Closure-interior duality over complete local rings

Integral closure, basically full closure, and duals of nonresidual closure operations

“…In this section, we define pair operations and several important properties. We describe the duality between closure operations and interior operations over a complete local ring as first given by the first two named authors in [ER21]. We then recall the definition of a Nakayama closure, cl-reductions, and the cl-core, and give some of their properties.…”

“…We had previously (see [ER21]) defined a duality between submodule selectors which we parlayed into a duality between residual closure operations and (absolute) interior operations. However, in this paper we extend this duality to the more general context of pair operations, allowing us to deal effectively with nonresidual closure operations such as integral closure and basically full closure.…”

“…Remark 2.4. Recall that in [ER21], an extensive operation e was called residual if whenever q ∶ M ↠ P were a surjection, we had (ker q) e M = q −1 (0 e P ). Note that this is equivalent to the definition given above.…”

“…Indeed it follows from the first isomorphism theorem that the current definition implies the older one. For the converse, suppose e is a residual operation in the sense of [ER21] and let L ⊆ N ⊆ M such that (N, M ), (0, M N ), (N L, M L) ∈ P. Let π ∶ M ↠ M L and q ∶ M L ↠ M N be the natural surjections. Then…”

“…In previous work [ER21,ERV20], the authors explored a duality operation that takes a residual closure operation on modules over a complete local ring (such as tight closure, module closures, or Frobenius closure) to an absolute interior operation (e.g. in the above cases, the tight interior explored in [ES14], trace as in [PR19], and the novel Frobenius interior, respectively).…”