We study, in the setting of algebraic varieties, finite-dimensional spaces of functions $V$ that are invariant under a ring $\Dc^V$ of differential operators, and give conditions under which $\Dc^V$ acts irreducibly. We show how this problem, originally formulated in physics \cite{kamran-Milson-Olver:invariant,turbiner:bochner}, is related to the study of principal parts bundles and Weierstrass points \cite{EGA4,laksov-thorup}, including a detailed study of Taylor expansions. Under some conditions it is possible to obtain $V$ and $\Dc^V$ as global sections of a line bundle and its ring of differential operators. We show that several of the published examples of $\Dc^V$ are of this type, and that there are many more - in particular, arising from toric varieties.

2.

Källström, Rolf

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.

Let (R,m,k) be a regular local k-algebra satisfying the weak Jacobian criterion, and such that k_{R}/k is an algebraic field extension. Let D be the ring of k-linear differential operators of R. We give an explicit decomposition of the DR-module D/Dm^(n+1) as a direct sum of simple modules, all isomorphic to D/Dm, where certain “Pochhammer” differential operators are used to describe generators of the simple components.

We consider dominant, generically algebraic (e. g. generically finite), and tamely ramified (if the characteristic is positive) morphisms pi : X/S -> Y/S of S-schemes, where Y, S are Noetherian and integral and X is a Krull scheme (e. g. normal Noetherian), and study the sheaf of tangent vector fields on Y that lift to tangent vector fields on X. We give an easily computable description of these vector fields using valuations along the critical locus. We apply this to answer the question when the liftable derivations can be defined by a tangency condition along the discriminant. In particular, if p is a blow-up of a coherent ideal I, we show that tangent vector fields that preserve the Ratliff-Rush ideal (equals [I(n+1) : I(n)] for high n) associated to I are liftable, and that all liftable tangent vector fields preserve the integral closure of I. We also generalise in positive characteristic Seidenberg's theorem that all tangent vector fields can be lifted to the normalisation, assuming tame ramification.

Let X/S be a noetherian scheme with a coherent -module M, and T_{X/S} be the relative tangent sheaf acting on M. We give constructive proofs that sub-schemes Y, with defining ideal I_{Y}, of points xX where or M_{x} is “bad”, are preserved by T_{X/S}, making certain assumptions on X/S. Here bad means one of the following: is not normal; has high regularity defect; does not satisfy Serre's condition (R_{n}); has high complete intersection defect; is not Gorenstein; does not satisfy (T_{n}); does not satisfy (G_{n}); is not n-Gorenstein; M_{x} is not free; M_{x} has high Cohen–Macaulay defect; M_{x} does not satisfy Serre's condition (S_{n}); M_{x} has high type. Kodaira–Spencer kernels for syzygies are described, and we give a general form of the assertion that M is locally free in certain cases if it can be acted upon by T_{X/S}.

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.

Purity of Branch and Critical locus2013In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 379, p. 156-178Article in journal (Refereed)

Abstract [en]

To a dominant morphism $X/S \to Y/S$ of N\oe therian integral $S$-schemes one has the inclusion $C_{X/Y}\subset B_{X/Y}$ of the critical locus in the branch locus of $X/Y$. Starting from the notion of locally complete intersection morphisms, we give conditions on the modules of relative differentials $\Omega_{X/Y}$, $\Omega_{X/S}$, and $\Omega_{Y/S}$ that imply bounds on the codimensions of $ C_{X/Y}$ and $ B_{X/Y}$. These bounds generalise to a wider class of morphisms the classical purity results for finite morphisms by Zariski-Nagata-Auslander, and Faltings and Grothendieck, and van der Waerden's purity for birational morphisms.

The tangential branch locus B_{X/Y}^{t}∩B_{X/Y} is the subset of points in the branch locus where the sheaf of relative vector fields T_{X/Y} fails to be locally free. It was conjectured by Zariski and Lipman that if V/k is a variety over a field k of characteristic 0 and B_{V/k}^{t}=∅, then V/k is smooth (= regular). We prove this conjecture when V/k is a locally complete intersection. We prove also that B_{V/k}^{t}=∅ implies codim_{X}B_{V/k}≤1 in positive characteristic, if V/k is the fibre of a flat morphism satisfying generic smoothness.

We consider modules M over Lie algebroids gA which are of finite type over a local noetherian ring A. Using ideals J ⊂ A such that gA ·J ⊂ J and the length ℓgA (M/JM) < ∞ we can define in a natural way the Hilbert series of M with respect to the defining ideal J. This notion is in particular studied for modules over the Lie algebroid of k-linear derivations gA = TA(I) that preserve an ideal I ⊂ A, for example when A = On, the ring of convergent power series. Hilbert series over Stanley-Reisner rings are also considered.