Complete intersection subvariety of projective variety

Question

I am not able to find any literature which studies complete intersection subvarieties of a projective variety, all good references consider CI in projective space.

My guess is, a subvariety X of pure codimension r in a projective variety Y is called complete intersection if the ideal of X is generated by r homogeneous polynomials coming from the coordinate ring of X. Embed Y into a suitable projective space.

My question is: With this definition of complete intersection, will we get a finite koszul resolution of the ideal of X, analogous to the case of projective space. In particular, I want the terms of the koszul resolution to be twists of the coordinate ring of Y? If this is not true, is there any other suitable definition of complete intersection, under which we have such a koszul resolution.

Any reference will be most welcome.

Answer 1

If $E$ is a vector bundle of rank $r$ on a Cohen--Macaulay (e.g., smooth) scheme $Y$ and $s \in \Gamma(Y,E)$ is a global section with zero locus $X = Z(s) \subset Y$ such that $\dim(X) = \dim(Y) - r$ then the Koszul complex $$ 0 \to \wedge^rE^\vee \to \wedge^{r-1}E^\vee \to \dots \to \wedge^2E^\vee \to E^\vee \to I_X \to 0 $$ is exact.

If additionally $E$ is a sum of line bundles $E = L_1 \oplus L_2 \oplus \dots \oplus L_r$ then every term of the Koszul complex is a sum of line bundles $$ \wedge^kE^\vee \cong \bigoplus_{i_1 < i_2 < \dots < i_k} L_{i_1}^\vee \otimes L_{i_2}^\vee \otimes \dots \otimes L_{i_k}^\vee. $$ Finally, if each $L$ is a power of a given line bundle, then the same is true for each summand above.