Graduate Seminar

Location:  RI 209

Speaker: Luis Islas Vizcarra

MSc Student supervised by Martin Frankland and Don Stanley

Title: From Commutative Rings to Balmer Spectra

 Abstract:

Commutative rings are fundamental objects in algebra, and there are several ways to study them. Two perspectives that will be central to this talk are the spectrum of a commutative ring, which associates a geometric object to an algebraic one, and the derived category of modules, which provides a homological framework for the study of chain complexes up to quasi-isomorphism.

This talk will introduce the concepts that bring these two perspectives together. The resulting framework consists of categories equipped with both a tensor product and a triangulated structure, known as tensor-triangulated categories. These categories serve as the input for a construction known as the Balmer spectrum, which provides a categorical analog of the spectrum of a commutative ring. In a natural example involving the derived category of modules over a given ring, the Balmer spectrum recovers the original spectrum of the ring.

Along the way, we will develop the notions of ideals and prime ideals in the categorical setting, and illustrate the theory through several examples.