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Free divisors form a celebrated class of hypersurfaces which has been extensively studied in the past fifteen years. Our main goal is to introduce four new families of homogeneous free divisors and investigate central aspects of the blowup algebras of their Jacobian ideals. For instance, for all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay -- a desirable feature in blowup algebra theory. Moreover, we raise the problem of when the analytic spread of the Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property with tools ranging from cohomology to asymptotic depth. In addition, as an application, we give an ideal-theoretic homological criterion for homaloidal divisors, i.e., hypersurfaces whose polar maps are birational., Comment: 31 pages. Submitted for publication. Dedicated with gratitude to the memory of Professor Wolmer V. Vasconcelos, mentor of generations of commutative algebraists

Our main goal in this paper is to answer new positive cases of the natural generalized version of Hartshorne's celebrated question on cofiniteness of local cohomology modules, and consequently of Huneke's conjecture on the finiteness of their sets of associated primes. Our approach, by means of which we extend several results from the literature, is essentially based on spectral sequence techniques and connections to numerical invariants such as the cohomological dimension and the Gorenstein projective dimension. We also provide, over a polynomial ring, a rather pathological example of a non weakly Laskerian module (i.e., it admits a quotient with infinitely many associated primes) whose first local cohomology module is non-zero and has finitely many associated primes., Comment: 26 pages. Submitted to a journal for publication

We prove new results on the connections between reduction numbers and the Castelnuovo-Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff-Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we generalize a result of A. Mafi on ideals in two-dimensional Cohen-Macaulay local rings, by extending it to arbitrary dimension (and allowing for the setting relative to a Cohen-Macaulay module). We derive a number of applications, including a characterization of (polynomial) ideals of linear type, progress on the theory of generalized Ulrich ideals, and improvements of results by other authors., 21 pages. Submitted for publication

We establish new results on (co)homology vanishing and Ext-Tor dualities, and derive a number of freeness criteria for finite modules over Cohen-Macaulay local rings. In the main application, we settle the long-standing Auslander-Reiten conjecture for the class of Cohen-Macaulay Burch rings, among other results toward this and related problems, e.g., the Tachikawa and Huneke-Wiegand conjectures. We also derive results on further topics of interest such as Cohen-Macaulayness of tensor products and Tor-independence, and inspired by a paper of Huneke and Leuschke we obtain characterizations of when a local ring is regular, or a complete intersection, or Gorenstein; for the regular case, we describe progress on some classical differential problems, e.g., the strong version of the Zariski-Lipman conjecture. Along the way, we generalize several results from the literature and propose various questions., 25 pages. Submitted for publication

We make use of the concepts of Tor-rigid and rigid-test modules, among others, to investigate the interplay between cohomology vanishing and the finiteness of several homological dimensions such as projective, injective and Gorenstein injective dimensions. Besides criteria for the freeness of modules, the applications include new characterizations of Gorenstein and regular local rings, which are also of interest in the realm of algebraic geometry., Comment: Several interesting suggestions have arrived concerning the manuscript, so it will be rewritten and largerly improved in a forthcoming version (possibly with a different title)

In their investigation of horizontal linkage of modules of finite Gorenstein dimension over a commutative, Noetherian, semiperfect (e.g., local) ring, Dibaei and Sadeghi introduced the class of reduced G-perfect modules, making use of Bass' concept of reduced grade. A few years later, the same authors extended this class by considering the relative property of reduced G$_C$-perfection, where $C$ is a semidualizing module, and studied linkage even further. In the present paper, we contribute to their theory and also generalize results of Auslander and Bridger as well as of Martsinkovsky and Strooker. Our investigation includes, for example, when reduced G$_C$-perfection is preserved by relative Auslander transpose, and how to numerically characterize horizontally linked modules under suitable conditions. Along the way, we show how to produce reduced $\textrm{G}_C$-perfect modules that are also $C$-$k$-torsionless (for a given integer $k\geq 0$) but fail to be $\textrm{G}_C$-perfect, and moreover we illustrate that, in contrast to the usual grade, the relative reduced grade does depend on the choice of $C$., Comment: 23 pages

In this paper we further develop the theory of generalized Ulrich modules introduced in 2014 by Goto et al. Our main goal is to address the problem of when the operations of taking the Hom functor and horizontal linkage preserve the Ulrich property. One of the applications is a new characterization of quadratic hypersurface rings. Moreover, in the Gorenstein case, we deduce that applying linkage to sufficiently high syzygy modules of Ulrich ideals yields Ulrich modules. Finally, we explore connections to the theory of modules with minimal multiplicity, and as a byproduct we determine the Chern number of an Ulrich module as well as the Castelnuovo-Mumford regularity of its Rees module., Comment: Final version, to appear in Pacific J. Math

Let $S$ be a Cohen-Macaulay ring which is local or standard graded over a field, and let $I$ be an unmixed ideal that is also generically a complete intersection. Our goal in this paper is multi-fold. First, we give a multiplicity-based characterization of when an unmixed subideal $J \subseteq I^{(m)}$ equals the $m$-th symbolic power $I^{(m)}$ of $I$. Second, we provide a saturation-type formula to compute $I^{(m)}$ and employ it to deduce a theoretical criterion for when $I^{(m)}=I^m$. Third, we establish an explicit linear bound on the exponent that makes the saturation formula effective, and use it to obtain lower bounds for the initial degree of $I^{(m)}$. Along the way, we prove a conjecture (in fact, a generalized version of it) due to Eisenbud and Mazur about ${\rm ann}_S(I^{(m)}/I^m)$, and we propose a conjecture connecting the symbolic defect of an ideal to Jacobian ideals., Comment: 16 pages, comments welcome