arXiv subjects

Aidan Schofield

Publications and source records attributed to Aidan Schofield.

7 recordsLinked to original sources

arXiv2007

Universal localisations of hereditary rings

We describe all possible universal localisations of a hereditary ring in terms of suitable full subcategories of the category of finitely presented modules. For these universal localisations we then identify the category of finitely presented bound modules over the universal localisation as being equivalent to a certain full subcategory of the category of finitely presented bound modules over the original ring. We also describe the abelian monoid of finitely generated projective modules over the universal localisation.

math.RA↗

arXiv2007

Severe right Ore sets and universal localisation

We introduce the notion of a severe right Ore set in the main as a tool to study universal localisations of rings but also to provide a short proof of P. M. Cohn's classification of homomorphisms from a ring to a division ring. We prove that the category of finitely presented modules over a universal localisation is equivalent to a localisation at a severe right Ore set of the category of finitely presented modules over the original ring. This allows us to describe the structure of finitely presented modules over the universal localisation as modules over the original ring.

math.RA↗

arXiv2005

On the number of subrepresentations of a general quiver representation

It is well-known that the intersection multiplicities of Schubert classes in the Grassmanian are Littlewood-Richardson coefficients. We generalize this statement in the context of quiver representations. Here the intersection multiplicity of Schubert classes is replaced by the number of subrepresentations of a general quiver reprsentation, and the Littlewood-Richardson coefficients are replaced by the the dimension of a certain space of semiinvariants.

math.AG↗

arXiv2002

Representations of algebras as universal localizations

Every finitely presented algebra S is shown to be Morita equivalent to the universal localization σ^{-1}R of a finite dimensional algebra R. The construction provides many examples of universal localizations which are not stably flat, i.e. Tor^R_i(σ^{-1}R,σ^{-1}R) is non-zero for some i>0. It is also shown that there is no algorithm to determine if two Malcolmson normal forms represent the same element of σ^{-1}R.

math.RA↗

arXiv1999

Birational classification of moduli spaces of vector bundles over $\mathbb{P}^{2}$

The depth of a vector bundle E over the projective plane P^2 is the largest integer h such that [E]/h is in the Grothendieck group of coherent sheaves on P^2 where [E] is the class of E in this Grothendieck group. We show that a moduli space of vector bundles is birational to a suitable number of h by h matrices up to simultaneous conjugacy where h is the depth of the vector bundles classified by the moduli space. In particular, such a moduli space is a rational variety if h <= 4 and is stably rational when h divides 420.

math.AG↗

arXiv1999

Birational classification of moduli spaces of representations of quivers

Let αbe a Schur root; let h=hcf_v(α(v)) and let p = 1 - < α/h,α/h >. Then a moduli space of representations of dimension vector αis birational to p h by h matrices up to simultaneous conjugacy. Therefore, if h=1,2,3 or 4, then such a moduli space is a rational variety and if h divides 420 it is a stably rational variety.

math.AG↗

arXiv1999

Semi-invariants of quivers for arbitrary dimension vectors

The representations of dimension vector $α$ of the quiver Q can be parametrised by a vector space $R(Q,α)$ on which an algebraic group $\Gl(α)$ acts so that the set of orbits is bijective with the set of isomorphism classes of representations of the quiver. We describe the semi--invariant polynomial functions on this vector space in terms of the category of representations. More precisely, we associate to a suitable map between projective representations a semi-invariant polynomial function that describes when this map is inverted on the representation and we show that these semi-invariant polynomial functions form a spanning set of all semi-invariant polynomial functions in characteristic 0. If the quiver has no oriented cycles, we may replace consideration of inverting maps between projective representations by consideration of representations that are left perpendicular to some representation of dimension vector $α$. These left perpendicular representations are just the cokernels of the maps between projective representations that we consider.

math.RA↗