### The TestIdeals package for Macaulay2

Alberto F. Boix, Daniel Hernández, Zhibek Kadyrsizova, Mordechai Katzman, Sara Malec, Marcus Robinson, Karl Schwede, Daniel Smolkin, Pedro Teixeira, Emily Witt
2019 The Journal of Software for Algebra and Geometry
We describe a Macaulay2 package for computations in prime characteristic commutative algebra. This includes those for Frobenius powers and roots, p −e -linear and p e -linear maps, singularities defined in terms of these maps, different types of test ideals and modules, and ideals compatible with a given p −e -linear map. After the test element c has been identified, we pull back the ideal ω R to an ideal J ⊆ S. Next, we compute the following ascending sequence of ideals where u represents T :
more » ... e u represents T : ω R 1/ p e − → ω R as above: . . . (4-1) As soon as this ascending sequence of ideals stabilizes, we are done. In fact, because this strategy is used in several contexts, the user can call it directly for a chosen ideal J and u with the function ascendIdeal (this is done for test ideals below). We can also compute parameter test modules of pairs (ω R , f t ) with t ∈ ‫ޑ‬ ≥0 . This is done by modifying the element u when the denominator of t is not divisible by p. When t has p in its denominator, we rely on the fact (see [Blickle et al. 2008 ; Schwede and Tucker 2014a]) that where the second line roughly explains how this is accomplished internally. Here I 1 is τ (ω R , f a ) pulled back to S and, likewise, I 2 defines τ (ω R , f a/ p ) modulo the defining ideal of R. Remark 4.3 (optimizations in ascendIdeal and other testModule computations). Throughout the computations described above, we very frequently use the following fact