A Briançon-Skoda type result for a non-reduced analytic space

A comparison formula for residue currents

“…The theory of residue currents of Coleff-Herrera, Dickenstein-Sessa, Passare-Tsikh-Yger, Andersson-Wulcan and others has provided a strong tool for proving different results. For example, it has been used to prove results about membership problems in commutative algebra, including Briançon-Skoda type results in [ASS,AW3,S]. However, there are similar results which appear natural to approach by such methods, but which have so far not been possible to prove in this way due to lack of precise enough description of the involved residue currents.…”

A global Briançon-Skoda-Huneke-Sznajdman theorem

“…An analytic proof of this statement appeared in [3]. A nice variant for a non-reduced X of pure dimension is formulated and proved in [28].…”

“…Theorem 1.1 (Sznajdman, [28]). Given x ∈ X, there is a finite set L α of Noetherian operators at x and a number ν such that for each ideal (a) = (a 1 , .…”

“…Following the ideas in [7] we then form the "product" R f ∧R X , where R f is the current of Bochner-Martinelli type introduced in [1], inspired by [26]. By computations as in [28], the condition (1.4) ensures that φ annihilates this current at each point x ∈ V red . If ρ is large enough, this is reflected by the first entry of the right hand side of (1.5), then a geometric estimate from [13] ensures that the ρ-homogenization φ of Φ indeed satisfies a condition like (1.4) even at infinity.…”

Residue currents with prescribed annihilator ideals on singular varieties

“…The applications described for Coleff-Herrera products have been generalized in various ways to Andersson-Wulcan currents, thereby being able to remove assumptions about complete intersection, see for example [AS, AS2, ASS, AW, AW3,Sz].…”