) and \(R^ig_*\) with shift \(0\). Now we turn to the construction of intersection product. A trivial definition goes like below.

[Trivial version, intersetion product by \(Tor\) formula] Denote two integral subschemes \(Y,Z\hookrightarrow X\) of codimension \(r\) and \(s\) respectively, then the intersection product of \([Y]_{dimX-r}\) and \([Z]_{dimX-s}\) is defined as \[ \sum_{i=0}(-1)^i[Tor_i(\mathcal{O}_Y,\mathcal{O}_Z)]_{dimX-r-s}. \]

Note that there are only finitely many terms \(Tor_i(\mathcal{O}_Y,\mathcal{O}_Z)\) that do not vanish, so the sum make sense. However, this is not the right version since there’s no guarantee that \(Tor_i(\mathcal{O}_Y,\mathcal{O}_Z)\) has a support with codimension \(r+s\), that is, this definition does not work since we want an operation that respects filtration. To get a correct definition, we need the lemma below.

[Lemma 1] If \(A\) is a regular local ring, \(M\) and \(N\) be two finitely generate \(A\)-modules and \(\mathfrak{q}\) a minimal prime ideal of \(supp(M\otimes_A N)\), then

1. \(\chi_{\mathfrak{q}}(M,N)=\sum_i(-1)^ilength_{\mathfrak{q}}(Tor_i^A(M,N))\) is non-negative;
2. \(dim_{A_{\mathfrak{q}}}M_{\mathfrak{q}}+dim_{A_{\mathfrak{q}}}N_{\mathfrak{q}}\leq ht_A\mathfrak{q}\);
3. \(\chi_{\mathfrak{q}}(M,N)=0\) if \(dim_{A_{\mathfrak{q}}}M_{\mathfrak{q}}+dim_{A_{\mathfrak{q}}}N_{\mathfrak{q}}<ht_A\mathfrak{q}\). {#l1}

The version where \(A\) is equicharacteristic is proved in SerreLocal, and see GiS for further discussion.

This lemma suggests that we can slightly alter the definition above as \[ \sum_{i=0}(-1)^i[Tor_i(\mathcal{O}_Y,\mathcal{O}_Z)]^\sim. \] where \([\cdot]\) sends \(Coh(X)\) into \(K_0'(X)\) and \(\sim\) means taking quotient by the \(K_0\) of serre subcategory \(Coh_{\leq dimX-r-s-1}(X)\). The third statement guarantees that the sum do lies in \(gr_{dimX-r-s}K_0'(X)\), in other words, taking intersection with a codimension \(r\) cycles respects filtration with shift \(-r\).

…Then to the derived category

To do intersection theory cleverly we shall pass to the derived category \(D^bK_0'(X)\). In fact, the functor \([\cdot]\) sending an abelian category \(\mathcal{A}\) to it’s \(K_0\) group factors through the bounded derived category \(D^b(\mathcal{A})\) as \[ \begin{aligned}[] \mathcal{A}\rightarrow & D^b(\mathcal{A})\rightarrow & K_0(\mathcal{A}) \\ S\mapsto & (0\rightarrow S\rightarrow 0) & \\ & T\mapsto & [T]=\sum (-1)^i[H^i(T)] \end{aligned}. \]

Note that this construction is additive, that is, for a distinguished triangle \(P\rightarrow Q\rightarrow R\rightarrow\Sigma\), we have \([P]-[Q]+[R]=0\). Now the intersection product of two (equidimensional) cycles on \(X\) represented by \([M]\) and \([N]\) respectively can be given as \([M\otimes_{\mathcal{O}_X}^LN]\). Moreover, we say an exact functor \(T:D^bCoh(X)\rightarrow D^bCoh(Y)\) respects filtration with shift \(r\) if \([TM]\in K_0(Coh_{\leq k+r}(Y))\) whenever \([M]\in K_0(Coh_{\leq k}(X))\). Such a functor induces a homomorphism between graded \(K_0\) groups as \(gr_kK_0'(X)\rightarrow gr_{k+r}K_0'(Y)\).

[Lemma 2] For \(M\) and \(N\) in an (essentially small) Abelian category \(\mathcal{A}\) such that \([M]=[N]\) in \(K_0(\mathcal{A})\) and for any exact functor \(T:D^b\mathcal{A}\rightarrow D^b\mathcal{B}\), we have \([TM]=[TN]\) in \(K_0(\mathcal{B})\).

proof This is true since relationships in \(K_0(\mathcal{A})\) are generated by short exact sequences, which imples distinguished traingles immediately, and are preserved by exact functors.

[Theorem 3] For a proper (this implies being of finite type) morphism \(f:X\rightarrow S\) and a coherent sheaf \(M\) on \(X\) supported on a dimension \(k\) subscheme, we have \([f_*M]_k=[Rf_*M]_k\).

proof SerreLocal page 117, proposition 2 gives some comment on proving this lemma, but it’s unknown that if his approach works over fields that are not algebracially closed. However, we can give an alternative proof using our theory constructed here. First we suppose \(S\) being affine over \(k\) (hence Noetherian), and according to Chow’s lemma 0202 there is a proper cover \(\pi:X'\rightarrow X\) and an immersion \(i:X'\rightarrow\mathbb{P}^n_S\)

Since \([f_*M]_k=f_*[M]_k\) and \(Rf_*\) preserves distinguished traingles, the case can be reduced to that \([M]_k\) being an integral closed subscheme of \(X\). Hence there is an elementary cycle \([M']_k\in X'\) and a non-zero integer \(d\) such that \(\pi_*[M']_k=d[M]_k\). Note that tensoring with a line bundle \(\mathcal{L}\) is exact and does not affect the length at a certain point, we have \[ \begin{aligned}[] df_*[M]_k&=f_*\pi_*[M']_k=f_*'[M']_k \\ &=[f_*'(M'\otimes\mathcal{L})]_k \end{aligned}. \] Now take \(\mathcal{L}\) being an ample bundle twisted after a sufficient large times, say \((i^*\mathcal{O}(1))^{\otimes m}\), and according to 0897 we see that \(f_*'(M'\otimes\mathcal{L})\) is quasi-isomorphic to \(Rf_*'(M'\otimes^L\mathcal{L})\) and \(\pi_*(M'\otimes\mathcal{L})\) is quasi-isomorphic to \(R\pi_*(M'\otimes^L \mathcal{L})\). According to [Lemma 2] we have \([Rf_*'(M'\otimes^L \mathcal{L})]=[Rf_*'M']\) so \[ \begin{aligned}[] [f_*'(M'\otimes\mathcal{L})]_k&=[Rf_*'(M'\otimes\mathcal{L})] \\ &=[Rf_*(R\pi_*(M'\otimes\mathcal{L}))] \\ &=[Rf_*(\pi_*(M'\otimes\mathcal{L}))] \\ &=d[Rf_*M]&use\ [Lemma\ 2]\ again \end{aligned}. \] Now this holds in affine locals of \(S\), and the global version (lost due to laziness).

[Lemma 2] has many other corollaries, for example

[Theorem 3] The intersection product is an associative operation.

proof Let \(M\), \(N\) and \(P\) in \(Coh(X)\). Since taking derived tensor product is exact, any coherent sheaf \(L\) representing the same cycle \([N\otimes^LP]_*\) results in the same cycle \([M\otimes^LN\otimes^LP]_*\).

The projection formula

Following 01e6, for a morphism \(f:X\rightarrow Y\) and \(E\), \(K\) in \(D(\mathcal{O}_X)\) and \(D(\mathcal{O}_Y)\) respectively, there is a canonical morphism \[ Rf_*E\otimes_{\mathcal{O}_Y}^LK\rightarrow Rf_*(E\otimes_{\mathcal{O}_X}^LLf^*K). \]

[Lemma 0B54] If \(K\) is perfect, then the morphism above is an isomorphism in \(D(\mathcal{O}_Y)\).

[Lemma 4] Any coherent sheaf on nonsingular scheme \(X\) is perfect.

proof See Fulton, intersection theory Appendix B.8.3.

This is temporarily postponed until I find out a slightly better way to relate the local \(K_0\) to the global one.

References

[GiS] Gillet, H., and C. Soulé. “Intersection Theory Using Adams Operations.” Inventiones Mathematicae 90, no. 2 (June 1987): 243–77. https://doi.org/10.1007/BF01388705. {#ref0}