Flat extension and phantom homology

Phantom homology arises in tight closure theory due to small non-exactness when ‘kernel’ is not equal to ‘image’ but ‘kernel’ is in the tight closure of the ‘image’. In this paper we study a typical flat extension, which we call $*$-flat extension, such that upon tensoring which preserves phantom homology. Along with other properties, we observe that $*$-flat extension preserves ghost regular sequence, which is a typical ‘tight closure’ generalization of regular sequence. We also show that in some situations, under $*$-flat extension, test ideal of the $*$-flat algebra is the expansion of the test ideal of the base ring.