Conference topic
This is from discussions at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Zijian Yao for explaining the material.
Consider a morphism of schemes φ:S′→S and coherent sheaves F,G over S. Consider also a map of sheaves f:F→G and a map f′ between the pullbacks of F and G, as described by the diagram below.
There are two natural questions to ask.
Remark: If 1. is true, then p∗1(f′)=p∗2(f′). If the previous statement is an equivalence, then φ is a morphism of descent.
Remark: If 2. is true, then there exists α:p∗1(G′)→p∗2(G′) such that π∗32(α)π∗21(α)=π∗31(α) and π∗(Δ)=α. If the previous statement is an equivalence, then φ is effective.
Consider a morphism of schemes φ:S′→S and coherent sheaves F,G over S. Consider also a map of sheaves f:F→G and a map f′ between the pullbacks of F and G, as described by the diagram below.
There are two natural questions to ask.
- When is f′=φ∗f?
- If we start with G′ over S′, when is G′=φ∗G?
Remark: If 1. is true, then p∗1(f′)=p∗2(f′). If the previous statement is an equivalence, then φ is a morphism of descent.
Remark: If 2. is true, then there exists α:p∗1(G′)→p∗2(G′) such that π∗32(α)π∗21(α)=π∗31(α) and π∗(Δ)=α. If the previous statement is an equivalence, then φ is effective.
No comments:
Post a Comment