[Lecture] Finite generations of Yoneda algebras for complete intersections

Jan. 25, 2024

Given a finitely generated commutative algebra A and an ideal I with residue ring k = A/I, we attach two graded algebras Tor'(k, k) and Ext"(k, k). The first algebra is always graded commutative and plays important role in algebraic geometry and computing the intersection multiplicities. The second algebra, called the Yoneda algebra, is neither graded commutative and nor finitely generated as k-algebra in general. In this talk we will discuss the graded commutativity of the Yenoda algebra and finite generations properties. It turns out the finite generation is equivalent to the condition that the pair (A, I) is a complete intersection. We will give a complete description of the Yoneda algebras in terms of generator relations when the pair (A, I) is a complete intersection over k. In algebraic geometry, given an algebraic variety X and closed subvariety Z, the algebras TorOx (Oz, Oz) and Extox (Oz, Oz) are closely related to the conormal complex and normal complexes of Z in X. The approach is based on Tate's construction of a resolution T(A/I) of the chain complex0- I+A-+ 0 with I in degree one. It turns out that T(A/I) isa minimal free divided power algebra over the divided power ring (A,I,y). We will show that if a divided power algebra has a finitely generated minimal free divided power resolution, then the Yoneda algebra is finitely generated as algebra over A/I. This is a joint work with Cuipo Jiang and Antoine Caradot in an attempt to understand the cohomological varieties of vertex operator algebras.