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This is a joint work with E.

Ballico, F. Malaspina and J. For instance, the existence of Ulrich bundles on a given projective variety implies that the cone of cohomology tables has the same structure as the one of the projective space of the same dimension.

Unfortunately, the existence problem seems to be difficult even for smooth surfaces. In this talk, I will review a few known constructions of special Ulrich bundles using the Brill-Noether theory of curves by Beauville and Aprodu-Farkas-Ortega.

And I will introduce a relation between special Ulrich bundles on a blown-up and the original surface. In this talk, I show a geometric characterization of smooth projective varieties with 2-regular structure sheaves using adjunction theory. Then I study syzygies of section rings of projected varieties with 2-regular structure sheaves, and give a sharp bound for Castelnuovo-Mumford regularity of such varieties.

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This is joint work in progress with Sijong Kwak. We may therefore form its right derived functors. The definition using derived functors is conceptually very clear, but for concrete applications, the following computations, which some authors also use as a definition, are often helpful.

This is an abelian group; its elements are called the inhomogeneous n -cochains. The coboundary homomorphisms. It can be shown that the above-mentioned definition of group cohomology in terms of derived functors is isomorphic to the cohomology of this complex. Therefore, as Ext functors are the derived functors of Hom , there is a natural isomorphism.

Low Order Cohomology And Applications

We recall the definition of Ext more explicitly for this context. This construction initially leads to a coboundary operator that acts on the "homogeneous" cochains. Assigning to M its so-called coinvariants , the quotient. Its left derived functors are by definition the group homology. Specifically, the homology groups H n G , M can be computed as follows.

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Group homology and cohomology can be treated uniformly for some groups, especially finite groups , in terms of complete resolutions and the Tate cohomology groups. The first cohomology group is the quotient of the so-called crossed homomorphisms , i. This follows from the definition of cochains above. If M is a trivial G -module i. An example of a second group cohomology group is the Brauer group : it is the cohomology of the absolute Galois group of a field k which acts on the invertible elements in a separable closure:.

The so-called connecting homomorphisms ,. This map is called the restriction map. If the index of H in G is finite, there is also a map in the opposite direction, called transfer map , [6]. Similarly to other cohomology theories in topology and geometry, such as singular cohomology or de Rham cohomology , group cohomology enjoys a product structure: there is a natural map called cup product :. For example, let G be the group with two elements, under the discrete topology. As for other cohomology theories, such as singular cohomology , group cohomology and homology are related to one another by means of a short exact sequence [8].

From it, one gets the inflation-restriction exact sequence. Group cohomology is closely related to topological cohomology theories such as sheaf cohomology , by means of an isomorphism.

Christian Liedtke: Crystalline cohomology, period maps, and applications to K3 surfaces

The expression BG at the left is a classifying space for G. However, BG does not usually have an easily amenable geometric description.

More generally, one can attach to any G -module M a local coefficient system on BG and the above isomorphism generalizes to an isomorphism [11]. Indeed, by Maschke's theorem the category of representations of a finite group is semi-simple over any field of characteristic zero or more generally, any field whose characteristic does not divide the order of the group , hence, viewing group cohomology as a derived functor in this abelian category, one obtains that it is zero.

Low Order Cohomology And Applications

The other argument is that over a field of characteristic zero, the group algebra of a finite group is a direct sum of matrix algebras possibly over division algebras which are extensions of the original field , while a matrix algebra is Morita equivalent to its base field and hence has trivial cohomology. Tate cohomology groups combine both homology and cohomology of a finite group G :.

Veranstaltung zum Jahr der Mathematik und zum The Farrell Conjecture for algebraic K-theory with arbitrary coefficients for word-hyperbolic groups. The algebraic K-theory of the group ring of a word hyperbolic group with arbitrary coefficient ring.

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