73 research outputs found
Morita Contexts, Idempotents, and Hochschild Cohomology - with Applications to Invariant Rings -
We investigate how to compare Hochschild cohomology of algebras related by a
Morita context. Interpreting a Morita context as a ring with distinguished
idempotent, the key ingredient for such a comparison is shown to be the grade
of the Morita defect, the quotient of the ring modulo the ideal generated by
the idempotent. Along the way, we show that the grade of the stable
endomorphism ring as a module over the endomorphism ring controls vanishing of
higher groups of selfextensions, and explain the relation to various forms of
the Generalized Nakayama Conjecture for Noetherian algebras. As applications of
our approach we explore to what extent Hochschild cohomology of an invariant
ring coincides with the invariants of the Hochschild cohomology.Comment: 28 pages, uses conm-p-l.sty. To appear in Contemporary Mathematics
series volume (Conference Proceedings for Summer 2001 Grenoble and Lyon
conferences, edited by: L. Avramov, M. Chardin, M. Morales, and C. Polini
Linear free divisors and quiver representations
Linear free divisors are free divisors, in the sense of K.Saito, with linear
presentation matrix (example: normal crossing divisors). Using techniques of
deformation theory on representations of quivers, we exhibit families of linear
free divisors as discriminants in representation spaces for real Schur roots of
a finite quiver. We review some basic material on quiver representations, and
explain in detail how to verify whether the discriminant is a free divisor and
how to determine its components and their equations, using techniques of A.
Schofield. As an illustration, the linear free divisors that arise as the
discriminant from the highest roots of Dynkin quivers of type E7 and E8 are
treated explicitly.Comment: 27 pages; to appear in Singularities and Computer Algebra, papers in
honour of G.-M.Greuel's 60th birthda
Hilbert-Kunz functions of cubic curves and surfaces
We determine the Hilbert-Kunz function of plane elliptic curves in odd
characteristic, as well as over arbitrary fields the generalized Hilbert-Kunz
functions of nodal cubic curves. Together with results of K. Pardue and P.
Monsky, this completes the list of Hilbert-Kunz functions of plane cubics.
Combining these results with the calculation of the (generalized) Hilbert-Kunz
function of Cayley's cubic surface, it follows that in each degree and over any
field of positive characteristic there are curves resp. surfaces taking on the
minimally possible Hilbert-Kunz multiplicity.Comment: LaTex 2e with Xy-pic v3.2 for commutative diagram
New free divisors from old
We present several methods to construct or identify families of free divisors
such as those annihilated by many Euler vector fields, including binomial free
divisors, or divisors with triangular discriminant matrix. We show how to
create families of quasihomogeneous free divisors through the chain rule or by
extending them into the tangent bundle. We also discuss whether general
divisors can be extended to free ones by adding components and show that adding
a normal crossing divisor to a smooth one will not succeed
Lifting free divisors
Let be a morphism between smooth complex analytic spaces,
and let define a free divisor on . We prove that if the deformation
space of is a Cohen-Macaulay -module of
codimension 2, and all of the logarithmic vector fields for lift via
, then defines a free divisor on ; this is
generalized in several directions.
Among applications we recover a result of Mond-van Straten, generalize a
construction of Buchweitz-Conca, and show that a map
with critical set of codimension
has a with the desired properties. Finally, if is a
representation of a reductive complex algebraic group and is the
algebraic quotient with smooth, we describe sufficient
conditions for to be Cohen-Macaulay of codimension . In one such
case, a free divisor on lifts under the operation of
"castling" to a free divisor on , partially generalizing
work of Granger-Mond-Schulze on linear free divisors. We give several other
examples of such representations.Comment: 30 pages. Many minor changes from v1 in response to a thorough review
process. To appear in Proc. London Math. Soc. This version differs from the
final published versio
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