The Stacks project

43.17 Intersection product using Tor formula

Let $X$ be a nonsingular variety. Let $\alpha = \sum n_ i [W_ i]$ be an $r$-cycle and $\beta = \sum _ j m_ j [V_ j]$ be an $s$-cycle on $X$. Assume that $\alpha $ and $\beta $ intersect properly, see Definition 43.13.5. In this case we define

\[ \alpha \cdot \beta = \sum \nolimits _{i,j} n_ i m_ j W_ i \cdot V_ j. \]

where $W_ i \cdot V_ j$ is as defined in Section 43.14. If $\beta = [V]$ where $V$ is a closed subvariety of dimension $s$, then we sometimes write $\alpha \cdot \beta = \alpha \cdot V$.

Lemma 43.17.1. Let $X$ be a nonsingular variety. Let $a, b \in \mathbf{P}^1$ be distinct closed points. Let $k \geq 0$.

  1. If $W \subset X \times \mathbf{P}^1$ is a closed subvariety of dimension $k + 1$ which intersects $X \times a$ properly, then

    1. $[W_ a]_ k = W \cdot X \times a$ as cycles on $X \times \mathbf{P}^1$, and

    2. $[W_ a]_ k = \text{pr}_{X, *}(W \cdot X \times a)$ as cycles on $X$.

  2. Let $\alpha $ be a $(k + 1)$-cycle on $X \times \mathbf{P}^1$ which intersects $X \times a$ and $X \times b$ properly. Then $pr_{X,*}( \alpha \cdot X \times a - \alpha \cdot X \times b)$ is rationally equivalent to zero.

  3. Conversely, any $k$-cycle which is rationally equivalent to $0$ is of this form.

Proof. First we observe that $X \times a$ is an effective Cartier divisor in $X \times \mathbf{P}^1$ and that $W_ a$ is the scheme theoretic intersection of $W$ with $X \times a$. Hence the equality in (1)(a) is immediate from the definitions and the calculation of intersection multiplicity in case of a Cartier divisor given in Lemma 43.16.4. Part (1)(b) holds because $W_ a \to X \times \mathbf{P}^1 \to X$ maps isomorphically onto its image which is how we viewed $W_ a$ as a closed subscheme of $X$ in Section 43.8. Parts (2) and (3) are formal consequences of part (1) and the definitions. $\square$

For transversal intersections of closed subschemes the intersection multiplicity is $1$.

Lemma 43.17.2. Let $X$ be a nonsingular variety. Let $r, s \geq 0$ and let $Y, Z \subset X$ be closed subschemes with $\dim (Y) \leq r$ and $\dim (Z) \leq s$. Assume $[Y]_ r = \sum n_ i[Y_ i]$ and $[Z]_ s = \sum m_ j[Z_ j]$ intersect properly. Let $T$ be an irreducible component of $Y_{i_0} \cap Z_{j_0}$ for some $i_0$ and $j_0$ and assume that the multiplicity (in the sense of Section 43.4) of $T$ in the closed subscheme $Y \cap Z$ is $1$. Then

  1. the coefficient of $T$ in $[Y]_ r \cdot [Z]_ s$ is $1$,

  2. $Y$ and $Z$ are nonsingular at the generic point of $T$,

  3. $n_{i_0} = 1$, $m_{j_0} = 1$, and

  4. $T$ is not contained in $Y_ i$ or $Z_ j$ for $i \not= i_0$ and $j \not= j_0$.

Proof. Set $n = \dim (X)$, $a = n - r$, $b = n - s$. Observe that $\dim (T) = r + s - n = n - a - b$ by the assumption that the intersections are proper. Let $(A, \mathfrak m, \kappa ) = (\mathcal{O}_{X, \xi }, \mathfrak m_\xi , \kappa (\xi ))$ where $\xi \in T$ is the generic point. Then $\dim (A) = a + b$, see Varieties, Lemma 33.20.3. Let $I_0, I, J_0, J \subset A$ cut out the trace of $Y_{i_0}$, $Y$, $Z_{j_0}$, $Z$ in $\mathop{\mathrm{Spec}}(A)$. Then $\dim (A/I) = \dim (A/I_0) = b$ and $\dim (A/J) = \dim (A/J_0) = a$ by the same reference. Set $\overline{I} = I + \mathfrak m^2/\mathfrak m^2$. Then $I \subset I_0 \subset \mathfrak m$ and $J \subset J_0 \subset \mathfrak m$ and $I + J = \mathfrak m$. By Lemma 43.14.3 and its proof we see that $I_0 = (f_1, \ldots , f_ a)$ and $J_0 = (g_1, \ldots , g_ b)$ where $f_1, \ldots , g_ b$ is a regular system of parameters for the regular local ring $A$. Since $I + J = \mathfrak m$, the map

\[ I \oplus J \to \mathfrak m/\mathfrak m^2 = \kappa f_1 \oplus \ldots \oplus \kappa f_ a \oplus \kappa g_1 \oplus \ldots \oplus \kappa g_ b \]

is surjective. We conclude that we can find $f_1', \ldots , f_ a' \in I$ and $g'_1, \ldots , g_ b' \in J$ whose residue classes in $\mathfrak m/\mathfrak m^2$ are equal to the residue classes of $f_1, \ldots , f_ a$ and $g_1, \ldots , g_ b$. Then $f'_1, \ldots , g'_ b$ is a regular system of parameters of $A$. By Algebra, Lemma 10.106.3 we find that $A/(f'_1, \ldots , f'_ a)$ is a regular local ring of dimension $b$. Thus any nontrivial quotient of $A/(f'_1, \ldots , f'_ a)$ has strictly smaller dimension (Algebra, Lemmas 10.106.2 and 10.60.13). Hence $I = (f'_1, \ldots , f'_ a) = I_0$. By symmetry $J = J_0$. This proves (2), (3), and (4). Finally, the coefficient of $T$ in $[Y]_ r \cdot [Z]_ s$ is the coefficient of $T$ in $Y_{i_0} \cdot Z_{j_0}$ which is $1$ by Lemma 43.14.3. $\square$

Comments (2)

Comment #7518 by Hao Peng on

There is previous defined intersection product in 42.62, what does that relate to the notion defiend here?

Comment #7650 by on

OK, yes, these two intersection products are the same. But we do not want to say this here, because the point of this chapter is to show how one can use the Tor formula to define an intersection product using the moving lemma, see the introduction to this chapter: Section 43.1. What we probably should do is at the end of this chapter add a section with comparisons between the various constructions. Anybody want to write it?

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