Addition and Subtraction of Ideals

LetAbe a Noetherian ring, andIbe comaximal ideals ofA, andPbe a projectiveA-module. “Addition” refers to being given a surjectionP and producing a surjectionP ∩ I. This is useful, for example, whenPis free, to determine how many elements it takes to generate the ideal ∩ I. “Subtraction” refers to being givenP ∩ Iand producing someP . A major use of this is when = A, to show a projective module has a unimodular element. Here we extend certain addition and subtraction results of R. Sridharan (1995,J. Algebra176,947–958) and S. Mandal and R. Sridharan (1996,J. Math. Kyoto Univ.36,No. 3, 453–470). In both addition and subtraction, we weaken the hypotheses imposed on and in a suitable fashion remove the hypothesis thatPmust have trivial determinant. With certain restrictions, we also now allow dim A/I ≤ 1, rather than ht I = dim A. WhenAis an affine algebra over a fieldF, Mandal and Sridharan had subtraction results for whenIis the intersection of finitely many maximal ideals whose residue fields are quadratically closed, when has this form (if char F ≠ 2), or whenIis the maximal ideal of anF-rational point. In our generalization, we allowIto be the intersection of finitely many maximal ideals, all of whose residue fields are quadratically closed, except possibly one which instead defines anF-rational point. When char F ≠ 2, we only need to require to have this form. In particular, we can subtract an ideal whose radical is the maximal ideal of anF-rational point; this has been used by S. M. Bhatwadekar and R. Sridharan (1998,Invent. Math.133,No. 1, 161–192) to construct a certain local complete intersection ideal which is not a complete intersection ideal.