Grothendieck's six operations

Definition

Grothendieck's six operations refer to a collection of functors that arise in the context of derived categories in algebraic geometry. These operations, denoted as $f^*, f_*, f_!, f^!, g^*, g_*$, provide a framework for manipulating sheaves and cohomology, facilitating the study of how these mathematical structures behave under various morphisms between schemes. This approach is fundamental for understanding both classical and derived algebraic geometry.

5 Must Know Facts For Your Next Test

1. The functor $f^*$ is known as the direct image functor, which pulls back sheaves along a morphism.
2. $f_*$ is the direct image sheaf functor, pushing forward sheaves from one scheme to another, preserving certain cohomological properties.
3. $f_!$ is the proper direct image functor that is used when the morphism is proper, ensuring good behavior of cohomological dimensions.
4. $f^!$ is the dualizing functor, which serves as a way to handle duality in algebraic geometry.
5. $g^*$ and $g_*$ are analogous functors defined for another morphism $g$, illustrating how sheaves transform under different mappings.

Review Questions

• How do Grothendieck's six operations relate to the manipulation of sheaves in derived categories?
  • Grothendieck's six operations provide essential tools for manipulating sheaves within derived categories by defining how these structures behave under various morphisms. Each operation—such as $f^*$ for pulling back sheaves or $f_*$ for pushing them forward—allows mathematicians to study their properties and relationships in a structured way. This manipulation is crucial for understanding cohomology theories and the deeper implications of derived algebraic geometry.
• Discuss the importance of proper direct image functor $f_!$ in the context of Grothendieck's six operations.
  • $f_!$, as the proper direct image functor, plays a critical role in preserving cohomological dimensions when dealing with proper morphisms between schemes. Its significance lies in its ability to maintain essential topological features while transforming sheaves, allowing for better control over how geometric structures behave under mappings. This operation helps bridge classical results with modern techniques in derived categories, making it a vital tool in contemporary algebraic geometry.
• Evaluate how Grothendieck's six operations influence modern developments in algebraic geometry and their applications.
  • Grothendieck's six operations have profoundly influenced modern algebraic geometry by providing a robust framework for understanding the interplay between geometry and algebra through derived categories. Their applications extend beyond pure mathematics into areas like mathematical physics and number theory, where cohomological methods are increasingly important. By facilitating deeper insights into the structure of schemes and their sheaves, these operations continue to shape ongoing research and applications in diverse mathematical fields.

Related terms

Derived Categories: A derived category is a construction in mathematics that allows for the formal manipulation of chain complexes, providing a setting where homological algebra can be applied more flexibly.

Sheaf Theory: Sheaf theory is a branch of mathematics that studies the local behavior of functions and sections defined over topological spaces, capturing how these objects can be 'glued' together.

Cohomology: Cohomology is a mathematical tool used to study the properties of spaces by associating algebraic invariants, often through the use of sheaves and derived functors.