We show that if the cochain complex computing Ext groups (in category of modules over Hopf algebroids) admits a cocyclic structure, then noncommutative Cartan calculus structure on Tor dualises in cyclic sense to Coext Cotor. More precisely, duals chain resp. spaces two classical derived functors lead complexes compute more exotic ones, giving opposite module an operad with multiplication induc...

In this paper, we consider a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question whether such can be extended $n$-Tor-rigid modules, and obtain an affirmative answer $2$-Tor-rigid that are generically free. Furthermore, in appendix, use Dao's eta function determine new classes hypersurfaces quotient un...

Introduction 5 0.1. What is Commutative Algebra? 5 0.2. Why study Commutative Algebra? 5 0.3. Acknowledgments 7 1. Commutative rings 7 1.1. Fixing terminology 7 1.2. Adjoining elements 10 1.3. Ideals and quotient rings 11 1.4. The monoid of ideals of R 14 1.5. Pushing and pulling ideals 15 1.6. Maximal and prime ideals 16 1.7. Products of rings 17 1.8. A cheatsheet 19 2. Galois Connections 20 2...

We study the following properties about primary decomposition over a Noetherian ring R: (1) For finitely generated modules N ⊆ M and a given subset X = {P1, P2, . . . , Pr} ⊆ Ass(M/N), we define an X-primary component of N ( M to be an intersection Q1 ∩ Q2 ∩ · · · ∩ Qr for some Pi-primary components Qi of N ⊆ M and we study the maximal X-primary components of N ⊆ M ; (2) We give a proof of the ...

This paper studies extension groups between certain Weyl modules for the algebraic group GLn over the integers. Main results include: (1) A complete determination of Ext groups between Weyl modules whose highest weights differ by a single root and (2) Determination of Ext between an exterior power of the defining representation and any Weyl module. The significance of these results for modular ...