Progress in Commutative Algebra 2

This is the second of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains surveys on aspects of closure operations, finiteness conditions and factorization. Closure operations on ideals and modules are a bridge between noetherian and nonnoetherian commutative algebra. It contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and more

Representations of rational Cherednik algebras: Koszulness and localisation

An algebra is a typical object of study in pure mathematics. Take a collection of numbers (for example, all whole numbers or all decimal numbers). Inside, you can add and multiply, but with respect to these operations different collections can behave differently. Here is an example of what I mean by this. The collection of whole numbers is called Z. Starting anywhere in Z you can get to anywhere else by adding other members of the collection: 9 + (-3) + (-6) = 0. This is not true with multiplication; to get from 5 to 1 you would need to multiply by 1/5 and 1/5 doesn’t exist in the restricted universe of Z. Enter R, the collection of all numbers that can be written as decimals. Now, if you start anywhere—apart from 0—you can get to anywhere else by multiplying by members of R—if you start at zero you’re stuck there. By adjusting what you mean by ‘add’ and ‘multiply’, you can add and multiply other things too, like polynomials, transformations or even symmetries. Some of these collections look different, but behave in similar ways and some look the same but are subtly different. By defining an algebra to be any collection of things with a rule to add and multiply in a sensible way, all of these examples (and many more you can’t imagine) can be treated in general. This is the power of abstraction: proving that an arbitrary algebra, A, has some property implies that every conceivable algebra (including Z and R) has that property too. In order to start navigating this universe of algebras it is useful to group them together by their behaviour or by how they are constructed. For example, R belongs to a class called simple algebras. There are mental laboratories full of machinery used to construct new and interesting algebras from old ones. One recipe, invented by Ivan Cherednik in 1993, produces Cherednik algebras. Attached to each algebra is a collection of modules (also called representations). As shadows are to a sculpture, each module is a simplified version of the algebra, with a taste of its internal structure. They are not algebras in their own right: they have no sense of multiplication, only addition. Being individually simple, modules are often much easier to study than the algebra itself. However, everything that is interesting about an algebra is captured by the collective behaviour of its modules. The analogy fails here: for example, shadows encode no information about colour. Sometimes the interplay between its modules leads to subtle and unexpected insights about the algebra itself. Nobody understands what the modules for Cherednik algebras look like. One first step is to simplify the problem by only considering modules which behave ‘nicely’. This is what is referred to as category O. Being Koszul is a rare property of an algebra that greatly helps to describe its behaviour. Also, each Koszul algebra is mysteriously linked with another called its Koszul dual. One of the main results of the thesis is that, in some cases, the modules in category O behave as if they were the modules for some Koszul algebra. It is an interesting question to ask, what the Koszul dual might be and what this has to do with Cherednik’s recipe. Geometers study tangled, many-dimensional spaces with holes. In analogy with the algebraic world, just as algebraists use modules to study algebras, geometers use sheaves to study their spaces. Suppose one could construct sheaves on some space whose behaviour is precisely the same as Cherednik algebra modules. Then, for example, theorems from geometry about sheaves could be used to say something about Cherednik algebra modules. One way of setting up this analogy is called localisation. This doesn’t always work in general. The last part of the thesis provides a rule for checking when it does

Representation Theory of Quivers and Finite Dimensional Algebras

Methods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras

Simple artinian rings, hereditary rings and skew fields

In this dissertation, Cohn's methods for constructing skew fields from firs are extended to methods for constructing simple artinian rings from hereditary rings; the flexibility this gives is used in order to prove results about the skew field coproduct, and other skew fields originally investigated by Cohn. Part I is devoted to developing the techniques needed; this has two main themes; the first is a detailed study of the finitely generated projective modules over an hereditary ring; the second is an investigation of the ring construction, universal localisation. The construction of universal homomorphisms from suitable hereditary rings to simple artinian rings leads to the simple artinian coproduct with amalgamation of simple artinian rings, which is the natural generalisation of the skew field coproduct of Cohn. Part II is a detailed study of the skew fields and simple artinian rings that are constructed in this way from firs or more generally from hereditary rings. The finite dimensional division subalgebras are classified (apart from the case of the skew field coproduct of two quadratic extensions of a skew field), the transcendence degree of the commutative subfields is bounded, and the centralisers of transcendental elements is studied in special cases. In addition, there are isolated results on the isomorphism classes of skew field coproducts. Part III is distinct from the rest of the dissertation; it consists of a description of generic solutions to the. partial splitting for finite dimensional central simple algebras. These are twisted forms of grassmannian varieties; this is used to show that over number fields, these varieties satisfy the Hasse principle.<p

Seshadri Constants and Fujita's Conjecture via Positive Characteristic Methods

In 1988, Fujita conjectured that there is an effective and uniform way to turn an ample line bundle on a smooth projective variety into a globally generated or very ample line bundle. We study Fujita's conjecture using Seshadri constants, which were first introduced by Demailly in 1992 with the hope that they could be used to prove cases of Fujita's conjecture. While examples of Miranda seemed to indicate that Seshadri constants could not be used to prove Fujita's conjecture, we present a new approach to Fujita's conjecture using Seshadri constants and positive characteristic methods. Our technique recovers some known results toward Fujita's conjecture over the complex numbers, without the use of vanishing theorems, and proves new results for complex varieties with singularities. Instead of vanishing theorems, we use positive characteristic techniques related to the Frobenius-Seshadri constants introduced by Mustata-Schwede and the author. As an application of our results, we give a characterization of projective space using Seshadri constants in positive characteristic, which was proved in characteristic zero by Bauer and Szemberg.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/149842/1/takumim_1.pd

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum