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Homology of artinian and Matlis reflexive modules, I
Let R be a commutative local noetherian ring, and let L and L' be R-modules.
We investigate the properties of the functors Tor_i^R(L,-) and Ext^i_R(L,-).
For instance, we show the following:
(a) if L is artinian and L' is noetherian, then Hom_R(L,L') has finite
length;
(b) if L and L' are artinian, then the tensor product L \otimes_R L' has
finite length;
(c) if L and L' are artinian, then Tor_i^R(L,L') is artinian, and
Ext^i_R(L,L') is noetherian over the completion \hat R; and
(d) if L is artinian and L' is Matlis reflexive, then Ext^i_R(L,L'),
Ext^i_R(L',L), and Tor_i^R(L,L') are Matlis reflexive.
Also, we study the vanishing behavior of these functors, and we include
computations demonstrating the sharpness of our results.Comment: 24 page
When is each proper overring of R an S(Eidenberg)-domain?
A domain R is called a maximal "non-S" subring of a field L if R [containded in] L, R is not an S-domain and each domain T such that R [containded in] T [contained in or equal] L is an S-domain. We show that maximal "non-S" subrings R of a field L are the integrally closed pseudo-valuation domains satisfying dim(R) = 1, dimv(R) = 2 and L = qf(R)
Rank Bounded Hibi Subrings for Planar Distributive Lattices
Let be a distributive lattice and the associated Hibi ring. We
show that if is planar, then any bounded Hibi subring of has a
quadratic Gr\"obner basis. We characterize all planar distributive lattices
for which any proper rank bounded Hibi subring of has a linear
resolution. Moreover, if is linearly related for a lattice , we find
all the rank bounded Hibi subrings of which are linearly related too.Comment: Accepted in Mathematical Communication
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