Addition and subtraction of ideals

Let A be a Noetherian ring, Ĩ and I be comaximal ideals of A, and P be a projective A-module. "Addition" refers to being given a surjection P↠Ĩ and producing a surjection P↠Ĩ∩I. This is useful, for example, when P is free, to determine how many elements it takes to generate the ideal Ĩ∩I. "Subtraction" refers to being given P↠Ĩ∩I and producing some P↠Ĩ. A major use of this is when Ĩ=A, to show a projective module has a unimodular element. Here we extend certain addition and subtraction results of R. Sridharan (1995, J. Algebra 176, 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 Ĩ and in a suitable fashion remove the hypothesis that P must have trivial determinant. With certain restrictions, we also now allow dim A/I≤1, rather than htI=dim A. When A is an affine algebra over a field F, Mandal and Sridharan had subtraction results for when I is the intersection of finitely many maximal ideals whose residue fields are quadratically closed, when √I has this form (if char F≠2), or when I is the maximal ideal of an F-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 an F-rational point. When char F≠2, we only need to require √I to have this form. In particular, we can subtract an ideal whose radical is the maximal ideal of an F-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.