Investigate the conditions under which a functor's right derived functor preserves a complex's structure in the derived category | Step-by-Step Solution

TinyProf's Guide to Derived Functor Invariance

What We're Solving: We need to investigate whether the right derived functor RHom(F,C) is isomorphic to a complex C when we have a left exact functor F that preserves homology. This is asking about a fundamental invariance property in homological algebra!

The Approach: This problem touches on some deep concepts in category theory and homological algebra. We're essentially asking: "When does applying a derived functor leave our complex essentially unchanged?"

Step-by-Step Solution:

Step 1: Clarify the Setup We have:

• An abelian category C (think of this as a nice setting where we can do homological algebra)
• D(C) is the derived category (where we work with complexes up to quasi-isomorphism)
• F is a left exact functor (preserves finite limits and exact sequences on the left)
• The condition "F preserves homology" means F takes quasi-isomorphic complexes to quasi-isomorphic complexes

Step 2: Understanding RHom(F,C) The notation suggests examining:

• RF(C) ≅ C (applying the right derived functor of F to complex C)
• Or RHom(A,C) for some object A related to F

Step 3: The Key Insight For a left exact functor F that preserves homology, we're looking at when derived functors behave "nicely." The crucial point is:

If F preserves homology AND we're working in the derived category, then F already captures the "derived" information without needing to derive it further!

Step 4: The Homology Preservation Condition When F preserves homology:

• F takes exact sequences to exact sequences (up to quasi-isomorphism)
• The derived functor RF essentially becomes F itself on the level of D(C)
• This is because F is already "doing the job" that the derived functor RF was designed to do

The Answer: Yes, under the given conditions, we typically get an isomorphism! When F is left exact and preserves homology, then in the derived category D(C), the right derived functor behaves as an equivalence on the objects it acts upon.

More precisely: If F preserves homology, then the natural morphism F(C) → RF(C) is a quasi-isomorphism, giving us RF(C) ≅ F(C) ≅ C in D(C) under appropriate conditions.

Memory Tip: Think of it this way: "If F already preserves homology, then deriving F doesn't add new information - it just confirms what F was already doing correctly!" The derived functor becomes redundant when the original functor already behaves well with respect to homology.

Great question! This touches on some beautiful connections between functors and their derived versions. The key insight is that when a functor already "behaves well" (preserves homology), the derived version doesn't need to "fix" anything - they essentially agree!