Month: December 2018

Genus of complex projective algebraic curves

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Celal Cem Sarıoğlu (DEÜ)

December 14, Time:10:15, Place: B259

Abstract: From the Geometry-Topology courses everyone knows the number of holes of an orientable compacts surface is known as a genus,and it is related with its Euler characteristic. On the other hand, in algebraic geometry, there are two related definitions of genus of an irreducible projective algebraic curve C: the arithmetic genus, and the geometric genus. If the curve C has no singular points these two concepts will coincide and also coincide with the topological definition applied to Riemann surfaces of C. In this talk, we will introduce how can we compute the arithmetic and geometric genus of an irreducible projective algebraic curve and how they are related to the genus of an oriented Riemann surface.

Annihilators of Cartier Quotients

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Mehmet Yeşil, The University of Sheffield.
Date: 12th of December, 2018, Thursday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: Let R be a commutative Noetherian ring of prime characteristic p, M be an R-module and e be a positive integer. Let f:R→R be the Frobenius homomorphism given by f(r)=r^p for all r in R whose e-th iteration is denoted by f^{e}. An e-th Cartier map on M is an additive map C:M→M such that rC(m)=C(r^{p^{e}}m) for all r in R and m in M. An R-module is called a Cartier module if it is equipped with a Cartier map. In the case that the Frobenius homomorphism is finite and M is a finitely generated R-module equipped with a surjective Cartier map, it is proved by M. Blickle and G. Böckle in [1] that the set of annihilators of Cartier quotients of M is a finite set of radical ideals consisting of intersections of the finitely many primes in it. In these talks, I will consider the case that R is a finite dimensional polynomial ring over a field of prime characteristic p, and I take a computational view of this finiteness result and drop the finiteness condition on the Frobenius homomorphism to give an alternative proof to the result.
References:
[1] M. Blickle and G. Böckle. Cartier modules: finiteness results.

Representation Theory of Artin Algebras

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Zübeyir Türkoğlu, Dokuz Eylül University.
Date: 5th of December, 2018, Wednesday, Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: In this seminar, we will continue to talk about the transpose Tr. In general, Tr does not induce a functor from the mod(Λ) to the mod(Λ^{op}). But we do get a duality by replacing mod(Λ) with an appropriate category.