Projective Euler values from a single generating degree

A graded projective generated in degree d has maps into a shift of a degree-zero module only for shift d. Its q-Euler value is therefore a single Laurent monomial of exponent -d. This computes projective multiplicities from maps to degree-zero simples, without requiring a cohomological bound on resolutions of those simples.

References

• A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, Section 1.2, for generating degrees and maps to shifted simples.
• Z. Dancso and A. Licata, Koszul algebras and flow lattices, Section 2.2, for the q-antilinear convention in the first variable.

theorem TauCeti.GradedModuleCat.gradedExtEuler_eq_C_mul_T_of_isGeneratedInDegree {k : Type uk} [Field k] {A : Type uA} [Ring A] [Algebra k A] {𝒜 : ℤ → Submodule k A} [CategoryTheory.HasExt (GradedModuleCat 𝒜)] (P N : GradedModuleCat 𝒜) [CategoryTheory.Projective P] {d : ℤ} (hP : P.grading.IsGeneratedInDegree A d) (hN : ∀ (p : ℤ), p ≠ 0 → N.grading.piece ℤ p = ⊥) (h : IsGradedEulerAdmissible k (shift 𝒜) P N) : gradedExtEuler k (shift 𝒜) h = LaurentPolynomial.C ↑(Module.finrank k (P ⟶ N.shiftObj d)) * LaurentPolynomial.T (-d)

A projective generated in degree d pairs with a degree-zero target by the monomial dim Hom(P,N{d}) q⁻ᵈ. The admissibility witness concerns only this projective pair.