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

R : CM(R⊗ k V ) → CM(R) defined by taking −⊗ R⊗k V (R⊗ k V )/t(R⊗ k V ) = −⊗ V V/tV .
Now we define <strong>the</strong> stable degeneration <strong>of</strong> CM modules.
Definition 24. Let M, N ∈ CM(R). We say that M stably degenerates to N if <strong>the</strong>re
is a Cohen-Macaulay module Q ∈ CM(R⊗ k V ) such that L(Q) ∼ = M ⊗ k K in CM(R⊗ k K)
and R(Q) ∼ = N in CM(R).
Lemma 25. [15, Lemma 4.2, Proposition 4.3]
(1) Let M, N ∈ CM(R). If M degenerates to N, <strong>the</strong>n M stably degenerates to N.
(2) Suppose that <strong>the</strong>re is a triangle in CM(R);
L
α
−−−→ M
β
−−−→ N
Then M stably degenerates to L ⊕ N.
γ
−−−→ L[1].
Lemma 26. [15, Proposition 4.4] Let M, N ∈ CM(R) and suppose that M stably degenerates
to N. Then <strong>the</strong> following hold.
(1) M[1] (resp. M[−1]) stably degenerates to N[1] (resp. N[−1]).
(2) M ∗ stably degenerates to N ∗ , where M ∗ denotes <strong>the</strong> R-dual Hom R (M, R).
Lemma 27. [15, Proposition 4.5] Let M, N, X ∈ CM(R). If M ⊕ X stably degenerates
to N, <strong>the</strong>n M stably degenerates to N ⊕ X[1].
Remark 28. The zero object in CM(R) can stably degenerate to a non-zero object. In
fact, in Example 13 <strong>the</strong> free module R degenerates to an ideal N. Hence it follows from
Proposition 25(1) that 0 = R stably degenerates to N.
For ano<strong>the</strong>r example, note that <strong>the</strong>re is a triangle
X −−−→ 0 −−−→ X[1]
1
−−−→ X[1],
for any X ∈ CM(R). Hence 0 stably degenerates to X ⊕ X[1] by Proposition 25(2).
Let (R, m, k) be a Gorenstein complete local k-algebra and assume for simplicity that
k is an infinite field. For Cohen-Macaulay R-modules M and N we consider <strong>the</strong> following
four conditions:
(1) R m ⊕ M degenerates to R n ⊕ N for some m, n ∈ N.
(2) There is a triangle Z (φ ψ)
→ M ⊕ Z → N → Z[1] in CM(R), where ψ is a nilpotent
element <strong>of</strong> End R (Z).
(3) M stably degenerates to N.
(4) There exists an X ∈ CM(R) such that M ⊕ R m ⊕ X degenerates to N ⊕ R n ⊕ X
for some m, n ∈ N.
In [15] we proved <strong>the</strong> following implications and equivalences <strong>of</strong> <strong>the</strong>se conditions:
Theorem 29. (i) In general, (1) ⇒ (2) ⇒ (3) ⇒ (4) holds.
(ii) If dim R = 0, <strong>the</strong>n (1) ⇔ (2) ⇔ (3) holds.
(iii) If R is an isolated singularity <strong>of</strong> any dimension, <strong>the</strong>n (2) ⇔ (3) holds.
(iv) There is an example <strong>of</strong> isolated singularity <strong>of</strong> dim R = 1 for which (2) ⇒ (1) fails.
(v) There is an example <strong>of</strong> dim R = 0 for which (4) ⇒ (3) fails.
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