Pushforward

Definition

The pushforward is a fundamental concept in algebraic geometry that describes how a function, particularly a morphism of varieties, can induce a transformation on the associated cohomology or Chow groups. It allows for the transfer of geometric properties from one space to another, particularly when analyzing intersections and cycles within the context of Chow rings.

5 Must Know Facts For Your Next Test

1. The pushforward operation is typically denoted as $f_*: A_*(X) \to A_*(Y)$, where $f: X \to Y$ is a morphism of varieties and $A_*$ denotes the Chow groups.
2. When calculating pushforwards, one must consider how cycles behave under the morphism, particularly focusing on how they are 'pushed forward' into the target space.
3. The pushforward is compatible with intersection products in Chow rings, meaning that if you have two cycles, their pushforwards can interact in a meaningful way regarding their intersections.
4. In the case of a proper morphism, the pushforward of a cycle behaves well and is often well-defined, allowing for nice properties in algebraic geometry.
5. The pushforward can be used to define degrees of morphisms between varieties, which gives valuable geometric insight into how these varieties relate to each other.

Review Questions

• How does the pushforward relate to the concept of morphisms between varieties in terms of cycles and Chow groups?
  • The pushforward connects morphisms between varieties and algebraic cycles by allowing one to translate cycles from a source variety to a target variety. When you have a morphism $f: X \to Y$, the pushforward operation $f_*$ takes a cycle from Chow group $A_*(X)$ and maps it into Chow group $A_*(Y)$. This transformation encapsulates how geometric features are carried over under the mapping, preserving information about intersections and other properties.
• Discuss how the pushforward interacts with intersection products in Chow rings and why this interaction is important.
  • The interaction between pushforwards and intersection products in Chow rings is crucial because it ensures that when you push forward multiple cycles, their intersections are preserved within the new context. Specifically, if you have two cycles $Z_1$ and $Z_2$ in $X$, their pushforwards will yield $f_* (Z_1) \cdot f_* (Z_2)$ in $Y$. This property allows for an analysis of how different geometrical features interact under morphisms, which is key for understanding various geometrical phenomena like intersection theory.
• Evaluate the significance of proper morphisms in relation to the behavior of pushforwards in algebraic geometry.
  • Proper morphisms play a significant role because they ensure that the pushforward operation behaves nicely and is well-defined. In cases where $f: X \to Y$ is proper, the pushforward of a cycle respects the structural properties of Chow groups and retains coherence in intersection theory. This means that we can make reliable conclusions about the behavior of cycles when they are pushed forward through proper maps, which is vital for many results in algebraic geometry regarding compactness and dimensionality.

Related terms

Pullback: The pullback is the operation that allows one to take a function defined on a target space and pull it back to the preimage in the source space, typically preserving the structure of cycles or classes.

Chow Ring: The Chow ring is an algebraic structure that encodes information about algebraic cycles on a variety, allowing for the study of their intersections and algebraic properties through operations like pushforward and pullback.

Cohomology: Cohomology is a mathematical tool used to study topological spaces, providing information about their structure through cohomology groups, which can be related to algebraic cycles via pushforwards and pullbacks.