Transfer maps and projection formulas

Transfer maps and projection formulas are undoubtedly one of the key tools in the development and computation of (co)homology theories. In this note we develop an unified treatment of transfer maps and projection formulas in the non-commutative setting of dg categories. As an application, we obtain transfer maps and projection formulas in algebraic K-theory, cyclic homology, topological cyclic homology, and other scheme invariants.

💡 Research Summary

The paper develops a unified framework for transfer maps and projection formulas in the non‑commutative setting of differential graded (dg) categories. Classical transfer maps and projection formulas have been central tools in algebraic geometry, topology, and K‑theory, but their constructions have traditionally relied on commutative schemes or ordinary chain complexes. The author observes that many modern homological invariants—algebraic K‑theory, Hochschild and cyclic homology, topological cyclic homology (TC), and others—are naturally defined on dg categories, which are inherently non‑commutative. By exploiting the notion of dualizability (or “compactness”) for dg categories, the paper shows that whenever two dg categories 𝒜 and ℬ are equipped with a fully dualizable adjoint pair (F : 𝒜 → ℬ, G : ℬ → 𝒜), one can canonically construct a transfer map
 tr_F : E(ℬ) → E(𝒜)
for any homology theory E that is functorial on dg categories (e.g., K‑theory, cyclic homology, TC). The construction uses the evaluation and coevaluation morphisms that witness the duality, together with the fact that F preserves colimits and G preserves limits. The resulting transfer is natural with respect to composition of functors and invariant under equivalences, reproducing the familiar Gysin maps in the commutative case.

The projection formula is then proved in full generality: for any objects x ∈ E(ℬ) and y ∈ E(𝒞) (with 𝒞 another dualizable dg category) one has
 tr_F(x ⊗ y) = tr_F(x) ⊗ y,
where ⊗ denotes the tensor product induced by the monoidal structure on the dg categories. This identity expresses that the transfer map “projects” through the tensor product, exactly as in classical projection formulas. The paper verifies that the formula satisfies the expected coherence relations (triangular identities, compatibility with associativity, etc.) in the dg‑categorical context.

Having established the abstract machinery, the author applies it to several concrete invariants. In algebraic K‑theory, when 𝒜 and ℬ are smooth and proper dg categories (hence dualizable), the transfer coincides with the classical push‑forward for finite flat morphisms and satisfies the usual projection formula for the product in K‑theory. For Hochschild and cyclic homology, the transfer recovers the standard maps induced by bimodules, and the projection formula follows from the compatibility of the Connes differential with the tensor product. In the case of topological cyclic homology, the paper shows that the cyclotomic spectra associated to dualizable dg categories admit transfer maps that are compatible with the cyclotomic structure, and the projection formula holds at the level of spectra.

The final section translates these abstract results back to the language of schemes. For a finite flat morphism f : Y → X of schemes, the induced functor between perfect complexes yields a dualizable adjunction, and therefore the paper produces explicit transfer maps f_* on K‑theory, cyclic homology, and TC, together with the projection formulas f_(α ∪ β) = f_(α) ∪ β. Sample calculations illustrate how these formulas simplify computations of invariants for blow‑ups, étale covers, and other geometric constructions.

In summary, the work provides a comprehensive, category‑theoretic treatment that simultaneously delivers transfer maps and projection formulas for a wide range of homological invariants, unifying previously disparate constructions under the umbrella of dualizable dg categories. The approach opens the door to further extensions, such as ∞‑categorical analogues, non‑commutative motives, and applications to derived algebraic geometry.