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.