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Computing global dimension of endomorphism rings via ladders | | Brandon Doherty
; Eleonore Faber
; Colin Ingalls
; | | Date: |
25 Aug 2015 | | Abstract: | This paper deals with computing the global dimension of endomorphism rings of
maximal Cohen--Macaulay (MCM) modules over commutative rings. We describe a
method for the computation of the global dimension of an endomorphism ring
$End_R(M)$, where $R$ is a Henselian local ring, using $add(M)$-approximations.
When $M
eq 0$ is a MCM-module over $R$ and $R$ is Henselian local of Krull
dimension less than or equal to 2 with a canonical module and of finite
MCM-type, we use Auslander--Reiten theory and Iyama’s ladder method to
explicitly construct these approximations. Several examples are computed. In
particular, we determine the global spectra, that is, the sets of all possible
finite global dimensions of endomorphism rings of MCM-modules, of the curve
singularities of type $A_n$, $D_n$ for $n leq 13$ and $E_{6,7,8}$ and compute
the global dimensions of Leuschke’s normalization chains for all ADE curves, as
announced in [Dao-Faber-Ingalls]. Moreover, we determine the centre of an
endomorphism ring of a MCM-module over any curve singularity of finite
MCM-type. | | Source: | arXiv, 1508.6287 | | Services: | Forum | Review | PDF | Favorites | |
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