Torsion and purity on non-integral schemes and singular sheaves in the fine Simpson moduli spaces of one-dimensional sheaves on the projective plane

Doctoral thesis (2016)

This thesis consists of two individual parts, each one having an interest in itself, but which are also related to each other. In Part I we analyze the general notions of the torsion of a module over a ... [more ▼]

This thesis consists of two individual parts, each one having an interest in itself, but which are also related to each other. In Part I we analyze the general notions of the torsion of a module over a non-integral ring and the torsion of a sheaf on a non-integral scheme. We give an explicit definition of the torsion subsheaf of a quasi-coherent O_X-module and prove a condition under which it is also quasi-coherent. Using the associated primes of a module and the primary decomposition of ideals in Noetherian rings, we review the main criteria for torsion-freeness and purity of a sheaf that have been established by Grothendieck and Huybrechts-Lehn. These allow to study the relations between both concepts. It turns out that they are equivalent in "nice" situations, but they can be quite different as soon as the scheme does not have equidimensional components. We illustrate the main differences on various examples. We also discuss some properties of the restriction of a coherent sheaf to its annihilator and its Fitting support and finally prove that sheaves of pure dimension are torsion-free on their support, no matter which closed subscheme structure it is given. Part II deals with the problem of determining "how many" sheaves in the fine Simpson moduli spaces M = M_{dm-1}(P2) of stable sheaves on the projective plane P2 with linear Hilbert polynomial dm-1 for d\geq 4 are not locally free on their support. Such sheaves are called singular and form a closed subvariety M' in M. Using results of Maican and Drézet, the open subset M0 of sheaves in M without global sections may be identified with an open subvariety of a projective bundle over a variety of Kronecker modules N. By the Theorem of Hilbert-Burch we can describe sheaves in an open subvariety of M0 as twisted ideal sheaves of curves of degree d. In order to determine the singular ones, we look at ideals of points on planar curves. In the case of simple and fat curvilinear points, we characterize free ideals in terms of the absence of two coeffcients in the polynomial defining the curve. This allows to show that a generic fiber of M0\cap M' over N is a union of projective subspaces of codimension 2 and finally that M' is singular of codimension 2. [less ▲]

Some unexpected facts about Lie Algebras

Speeches/Talks (2013)

Lie algebras and their morphisms do not behave as nicely as one might expect. We will discuss some of the most important illustrations. For example, the category of Lie algebras admits direct products ... [more ▼]

Lie algebras and their morphisms do not behave as nicely as one might expect. We will discuss some of the most important illustrations. For example, the category of Lie algebras admits direct products, but no coproducts, and the object called "direct sum" is actually not a direct sum in the categorical sense. It admits kernels and cokernels, but it is neither abelian, nor additive, and the image of a morphism is not a categorical image. Nevertheless, one can still define short exact sequences of Lie algebras. We finish by discussing the Splitting Lemma, which for Lie algebras, in general, only holds in a weaker form than in the abelian case. [less ▲]