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Dedicated to Professor Craig Huneke on the occasion of his sixty-fifth birthday.Abstract. We show the existence of F -thresholds in full generality. In addition, we study properties of standard graded algebras over a field for which F -pure threshold and F -threshold at the irrelevant maximal ideal agree. We also exhibit explicit bounds for the a-invariants and Castelnuovo-Mumford regularity of Frobenius powers of ideals in terms of F -thresholds and F -pure thresholds, obtaining the existence of related limits in certain cases.

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.

In this note we study the partition of R n ≥0 given by the regions where the mixed test ideals τ (a t 1 1 ...a tn n ) are constant. We show that each region can be described as the preimage of a natural number under a p-fractal function ϕ : R n ≥0 → N. In addition, we give some examples illustrating that these regions do not need to be composed of finitely many rational polytopes.

We establish a "second vanishing theorem" for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this, and new results on the mixed characteristic Lyubeznik numbers, we further study connectedness properties of the spectra of a certain class of rings.1 I (S) = E S (K) ⊕t−1 by induction on t, noting that the result follows from Theorem 3.8 if t = 1. Fix t > 1, and assume that for any ideal J of S for which dim(S/q) ≥ 3 for every minimal prime q of J, if Spec • (S/J) has t − 1 connected components, then H n−1 J (S) = E S (K) ⊕t−2 . In particular, this holds forConsider the following piece of the Mayer-Vietoris sequence in local cohomology associated to J and J t :

We investigate the Lyubeznik numbers, and the injective dimension of local cohomology modules, of finitely generated Z-algebras. We prove that the mixed characteristic Lyubeznik numbers and the standard ones agree locally for almost all reductions to positive characteristic. Additionally, we address an open question of Lyubeznik that asks whether the injective dimension of a local cohomology module over a regular ring is bounded above by the dimension of its support. Although we show that the answer is affirmative for several families of Z-algebras, we also exhibit an example where this bound fails to hold. This example settles Lyubeznik's question, and illustrates one way that the behavior of local cohomology modules of regular rings of equal characteristic and of mixed characteristic can differ.1 To avoid contemplating the dimension of ∅, we only consider property (3) for nonzero M . 1 F p ⊗ Z R = R/pR the reduction of R to characteristic p > 0. In this context, one may specialize Question A, and instead ask for which reductions of a given finitely generated Z-algebra do the standard and 2 In [NBW13, NBWZ14], these invariants are referred to as Lyubeznik numbers in mixed characteristic. 3 This terminology is not intended to imply that R must be of mixed characteristic in order to define λi,j (R) (this would be false), but is chosen to emphasize that the integers λi,j (R) arise from considering R (or R) as a quotient of a regular unramified ring of mixed characteristic.

Let R be a standard graded finitely generated algebra over an F -finite field of prime characteristic, localized at its maximal homogeneous ideal. In this note, we prove that that Frobenius complexity of R is finite. Moreover, we extend this result to Cartier subalgebras of R.

Abstract. We introduce two families of ideals, F -jumping ideals and F -Jacobian ideals, in order to study the singularities of hypersurfaces in positive characteristic. Both families are defined using the D-modules M α that were introduced by Blickle, Mustaţȃ and Smith. Using strong connections between F -jumping ideals and generalized test ideals, we give a characterization of F -jumping numbers for hypersurfaces. Furthermore, we give an algorithm that determines whether certain numbers are F -jumping numbers. In addition, we use F -Jacobian ideals to study intrinsic properties of the singularities of hypersurfaces. In particular, we give conditions for F -regularity. Moreover, F -Jacobian ideals behave similarly to Jacobian ideals of polynomials. Using techniques developed to study these two new families of ideals, we provide relations among test ideals, generalized test ideals, and generalized Lyubeznik numbers for hypersurfaces.