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Tight closure test ideals have been central to the classification of singularities in rings of characteristic p > 0 p>0 , and via reduction to characteristic p > 0 p>0 , in equal characteristic 0 as well. Their properties and applications have been described by Schwede and Tucker [Progress in commutative algebra 2, Walter de Gruyter, Berlin, 2012]. In this paper, we extend the notion of a test ideal to arbitrary closure operations, particularly those coming from big Cohen-Macaulay modules and algebras, and prove that it shares key properties of tight closure test ideals. Our main results show how these test ideals can be used to give a characteristic-free classification of singularities, including a few specific results on the mixed characteristic case. We also compute examples of these test ideals.

Let R be a commutative, local, Noetherian ring. In a past article, the first author developed a theory of R-algebras, termed seeds, that can be mapped to balanced big Cohen-Macaulay R-algebras. In prime characteristic p, seeds can be characterized based on the existence of certain colon-killers, integral extensions of seeds are seeds, tensor products of seeds are seeds, and the seed property is stable under base change between complete, local domains. As a result, there exist directed systems of big Cohen-Macaulay algebras over complete, local domains. In this work, we will show that these properties can be extended to analogous results in equal characteristic zero. The primary tool for the extension will be the notion of ultraproducts for commutative rings as developed by Schoutens and Aschenbrenner.For a local ring (R, m), a big Cohen-Macaulay R-algebra B is an R-algebra such that some system of parameters for R forms a regular sequence on B with the extra property that mB = B to ensure that B is not trivial. If this is true for every system of parameters for R, then B is a balanced big Cohen-Macaulay algebra. Big Cohen-Macaulay algebras were first shown to exist in [HH92] using characteristic p methods related to tight closure theory (see [HH90]). Hochster and Huneke proved that the absolute integral extension R + is a balanced big Cohen-Macaulay R-algebra when R is an excellent, local, domain of prime characteristic. These results were extended in further articles. For example, in [Ho94] Hochster makes explicit use of tight closure to provide an alternative proof of the existence of big Cohen-Macaulay algebras in prime characteristic based on the notion of algebra modifications. In [HH95] Hochster and Huneke proved the existence of balanced big Cohen-Macaulay algebras for rings containing a field of characteristic zero and proved the "weakly functorial" existence of big Cohen-Macaulay algebras, i.e., given complete local domains of equal characteristic R → S, there exists a balanced big Cohen-Macaulay R-algebra B and a balanced big Cohen-Macaulay S-algebra C such that B → C extends the map R → S. The results in equal characteristic zero are based on reductions to prime characteristic that rely on Artin approximation.In [D07], the first author introduced the notion of seed algebras. Given a local, Noetherian ring R, a seed algebra over R is an R-algebra S such that there exists an R-algebra map S → B where B is a balanced big Cohen-Macaulay Ralgebra. Some of the noteworthy results in that paper for rings of prime characteristic are Theorem 4.8 (which characterizes seeds based on durable, colon-killers),