Local Algebra
Regular sequence and regular immersions
067M. A sequence \((f_1,\cdots, f_r)\) in a ring \(R\) is said to be regular if
- \(f_i\) is a nondivisor of \(R/(f_1,\cdots,f_{i-1})\),
- \(R/(f_1,\cdots,f_r)\not=0\).
The second requirement induces that no \(f_i\) can be invertible. There are weaker properties a sequence can hold, and they’re independent on the indexing. For a sequence \((f_1,\cdots,f_r)\) in \(R\), the Koszul complex is defined to be the complex
\[ 0\rightarrow\wedge^r R^r\rightarrow\cdots\rightarrow\wedge^2 R^r\rightarrow\wedge^1 R^r\rightarrow\wedge^0 R^r\rightarrow 0 \]
where the morphism \(d:\wedge^1 R^r=R^r\rightarrow R=\wedge^0 R^r\) is defined to be \(d(e_i)=f_i\), and the higher differentials are defined to be
\[ d(e_1\wedge\cdots\wedge e_n)=\sum(-1)^{i-1}f_k e_1\wedge\cdots\hat{e_k}\cdots\wedge e_n. \]
A sequence is said to be Koszul-regular if the Koszul complex attached to which is exact, and to be \(H_1\)-regular if the Koszul complex attached to which is exact at \(\wedge^2\rightarrow\wedge^1\rightarrow\wedge^0\).
V.7 Pull-backs
Definition
\(f:X\rightarrow Y\) be a morphism between varieties, with \(Y\) non-singular, and \(x\) and \(y\) be cycles on \(X\) and \(Y\) resp. Set \(|x|=Supp(x)\), then
\[ dim|x|\cap f^{-1}(|y|)\geq dim|x| - codim|y|. \]
This is 0AZN. If the equality holds, then one defines \(x\cdot_fy\) with support contained in \(|x|\cap f^{-1}(|y|)\) by either one of the following methods:
- Reduction to a standard intersection: Assume \(X\) being affine (locally, and at least separable), then embed \(X\) in an affine space \(V\). The graph morphism embeds \(X\) into \(V\times Y\), and sends a cycle \(x\) on \(X\) to \(\gamma(x)\) on \(V\times Y\). Then one defines \(x\cdot_fy\) as the unique cycle on \(X\) s.t.
\[ \gamma(x\cdot_fy)=\gamma(x)\cdot (V\times y). \]
The intersection product being calculated on the non-singular scheme \(V\times Y\).
INDEPENDENT ON THE CHOSEN EMBEDDING check later!
- Homological computation: Choose coherent sheaves \(\mathcal{M}\) and \(\mathcal{N}\) over \(X\) and \(Y\), representing \(x\) and \(y\) respectively. Then define \(x\)