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2 votes
0 answers
73 views

From exact triangles in the stable category of maximal Cohen--Macaulay modules to short exac...

Let $R$ be a local Gorenstein ring. Let $\underline{\text{CM}}(R)$ be the stable category of maximal Cohen--Macaulay modules, it is known to carry a triangulated structure. My question is: If $M\to N\ …
Alex's user avatar
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6 votes
1 answer
322 views

When can we choose non-zero-divisor $x\in \mathfrak m$ in a reduced local ring $(R,\mathfrak...

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay reduced ring of dimension at least $2$. Then, can we find a non-zero-divisor $x\in \mathfrak m$ such that $R/xR$ is again a reduced ring? If needed, I a …
Alex's user avatar
  • 480
6 votes
2 answers
292 views

If Serre's intersection multiplicity $\chi(R/I, R/J)$ equals $\operatorname{length}_R (R/(I+...

Let $(R,\mathfrak m)$ be a regular local ring. Let $I,J$ be proper ideals of $R$ such that $R/(I+J)$ has finite length i.e. $\sqrt{I+J}=\mathfrak m.$ Since $I+J$ annihilates $\text{Tor}_n^R(R/I, R/J)$ …
Alex's user avatar
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2 votes

Grade is not equal to injective dimension

https://math.stackexchange.com/questions/3459133/quotient-of-a-local-cohen-macaulay-ring-by-a-minimal-prime gives example of a local Gorenstein ring $R$ with a minimal prime ideal $P$ such that $R/P$ …
Alex's user avatar
  • 480
2 votes
1 answer
371 views

When is Hilbert-Samuel multiplicity of a local ring non-increasing along localization at pri...

For Noetherian local ring $(R,\mathfrak m)$, let $e(R)$ denote the Hilbert-Samuel multiplicity of $R$ with respect to $\mathfrak m$ (https://en.m.wikipedia.org/wiki/Hilbert%E2%80%93Samuel_function#Mul …
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  • 480