Hagen Meltzer

Nilpotentoperators and weighted projective lines

We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), obtained by Serre construction from the triangle singularity x^2+y^3+z^p. For p=6 the Ringel-Schmidmeier classification is thus covered by the classification of vector bundles for tubular type (2,3,6), and then is closely related to Atiyahs classification of vector bundles on a smooth elliptic curve. Returning to the general case, we establish that the stable categories associated to vector bundles or invariant subspaces of nilpotent operators may be naturally identified as triangulated categories. They satisfy Serre duality and also have tilting objects whose endomorphism rings play a role in singularity theory. In fact, we thus obtain a whole sequence of triangulated (fractional) Calabi-Yau categories, indexed by p, which naturally form an ADE-chain.

On equivalences of bernstein-gelfand-gelfand, beilinson and happel

Bernstein-Gelfand-Gelfand showed that the derived category of coherent sheaves on the projective n-space Db(cohPn) is equivalent to the stable ategory of the category of Z-graded finite-dimensional modules over the exterior algebra. At the same time Beilinson gave a description of Db(cohPn) which can be interpreted in terms of tilting theory. We compare these two characterizations using Happels description of the derived category of modules over a finite-dimensional algebra of fnite global dimension.

Exceptional Pairs in Hereditary Categories

We study the category 𝒞(X, Y) generated by an exceptional pair (X, Y) in a hereditary category ℋ. If r = dim k Hom(X, Y) ≥ 1 we show that there are exactly 3 possible types for 𝒞(X, Y), all derived equivalent to the category of finite dimensional modules mod(H r ) over the r-Kronecker algebra H r . In general 𝒞(X, Y) will not be equivalent to a module category. More specifically, if ℋ is the category of coherent sheaves over a weighted projective line 𝕏, then 𝒞(X, Y) is equivalent to the category of coherent sheaves on the projective line ℙ1 or to mod(H r ) and, if 𝕏 is wild, then every r ≥ 1 can occur in this way.

Exceptional Sequences Determined by their Cartan Matrix

We investigate complete exceptional sequences E=(E1,¨,En) in the derived category DbΛ of finite-dimensional modules over a canonical algebra, equivalently in the derived category DbX of coherent sheaves on a weighted projective line, and the associated Cartan matrices C(E)=(〈 [Ei],[Ej]〉). As a consequence of the transitivity of the braid group action on such sequences we show that a given Cartan matrix has at most finitely many realizations by an exceptional sequence E, up to an automorphism and a multi-translation (E1,¨,En)↦(E1[i1],¨,En[in]) of DbΛ. Moreover, we determine a bound on the number of such realizations. Our results imply that a derived canonical algebra A is determined by its Cartan matrix up to isomorphism if and only if the Hochschild cohomology of A vanishes in nonzero degree, a condition satisfied if A is representation-finite.

Matrix factorizations for domestic triangle singularities

Working over an algebraically closed field

Schofield Induction for Sheaves on Weighted Projective Lines

k

Triangle singularities, ADE-chains, and weighted projective lines

of any characteristic, we determine the matrix factorizations for the --- suitably graded --- triangle singularities

Exceptional Modules for Tubular Canonical Algebras

f=x^a+y^b+z^c

An algorithm for the construction of exceptional modules over tubular canonical algebras

of domestic type, that is, we assume that

Group algebras of finite representation type

(a,b,c)