Yuji Yoshino

Mutation in triangulated categories and rigid Cohen–Macaulay modules

We introduce the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander–Reiten–Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen–Macaulay modules over certain Veronese subrings.

Characterizing Cohen-Macaulay local rings by Frobenius maps

Let R be a commutative noetherian local ring of prime characteristic. Denote by e R the ring R regarded as an R-algebra through e-times composition of the Frobenius map. Suppose that R is F-finite, i.e., 1 R is a finitely generated R-module. We prove that R is Cohen-Macaulay if and only if the R-modules e R have finite Cohen-Macaulay dimensions for infinitely many integers e.

Linkage of Cohen-Macaulay modules over a Gorenstein ring

Let R be a Gorenstein complete local ring. We say that finitely generated modules M and N are linked if HomR/λR(M,R/λR)≅ΩR/λR1(N), where λ is a regular sequence contained in both of the annihilators of M and N. We shall show that the Cohen–Macaulay approximation functor gives rise to a map Φr from the set of even linkage classes of Cohen–Macaulay modules of codimension r to the set of isomorphism classes of maximal Cohen–Macaulay modules. When r=1, we give a condition for two modules to have the same image under the map Φ1. If r=2 and if R is a normal domain of dimension two, then we can show that Φ2 is a surjective map if and only if R is a unique factorization domain. Several explicit computations for hypersurface rings are also given.

Right and left modules over the Frobenius skew polynomial ring in the F -finite case

The main purposes of this paper are to establish and exploit the result that, over a complete (Noetherian) local ring R of prime characteristic for which the Frobenius homomorphism f is finite, the appropriate restrictions of the Matlis-duality functor provide an equivalence between the category of left modules over the Frobenius skew polynomial ring R [ x , f ] that are Artinian as R -modules and the category of right R [ x , f ]-modules that are Noetherian as R -modules.

Gröbner Bases for the Polynomial Ring with Infinite Variables and Their Applications

We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-to-one correspondences among various sets of partitions by using the division algorithm.

Examples of degenerations of Cohen-Macaulay modules

We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of such degenerations. It is proved that such degenerations over an even-dimensional simple hypersurface singularity of type

Tate–Vogel Completions of Half-Exact Functors

(A_n)

On the existence of embeddings into modules of finite homological dimensions

are given by extensions. We also prove that all extended degenerations of maximal Cohen-Macaulay modules over a Cohen-Macaulay complete local algebra of finite representation type are obtained by iteration of extended degenerations of Auslander-Reiten sequences.

Localization functors and cosupport in derived categories of commutative Noetherian rings

We provide a general method to construct the Tate–Vogel homology theory for a general half-exact functor with one variable, aiming at a good generalization of Cohen–Macaulay approximations of modules over commutative Gorenstein rings. For a half exact functor F, using the left and right satellites (Sn and Sn), we define F∨(X)=lim →SnSnF(X) and F∧(X)=lim ←SnSnF(X), and call F∨ and F∧ the Tate–Vogel completions of F. We provide several properties of F∨ and F∧, and their relations with the G-dimension and the projective dimension of the functor F. A comparison theorem of Tate–Vogel completions with ordinary Tate–Vogel homologies is proved. If F is a half exact functor over the category of R-modules, where R is a commutative Noetherian local ring inspired by Martsinkovskys works, we can define the invariants ξ(F) and η(F) of F. If F=ExtRi(M, ), then they coincide with Martsinkovskys ξ-invariants and Auslanders delta invariants. Our advantage is that we can consider these invariants for any half exact functors. We also compute these invariants for the local cohomology functors.

Noncommutative resolutions using syzygies: NONCOMMUTATIVE RESOLUTIONS USING SYZYGIES

Let R be a commutative Noetherian local ring. We show that R is Gorenstein if and only if every finitely generated R-module can be embedded in a finitely generated R-module of finite projective dimension. This extends a result of Auslander and Bridger to rings of higher Krull dimension, and it also improves a result due to Foxby where the ring is assumed to be Cohen-Macaulay.