RESEARCH STATEMENT BETHANY KUBIK My research is in the field of commutative algebra. My main area of interest is homological algebra. I have four projects that I am currently working on; three of these projects are discussed in the Section 1 and the fourth project is discussed in the Section 2. I am interested in properties of modules over a commutative local noetherian ring, (R, m, k). Much of the literature in this area focuses on R-modules that are finitely generated. Since R is noetherian, these are precisely the noetherian R-modules, in other words, the modules that satisfy the ascending chain condition on submodules. In my research I am working to apply techniques used to study noetherian modules to other classes of modules. One main focus of my work has been on artinian modules, those modules that satisfy the descending chain condition on submodules. Other classes of modules are discussed below. To summarize Section 1, given R-modules L and L, I am interested in understanding how properties of L and L translate to properties of their homomorphism set Hom R (L, L ), to their tensor product L R L, and to their cohomological counterparts Ext i R (L, L and Tor R i (L, L ). It is well-known that if N and N are noetherian R-modules, then the modules Ext i R (N, N ) and Tor R i (N, N ) are noetherian. Also, if A is an artinian module, we know that the modules Ext i R (N, A) and Tor R i (N, A) are artinian. In essence, the functors Ext i R (N, ) and TorR i (N, ) respect any class C of R-modules that satisfies a technical condition, namely that C is closed under extensions, submodules, and quotients. In particular, this includes the class of mini-max modules: an R-module M is mini-max if it has a noetherian submodule N M such that the quotient M/N is artinian. From the definition it is evident that the class of mini-max modules contains the class of noetherian modules as well as the class of artinian modules. For a non-noetherian module L, the functors Ext i R (L, ) and TorR i (L, ) do not always respect classes of R-modules in a similar manner. For example, if A is an artinian module, then the functor Tor R i (A, ) turns noetherian modules into artinian modules. However, under certain circumstances these classes of R-modules are respected. The three projects described in Section 1 investigate the functors Ext i R (M, ) and TorR i (M, ) where M is mini-max. 1
2 BETHANY KUBIK My fourth project, discussed in Section 2, investigates a class of modules known as quasidualizing modules. This class can be thought of as the artinian counterpart to the noetherian concept of a semidualizing module. One thread of the investigation focuses on how questions about semidualizing modules can be transformed into questions about quasidualizing modules and vice versa. Another thread of the investigation focuses on the properties of quasidualizing modules. An emphasis is placed on those properties that exist for quasidualizing modules but do not exist for semidualizing modules. My fourth project aims to understand the nature of quasidualizing modules. 1. Homology of Artinian and Mini-max Modules The first aspect of my research is joint work with my advisor Sean Sather-Wagstaff and Micah Leamer, a graduate student at University of Nebraska Lincoln; see [4], [5], [6]. Let R denote the m-adic completion of R, and let E = E R (k) denote the injective hull of k, with ( ) = Hom R (, E). An R-module M is said to be Matlis reflexive if M = M. In [5], we investigate the properties of Ext i R (M, ) and TorR i (M, ) where M is mini-max; e.g. when M is artiniain. The next theorem describes the properties of Ext i R (A, ) and TorR i (A, ) where A is artinian. Theorem 1.1. Let A be an artinian R-module and let M be a mini-max R-module. Fix an index i 0. Then the module Tor R i (A, M) is artinian and the module Ext i R (A, M) is noetherian over R. For our results, it is important to note that a module A is artinian over R if and only if A is artinian over R. Standard examples show that Ext i R (A, M) need not be noetherian over R or artinian and Tor R i (A, M) need not be noetherian over R or over R. In the next result, we consider Theorem 1.1 in the case where i = 0 and we replace the mini-max module M with either an artinian or a noetherian module. Theorem 1.2. Let A and A be artinian R-modules and let N be a noetherian R-module. Then the modules A R A and Hom R (A, N) have finite length. This result was unexpected because, when we began the investigation, it was not clear to us that Hom R (A, N) and A R A would be either artinian or noetherian, let alone both. In the next theorem we replace the artinian module A in Theorem 1.1 with a mini-max module.
RESEARCH STATEMENT 3 Theorem 1.3. Let A, M, and M be R-modules such that A is artinian and M and M are mini-max. Fix an index i 0. Then the module Tor R i (M, M ) is mini-max over R and the module Ext i R (M, A) is Matlis reflexive over R. A module M is Matlis reflexive over R when M = M vv where ( ) v = Hom R(, E). If we add another condition to the assumptions of Theorem 1.3, we can achieve better results as seen in the following corollary. Corollary 1.4. Let M and M be mini-max R-modules and let R/(Ann R (M) + Ann R (M )) be complete. Fix an index i 0. The modules Tor R i (M, M ) and Ext i R (M, M ) are Matlis reflexive over R and R. This corollary recovers some results from Belshoff [1]. It is important to note that the quotient R/(Ann R (M) + Ann R (M )) is complete whenever either M or M is Matlis reflexive. This follows from a result of Belshoff, Enochs, and García Rozas [2] who showed that a module M is Matlis reflexive if and only if M is mini-max and R/(Ann R (M)) is complete. Our work also describes the Matlis dual of the modules discussed in the theorems above. Theorem 1.5. Let A, M, and M be R-modules such that A is artinian and M and M are mini-max. Fix an index i 0. There is an isomorphism Ext i R(A, M ) v = Tor R i (A, M ) where ( ) v = Hom R(, E). If R/(Ann R (M) + Ann R (M )) is complete, then Ext i R(M, M ) v = Ext i R(M, M ) = Tor R i (M, M ). To compute the Matlis dual of Tor R i (M, M ), we have the well-known isomorphism Tor R i (M, M ) = Ext i R (M, M ) as a consequence of Hom-tensor adjointness; note that this isomorphism holds with no restrictions on M or M. In [4], we look at conditions under which the isomorphism in Theorem 1.5 holds when the module E is replaced by an arbitrary injective module. Also, we consider conditions under which the tensor evaluation morphism is an isomorphism, such as the following.
4 BETHANY KUBIK Theorem 1.6. Let R and S be rings with φ : R S a local flat homomorphism of rings. Let M and M be R-modules such that M is mini-max over R and M is Matlis reflexive over R. Then there is an isomorphism S R Hom R (M, M ) = Hom R (M, S R M ). In [6], we develop the tools to prove analogous results when R is not local. We apply the results of this section to the study of certain dualities described in the next section. 2. Quasidualizing Modules The second aspect of my research is a study of quasidualizing modules ; see [3]. The launching point for this investigation is the concept of a semidualizing module: an R-module C is semidualizing if it is noetherian and satisfies the conditions Hom R (C, C) = R and Ext i R (C, C) = 0 for each integer i > 0. For example, the module C = R is semidualizing. My research takes the tools and techniques used to understand semidualizing modules and applies them to a different class of modules, the quasidualizing modules: an R-module T is quasidualizing if it is artinian and satisfies the conditions Hom R (T, T ) = R and Ext i R (T, T ) = 0 for each integer i > 0; again R is the completion of R. For example, the injective hull E of the residue field of R is quasidualizing. Indeed, this example was the primary motivation for investigating these modules. Given the similarity in the definitions it is natural to ask when a quasidualizing module is semidualizing and vice versa. In the particular case when R is artinian, an R-module T is quasidualizing if and only if it is semidualizing. In general, though, these two types of R-modules are very different. We would like to understand the relations between these two types. Again, we take our cues from Matlis duality. It is well known that, when R is complete, this duality turns noetherian modules into artinian modules, and vice versa. This lays the foundation for my next theorem: Theorem 2.1. Assume that R is complete. Then the class of quasidualizing modules is in bijection with the class of semidualizing modules under Matlis duality: T is quasidualizing if and only if T is semidualizing, and T is semidualizing if and only if T is quasidualizing. My next two theorems explain the connections between this project and the ideas from Section 1. These theorems explain the relationship between certain classes determined by a quasidualizing module T and its Matlis dual T. The proof uses Theorem 1.5 and requires some definitions. If C is a semidualizing R-module, we say that a module M is totally C-reflexive, denoted M G C (R), if
RESEARCH STATEMENT 5 M is noetherian, the map M Hom R (Hom R (M, C), C) is an isomorphism and Ext i R (M, C) = 0 = Ext i R (Hom R(M, C), C) for each integer i > 0. If T is a quasidualizing R-module, we say that a module M is really T -reflexive, denoted M H T (R), if M is artinian, the map M Hom R (Hom R (M, T ), T ) is an isomorphism and Ext i R (M, T ) = 0 = Exti R (Hom R(M, T ), T ) for each integer i > 0. Naturally we wonder what the relationship is between totally C-reflexive modules and really T -reflexive modules. Because there exists a natural bijection in Theorem 2.1, it is natural to ask if a similar relationship exists between the classes of totally C-reflexive modules and really T -reflexive modules. Instead of a simple bijection linking these two classes of modules, I found the relationship to be a bit more complex. In particular, I found connections to the Auslander class and the Bass class: a module M is in the Bass class with respect to N, denoted B N (R), if the natural evaluation map Hom R (N, M) R N M is an isomorphism and Ext i R (N, M) = 0 = TorR i (N, Hom R (N, M)) for each i 0. A module M is in the Auslander class with respect to N, denoted A N (R), if the natural map M Hom R (N, N R M) is an isomorphism and Ext i R (N, N R M) = 0 = Tor R i (N, M) for each i 0. The following theorems describe how the relations work in the case of really T -reflexive modules. Theorem 2.2. Assume that R is complete. Then the class of really T -reflexive modules is in bijection with the class of noetherian modules in the Bass class B T (R) under Matlis duality: M is really T -reflexive if and only if M is noetherian and in the Bass class B T (R). Theorem 2.3. Assume that R is complete. Then the class of really T -reflexive modules equal to the class of artinian modules in the Auslander class A T (R) : M is really T -reflexive if and only if M is artinian and in the Auslander class A T (R). I found similar relationships between modules in the totally C-reflexive class, where C is a semidualizing module, and modules in the Auslander and Bass classes through Matlis duality. What is interesting is that properties of really T -reflexive modules do not always translate into similar properties for totally C-reflexive modules. A property that demonstrates this is the two of three condition. A class C satisfies the two of three condition if, when given a short exact sequence of modules such that two of the modules in the sequence belong to C, then the third module must also belong to C. Totally C-reflexive modules do not satisfy the two of three condition, however, I found the following: Theorem 2.4. Assume R is complete. Then H T (R) satisfies the two of three condition.
6 BETHANY KUBIK At the moment, many of the theorems in this project require R to be complete. I am investigating to find the conditions under which these results extend to the non-complete case. In the cases where completeness is a necessary assumption, I am working to find examples that prove the necessity of completeness. 3. Future Projects My future plans involve several aspects of the projects I have described in this statement. I am working to find other conditions guaranteeing that the Hom-evaluation map is an isomorphism. With a similar goal in mind, I am investigating the dual map tensor evaluation. With regard to my work with quasidualizing modules, I am continuing to compare and contrast these modules with semidualizing modules in an effort to better understand both classes of modules. I believe that these projects will deepen our understanding of Matlis duality. Through these projects we can gain new ways of using old tools and possibly even create new tools. References 1. R. Belshoff, Some change of ring theorems for Matlis reflexive modules, Comm. Algebra 22 (1994), no. 9, 3545 3552. MR 1278804 (95h:13010) 2. R. G. Belshoff, E. E. Enochs, and J. R. García Rozas, Generalized Matlis duality, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1307 1312. MR 1641645 (2000j:13015) 3. B. Kubik, Quasidualizing modules, in preparation. 4. B. Kubik, M. J. Leamer, and S. Sather-Wagstaff, Evaluation homomorphisms, in preparation. 5., Homology of artinian and mini-max modules, i, arxiv:1010.1278v1, submitted. 6., Homology of artinian and mini-max modules, II, in preparation.