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Hochster’s theta invariant is defined for a pair of finitely generated modules on a hypersurface ring having only an isolated singularity. Up to a sign, it agrees with the Euler invariant of a pair of matrix factorizations.
Working over the complex numbers, Buchweitz and van Straten have established an interesting connection between Hochster’s theta invariant and the classical linking form on the link of the singularity. In particular, they establish the vanishing of the theta invariant if the hypersurface is even-dimensional by exploiting the fact that the (reduced) cohomology of the Milnor fiber is concentrated in odd degrees in this situation.
We give purely algebraic versions of some of these results. In particular, we establish the vanishing of the theta invariant for isolated hypersurface singularities of even dimension in characteristic under some mild extra assumptions. This confirms, in a large number of cases, a conjecture of Hailong Dao.
We compute some motivic stable homotopy groups over . For , we describe the motivic stable homotopy groups of a completion of the motivic sphere spectrum. These are the first four Milnor–Witt stems. We start with the known groups over and apply the -Bockstein spectral sequence to obtain groups over . This is the input to an Adams spectral sequence, which collapses in our low-dimensional range.
We describe an arithmetic -valued invariant for longitudes of a link , obtained from an representation of the link group. Furthermore, we show a nontriviality on the elements, and compute the elements for some links. As an application, we develop a method for computing longitudes in representations for link groups, where is the universal covering group of .
Let be an affine group scheme over a noetherian commutative ring . We show that every -equivariant vector bundle on an affine toric scheme over with -action is equivariantly extended from for several cases of and .
We show that, given two affine schemes with group scheme actions, an equivalence of the equivariant derived categories implies isomorphism of the equivariant -theories as well as equivariant -theories.
We show that the theory of motivic complexes developed by Voevodsky over perfect fields works over nonperfect fields as well provided that we work with sheaves with transfers of -modules (). In particular we show that every homotopy invariant sheaf with transfers of -modules is strictly homotopy invariant.
We define a -theory for pointed right derivators and show that it agrees with Waldhausen -theory in the case where the derivator arises from a good Waldhausen category. This -theory is not invariant under general equivalences of derivators, but only under a stronger notion of equivalence that is defined by considering a simplicial enrichment of the category of derivators. We show that derivator -theory, as originally defined, is the best approximation to Waldhausen -theory by a functor that is invariant under equivalences of derivators.
We classify the split simple affine algebraic groups of types A and C over a field with the property that the Chow group of the quotient variety is torsion-free, where is a special parabolic subgroup (e.g., a Borel subgroup) and is a generic -torsor (over a field extension of the base field). Examples of include the adjoint groups of type A. Examples of include the Severi–Brauer varieties of generic central simple algebras.