Heart of a t-structure

Definition

The heart of a t-structure is a full subcategory of the triangulated category that captures the 'nice' objects that exhibit a certain stability under morphisms. It consists of objects that are not only in the category but also possess specific properties related to the t-structure's heart, essentially providing a bridge between homological algebra and the derived category framework.

5 Must Know Facts For Your Next Test

1. The heart of a t-structure is typically an abelian category, which allows for standard constructions like kernels and cokernels to be defined.
2. It serves as a crucial tool for understanding derived categories since many properties of derived categories can be studied through their hearts.
3. In practice, the heart helps in identifying 'bounded' or 'coherent' objects, providing a way to separate good behavior from pathological cases.
4. Many common examples of t-structures arise in the context of coherent sheaves on schemes, where the heart relates to coherent sheaves themselves.
5. The heart can change when the t-structure is altered, showing how sensitive this structure is to modifications in the triangulated category.

Review Questions

• How does the heart of a t-structure relate to the broader concepts in triangulated categories?
  • The heart of a t-structure acts as a critical subset within a triangulated category, representing objects that retain desirable properties under morphisms. By focusing on this heart, one can study various cohomological aspects, leveraging its abelian nature to define operations like limits and colimits. This relationship helps in understanding how triangulated categories behave and interact with homological techniques.
• What role does the heart play in connecting homological algebra and derived categories?
  • The heart of a t-structure serves as an essential link between homological algebra and derived categories by encapsulating objects that behave well under homological operations. By isolating this heart, one can apply familiar techniques from homological algebra, such as projectives and injectives, to analyze derived categories. This connection enables mathematicians to translate problems in derived categories into more manageable problems in their hearts.
• Evaluate how changing the t-structure affects the properties and examples of its heart in practice.
  • Altering the t-structure can significantly influence the properties and examples found within its heart. For instance, modifying the way we define distinguished triangles can lead to different objects being classified as part of the heart, thus changing our understanding of boundedness and coherence within that category. This sensitivity highlights how pivotal the choice of t-structure is in defining the categorical landscape and impacts both theoretical implications and practical computations.

Related terms

Triangulated Category: A category equipped with a translation functor and a distinguished class of triangles that satisfy certain axioms, allowing for the generalization of concepts like exact sequences.

t-structure:

A structure on a triangulated category that allows for the definition of cohomology theories, distinguishing between 'positive' and 'negative' objects.

Derived Category:

A construction that formalizes homological algebra by providing a framework for working with complexes of objects, leading to better handling of exact sequences.