Semidualizing DG Modules over Tensor Products

• Main Author: Altmann, Hannah Lee
• Format: Others
• Published: North Dakota State University 2018
• Online Access: https://hdl.handle.net/10365/27679

In this dissertation, we study rings: sets with addition, subtraction, and multiplication. One way to study a ring is by studying its modules: the algebraic objects the ring acts on. Since it is impractical to study all of its modules, I study its semidualizing modules. These modules have proven useful in the study of the composition of local ring homomorphisms of finite G-dimension and Bass numbers of local rings. Let R be a commutative, noetherian ring with identity. A finitely generated R-module C is semidualizing if the homothety map χ(R/C) : R → HomR(C,C) is an isomorphism and Ext(i/R)(C,C) = 0 for all i > 0. For example, the ring R is semidualizing over itself, as is a dualizing module, if R has one. In some sense the number of semidualizing modules a ring has gives a measure of the "complexity" of the ring. I am interested in that number. More generally in this dissertation we use the definition of semidualizing differential graded (DG) module, pioneered by Christensen and Sather-Wagstaff. In particular, I construct semidualizing DG modules over the tensor product of two DG k-algebras, say A' and A''. This gives us a lower bound on the number of semidualizing DG modules over the tensor product A' ⊗ k A''. Therefore, as far as semidualizing DG modules can detect, the singularity of A' ⊗ k A'' is at least as bad as the singularities of both A' and A'' combined.