Proceedings of the 44th Symposium on Ring Theory and ...

P = P ′
t by t is a projective R ⊗ k K-module. Then <strong>the</strong>re is a Cohen-Macaulay R-module
X with a triangle in CM(R ⊗ k V ) <strong>of</strong> <strong>the</strong> following form:
(3.1) X ⊗ k V −−−→ X ⊗ k V −−−→ P ′ −−−→ X ⊗ k V [1].
As a direct consequence <strong>of</strong> Theorem 29, we have <strong>the</strong> following corollary.
Corollary 32. Let (R 1 , m 1 , k) and (R 2 , m 2 , k) be Gorenstein complete local k-algebras.
Assume that <strong>the</strong> both R 1 and R 2 are isolated singularities, and that k is an infinite field.
Suppose <strong>the</strong>re is a k-linear equivalence F : CM(R 1 ) → CM(R 2 ) <strong>of</strong> triangulated categories.
Then, for M, N ∈ CM(R 1 ), M stably degenerates to N if and only if F (M) stably
degenerates to F (N).
Remark 33. Let (R 1 , m 1 , k) and (R 2 , m 2 , k) be Gorenstein complete local k-algebras as
above. Then it hardly occurs that <strong>the</strong>re is a k-linear equivalence <strong>of</strong> categories between
CM(R 1 ) and CM(R 2 ). In fact, if it occurs, <strong>the</strong>n R 1 is isomorphic to R 2 as a k-algebra.
(See [4, Proposition 5.1].)
On <strong>the</strong> o<strong>the</strong>r hand, an equivalence between CM(R 1 ) and CM(R 2 ) may happen for nonisomorphic
k-algebras. For example, let R 1 = k[[x, y, z]]/(x n +y 2 +z 2 ) and R 2 = k[[x]]/(x n )
with characteristic <strong>of</strong> k not being 2 and n ∈ N. Then, by Knoerrer’s periodicity ([10, Theorem
12.10]), we have an equivalence CM(k[[x, y, z]]/(x n + y 2 + z 2 )) ∼ = CM(k[[x]]/(x n )).
Since k[[x]]/(x n ) is an artinian Gorenstein ring, <strong>the</strong> stable degeneration <strong>of</strong> modules over
k[[x]]/(x n ) is equivalent to a degeneration up to free summands by Theorem 29(ii). Moreover
<strong>the</strong> degeneration problem for modules over k[[x]]/(x n ) is known to be equivalent to
<strong>the</strong> degeneration problem for Jordan canonical forms <strong>of</strong> square matrices <strong>of</strong> size n. (See Example
22.) Thus by virtue <strong>of</strong> Corollary 32, it is easy to describe <strong>the</strong> stable degenerations
<strong>of</strong> Cohen-Macaulay modules over k[[x, y, z]]/(x n + y 2 + z 2 ).
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