3.4.2 Cycles associated to closed subschemes
Before we can comfortably discuss flat pullback, we need to discuss how to associate an algebraic cycle to a closed subscheme \(Y\) of \(X\). This is obvious when \(Y\) is a closed subvariety of \(X\): then the algebraic cycle associated to \(Y\) is just \([Y]\in Z_* X\). The point is that when \(Y\) is not integral, we need to assign some multipicities to the various components \(Y\) could have depending on the structure sheaf.
We start with just \(X\) itself. Since \(X\) has finite type over \(k\), it has finitely many irreducible components \(X_1,\ldots , X_n\), which are closed subvarieties of \(X\) (possibly of different dimensions). For each \(i=1,\ldots , n\), let \(x_i\) denote the generic point of \(X_i\). Then the ring \(\mathcal{O}_{X, x_i}\) is Artinian, and we set \(m_i = \operatorname{\mathrm{length}}(\mathcal{O}_{X, x_i})\). The number \(m_i\) is called the multiplicity of \(X\) along \(X_i\). Then set \([X] = \sum m_i [X_i]\in Z_* X\).
Now, if \(Y\) is a closed subscheme of \(X\), then \(Z_* Y\subseteq Z_* X\) (because closed subvarieties in \(Y\) are closed subvarieties of \(X\)). So we let \([Y]\in Z_* X\) denote the image of \([Y]\in Z_* Y\) under the inclusion.
Suppose \(X = \mathbb {A}^1 = \operatorname{\mathrm{Spec}}\mathbb {C}[x]\) and \(Y = V(x^2)\). Then \(Y = \operatorname{\mathrm{Spec}}\mathbb {C}[x]/(x^2) = \mathcal{O}_{Y,y}\), where \(y=(x)\) is the generic point (and the closed point) of \(Y\). We have \(\operatorname{\mathrm{length}}\mathbb {C}[x]/(x^2) = 2\). So \([Y] = 2[y]\in Z_0 Y\subseteq Z_0 X\).