Finite injective dimension over rings with Noetherian cohomology

We study rings which have Noetherian cohomology under the action of a ring of cohomology operators. The main result is a criterion for a complex of modules over such a ring to have finite injective dimension. This criterion generalizes, by removing finiteness conditions, and unifies several previous results. In particular we show that for a module over a ring with Noetherian cohomology, if all higher self-extensions of the module vanish then it must have finite injective dimension. Examples of rings with Noetherian cohomology include commutative complete intersection rings and finite dimensional cocommutative Hopf algebras over a field.Comment: 10 page

The derived category of a graded Gorenstein ring

We give an exposition and generalization of Orlov's theorem on graded Gorenstein rings. We show the theorem holds for non-negatively graded rings which are Gorenstein in an appropriate sense and whose degree zero component is an arbitrary non-commutative right noetherian ring of finite global dimension. A short treatment of some foundations for local cohomology and Grothendieck duality at this level of generality is given in order to prove the theorem. As an application we give an equivalence of the derived category of a commutative complete intersection with the homotopy category of graded matrix factorizations over a related ring.Comment: To appear in the MSRI publications volume "Commutative Algebra and Noncommutative Algebraic Geometry (II)

Building modules from the singular locus

A finitely generated module over a commutative noetherian ring of finite Krull dimension can be built from the prime ideals in the singular locus by iteration of three procedures: taking extensions, direct summands, and cosyzygies. In 2003 Schoutens gave a bound on the number of iterations required to build any module, and in this note we determine the exact number. This building process yields a stratification of the module category, which we study in detail for local rings that have an isolated singularity.Comment: Minor corrections; final version to appear in Math. Scand; 8 p

Matrix factorizations over projective schemes

We study matrix factorizations of regular global sections of line bundles on schemes. If the line bundle is very ample relative to a Noetherian affine scheme we show that morphisms in the homotopy category of matrix factorizations may be computed as the hypercohomology of a certain mapping complex. Using this explicit description, we prove an analogue of Orlov's theorem that there is a fully faithful embedding of the homotopy category of matrix factorizations into the singularity category of the corresponding zero subscheme. Moreover, we give a complete description of the image of this functor

Untangling Attorney Retainers from Creditor Claims

Clients will often use a retainer to secure an attorney’s representation. But clients in economic distress may have creditors that are eager to access the client’s funds in the attorney’s hands. Attorneys, clients, courts, and regulators have struggled to understand who has the best claim to such retainer funds. In this Article, we attempt to untangle the most common areas of confusion. We conclude that Article 9 of the Uniform Commercial Code (UCC) offers strong protection for an attorney’s interest in client retainers through security interests, even though some courts have misapplied the UCC in this context. Further, we recommend that regulatory bodies create educational programs to help attorneys and courts understand how to apply Article 9 to security interest and also recommend that attorneys help clients understand the benefits and drawbacks of granting a security interest in retainer funds

Gluing Approximable Triangulated Categories

Given a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category is \emph{approximable} if this modified procedure is possible. Not surprisingly this has proved a powerful tool. For example: the fact that the derived category of a quasi compact, separated scheme is approximable has led to major improvements on old theorems due to Bondal, Van den Bergh and Rouquier. In this article we prove that, under weak hypotheses, the recollement of two approximable triangulated categories is approximable. In particular, this shows many of the triangulated categories that arise in noncommutative algebraic geometry are approximable.Comment: 18 page