Let T: 𝒜 →𝒞 be an exact additive functor between abelian...

Let $T: \mathcal{A} \rightarrow \mathcal{C}$ be an exact additive functor between abelian categories, and suppose that $P$ projective implies $T P$ projective. If $B \in \operatorname{obj}(\mathcal{A})$ and $\mathbf{P}_{B}$ is a deleted projective resolution of $B$, prove that $T \mathbf{P}_{T B}$ is a deleted projective resolution of $T B$.

Key Concepts

Abelian Categories

Abelian categories provide a framework for homological algebra and generalize the structure of modules and abelian groups. They are categories where morphisms have kernels and cokernels, every monomorphism and epimorphism is normal, and every morphism can be factored as an epimorphism followed by a monomorphism. This setting allows the development of exact sequences and homological invariants in a unified way.

Additive Functor

An additive functor between additive categories is a functor that preserves the addition of morphisms and the zero object. In the context of abelian categories, additive functors respect the underlying additive structure and play a crucial role in transferring algebraic information between categories without losing the additive relationships inherent to the objects and morphisms.

Exact Functor

An exact functor is one that preserves exact sequences, mapping short exact sequences in the source category to short exact sequences in the target category. This property is essential in homological algebra because it ensures that the functor behaves well with respect to the kernel–cokernel structure, allowing the transfer of homological properties like acyclicity and the correctness of resolutions.

Projective Object

A projective object in an abelian category is one that has the lifting property with respect to epimorphisms, meaning that any morphism from a projective object to a target can be lifted to a morphism from the projective object to the source of an epimorphism. Such objects are crucial for constructing projective resolutions and for facilitating computations in homological algebra.

Projective Resolution

A projective resolution of an object in an abelian category is an exact sequence consisting of projective objects that 'resolves' the object in question. It is a fundamental tool in homological algebra used to compute derived functors such as Ext and Tor, providing a method to replace a given object with a series of more manageable projective objects.

Deleted Projective Resolution

A deleted projective resolution is a variant of a projective resolution where the object being resolved is omitted from the complex. This complex still retains the essential chain homotopy properties, and when an exact functor is applied, it shows that key properties such as acyclicity and projectivity are preserved, thereby allowing the functor to carry over the resolution structure to the target category.

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