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Information about the cohomology ring of the group
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Here you can find the description of the cohomology ring of
the group G, including the polynomials and relations which describe it,
its Hilbert series, its Krull dimension, its depth, and a
regular sequence of maximal length. If the ring is not Cohen-Macaulay
(that is, if its Krull dimension is stricly larger than its depth), then
you can also find a homogeneous system of parameters and the
hypercohomology spectral sequence of the cohomology ring.
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The cohomology ring P = H*(G) is a quotient of
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a polynomial ring in the variables
z
,
y
,
x
,
w
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The respective degrees of these variables are
1
,
1
,
3
,
4
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The ideal of relations is generated by the polynomials
z2
,
zy2
,
zx
,
x2
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General information about the cohomology ring
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The Hilbert series of the cohomology ring (in factored form) is
1 / (1-t)2
(1+t2)
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The Krull dimension of the cohomology ring is
2
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A Regular Sequence of Maximal Length in the cohomology ring
consists of the element(s)
w
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The depth of the cohomology ring is
1
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Since the Krull dimension does not equal the depth in this case,
this cohomology ring is not a Cohen-Macaulay ring.
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A homogeneous System of Parameters in the cohomology ring
consists of the element(s)
w
,
y2
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This table describes the hypercohomology spectral sequence of
the cohomology ring.
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The hypercohomolgy spectral sequence has E2 term:
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(1):
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z
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zy
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(0):
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1
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y
, z
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zy
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x
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yx
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The spectral sequence satisfies Poincaré duality
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