The Brauer Monoid is a generalization of the Brauer Group using the Galois cohomological definition by allowing 0 to be in the image of the cocycles. These generalized cocycles are called weak 2-cocycles. This thesis introduces two definitions that help with the computation of the Brauer Monoid for certain idempotents, uses weak Galois cohomology to solve a problem in the theory of finite dimensional algebras, and includes a MATLAB program that produce non-tree idempotents for cyclic Groups.
The Brauer Monoid of a finite Galois field extension K/F with G = Gal(K, F) is denoted by M2( G,K). Each weak 2-cocycle ƒ has an associated weak crossed product algebra Aƒ constructed in an analogous method to that of crossed product algebras. To each idempotent e in the Brauer Monoid there is a group [special characters omitted] with identity e; the Brauer Monoid is the disjoint union of these groups. For a specific ring Re associated with an idempotent e, there is a chain complex of Re-modules, Me, that gives us that [special characters omitted]) . The idempotents e with inertial subgroup He are in one-to-one correspondence with lower subtractive partial orders on G/He and hence with a partial order graph.
This thesis builds on the above. We introduce the definition of a lower subtractive partially ordered subset S of (G, ≤ e) and its associated lower subtractive subgraph. We associate a complex MS to S that is a subcomplex of Me. For S and T lower subtractive partially ordered subsets of G, such that [special characters omitted](G, ≤e) and S, T≠ G, then there exists a split Mayer-Vietoris-like short exact sequence on the complexes:[special characters omitted]
This gives us a long exact sequence on cohomology, which can aid with the computation of H2(HomRe (Me,K ×))≇ [special characters omitted]
We also define a connected lower subtractive partially ordered subset of G and its associated connected lower subtractive subgraph. We provide two constructions and a theorem to determine when a nontrivial decomposition into connected lower subtractive subgraphs exists to be able to apply the short exact sequence to these subsets.
Given a finite Galois extension K/F and a central simple K –algebra B, we classify the F–algebras A that contain B and are weakly Azumaya with respect to B. We use weak Galois cohomology to do this classification.
We also include a MATLAB program that outputs matrices associated with idempotents on a cyclic Galois group such that each idempotent corresponds to a non-tree lower subtractive partial order graph on G. These idempotents contribute nontrivially to M2( G,K), whereas Br(K/F) is trivial when K/F is a finite Galois extension of finite fields.
|Commitee:||Larsen, Michael, Lindenstrauss, Ayelet, Strauch, Matthias|
|School Location:||United States -- Indiana|
|Source:||DAI-B 75/03(E), Dissertation Abstracts International|
|Keywords:||Azumaya algebras, Brauer Monoid, Galois extension|
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