| Description |
Test ideals are an important object of study in positive characteristic commutative algebra. The test ideals of a regular ring in positive characteristic enjoy a property known as subadditivity. This subadditivity property has numerous important applications, such as Ein-Lazarsfeld-Smith's argument showing a uniform boundedness in the growth of symbolic powers of ideals in regular rings. In [Tak06], Takagi finds a subadditivity formula for test ideals in the non-regular setting that uses the Jacobian ideal Jac(R) as a correction term. Both Takagi's proof, as well as the original proof of subadditivity in [HY03], uses the classical perspective on test ideals as annihilators of tight closure. In this dissertation, we use the more modern perspective of test ideals described in [HT04] and [Sch10] to find a new subadditivity formula for so-called "big" or "non-finitistic" test ideals, which are conjecturally the same as ordinary test ideals; this conjecture is known to hold for graded rings and for Q-Gorenstein rings. In particular, we use the theory of Cartier algebras introduced in [Sch11]. We introduce a new set of Cartier algebras, called the diagonal Cartier algebras, that measure the failure of R to be smooth. These Cartier algebras appear as correction terms in various versions of our subadditivity formula. This dissertation is organized as follows. In Chapter 1, we review the basic terminology of test ideals and Cartier algebras, as well as the process of "reduction modulo p." In Chapter 2, we introduce our subadditivity formula (Theorem 2.2.11) and show this formula yields sharper containments than Takagi's subadditivity formula (Theorem 2.3.1). Chapter 3 is devoted to showing that our subadditivity formula yields sharper containments than (the mod p reduction of) a subadditivity formula for multiplier ideals found by Eisenstein in [Eis10] (Theorem 3.2.2). To get there, we generalize earlier constructions of test ideals and multiplier ideals with good restriction properties (Definition 3.1.3, Definition 3.1.22) found in [Sch09, Tak10, Eis10]. In Chapter 4, we use our new subadditivity formula to generalize the symbolic power containment formulas of [ELS01, HH02] (Theorem 4.2.4, Proposition 4.3.5, Theorem 4.4.1). Finally, in Chapter 5, we provide a combinatorial description of the diagonal Cartier algebras of toric rings (Theorem 5.0.4). We briefly examine the singularities of rings with large diagonal Cartier algebras in Section 5.1. |
