# RECENT PAPERS OF ROGER WIEGAND

Factorization theory and decompositions of modules, preprint (with Nicholas Baeth).
Brauer-Thrall for totally reflexive modules, preprint (with L. W. Christensen, D. A. Jorgensen, H. Rahmati and Janet Striuli).
Prime ideals in Noetherian rings: a survey, in "Ring and Module Theory", T. Albu, G. F. Birkenmeier, A. Erdogan and A. Tercan, eds., Birkhaueser, Boston, MA, 2010, 175--193 (with S. Wiegand).
Direct-sum behavior
of modules over one-dimensional rings, in "Commutative Algebra", M. Fontana, S. Kabbaj, B. Olberding and I. Swanson, eds., Springer-Verlag, 2010 (with R. Karr).
Extended modules, J. Commutative Algebra 1 (2009), 481--506 (with W. Hassler).
Semigroups of
modules, in "Rings, Modules and Representations" (International Conference
on Rings and Things, 2007), Contemp. Math. 480, 335--349 (with S. Wiegand).
Ascent of
module structure, vanishing of Ext, and extended modules, Michigan Math. J.
57 (2008), 321--337 (with A. Frankild and S. Sather-Wagstaff).
Big
indecomposable modules and direct-sum relations, Illinois J. Math. 51
(2007), 99--122 (with W. Hassler, R. Karr and L. Klingler). **Comment:** This
paper extends the results in ``Large indecomposable modules over local rings"
and ``Indecomposable modules of large rank over Cohen-Macaulay local rings" and
treats the Cohen-Macaulay and non-Cohen-Macaulay cases in a unified way. One
consequence of the new approach is a complete set of invariants for the additive
monoid of isomorphism classes of finitely generated R-modules, where R is a
commutative Noetherian local ring of dimension one. **Erratum: **Parts (1)
and (3) of Theorem 6.3 are not quite correct as stated, since they do not take
into account the fact that some one-dimensional rings, e.g., k[[x,y]]/(x^2,xy)
can be proper homomorphic images of Dedekind-like rings. To fix this, change
"not Dedekind-like" in Theorem 6.3, Part (1), to "not a homomorphic image of a
Dedekind-like ring"; and "Dedekind-like, but not" in Theorem 6.3, Part (2), to
"a homomorphic image of a Dedekind-like ring, but is not".
What is
... a syzygy?, Notices Amer. Math. Soc. 53 (2006), 456--457.
Direct-sum
decompositions over one-dimensional Cohen-Macaulay rings, in "Multiplicative
Ideal Theory in Commutative Algebra: a tribute to the work of Robert Gilmer", J.
Brewer, S. Glaz, W. Heinzer, eds., Springer, 2006 (with A. Facchini, W. Hassler
and L. Klingler).
Large
indecomposable modules over local rings, J. Algebra 303 (2006), 202--215
(with W. Hassler, R. Karr and L.Klingler).
Big
indecomposable mixed modules over hypersurface singularities, in "Abelian
Groups, Rings, and Modules", P. Goeters and O. Jenda eds., Lecture Notes in Pure
and Appl. Math. 249, CRC/Taylor & Francis Books, 2006 (with W. Hassler).
**Advice:** Read this web version rather than the published version, which
somehow got a bit mangled.
Indecomposable modules
of large rank over Cohen-Macaulay local rings, Trans. Amer. Math. Soc. 360
(2008), 1391--1406 (with W. Hassler, R. Karr and L. Klingler).
Direct-sum
cancellation for modules over one-dimensional rings, J. Algebra 283 (2005),
93--124 (with W. Hassler).
Direct-sum
decompositions of modules with semilocal endomorphism rings, J. Algebra 274
(2004), 689-707 (with A. Facchini). Erratum: There is a
false statement in the introduction to the paper. This statement does not affect
the validity of the results in the paper.
Local rings
of bounded Cohen-Macaulay type, Algebr. Represent. Theory 8 (2005), 225 --
238 (with G. Leuschke). **Erratum:** The proof of Theorem 1.5 needs a small
repair.
Hypersurfaces of
bounded Cohen-Macaulay type, Algebr. Represent. Theory 8 (2005), 225 -- 238
(with G. Leuschke). **Comment:** Don't miss the **Note added in
proof** just before the references: The normal form needs a little adjustment
when the residue field is not algebraically closed. The adjustment is reproduced
here in a
more user-friendly form. **Erratum:** While the statement of the main theorem
classifying hypersurfaces of bounded CM type is correct, the argument in the
paper does not cover the rings k[[x_0, ..., x_d]]/(x_d^2), where d > 1.
Indeed, these rings do *not *have bounded CM type. Here is a proof
of a more general result.
The Tor
Game, in "Commutative Ring Theory: Proceedings of the Fourth International
Conference", M. Fontana, S.-E. Kabbaj and S. Wiegand, eds., Marcel Dekker, 2002,
289-300 (with C. Huneke).
Vanishing
theorems for complete intersections, J. Algebra 238 (2001), 684-702 (with C.
Huneke and D. Jorgensen).
Direct-sum
decompositions over local rings, J. Algebra 240 (2001), 83-97.
Ascent of
finite Cohen-Macaulay type, J. Algebra 228 (2000), 674-681 (with G.
Leuschke). **Erratum:** The proof of Proposition 1.6 needs a small repair.

# ERRATA

The paper "Tensor products of modules and the rigidity of Tor"
(by C. Huneke and R. Wiegand, Math. Ann. 299 (1994), 449--476) has errors caused
by an incorrect convention concerning the depth of the zero module. These errors
do not affect the validity of the main results of the paper, but several proofs
need to be modified. See the erratum for
details.
The paper "Bounds for one-dimensional rings with finite Cohen-Macaulay type"
(by R. Wiegand and S. Wiegand, J. Pure Appl. Algebra 93 (1994), 311-342)
asserts, in Example 3.12, that there is a local one-dimensional ring with finite
Cohen-Macaulay type, having a finitely generated indecomposable torsion-free
module of rank 4. Nicholas Baeth has shown, in Section 6 of his paper A Krull-Schmidt
for one-dimensional rings of finite Cohen-Macaulay Type, MR 2283435, that
the module in question is in fact a direct sum of two modules of rank two. In
fact, his results imply, at least in the equicharacteristic-zero case, that the
sharp bound on the ranks of the indecomposables is three. One of the main
results of the Wiegand /Wiegand paper is that, over a one-dimensional ring (not
necessarily local) with bounded Cohen-Macaulay type, each indecomposable
torsion-free module has rank 1, 2, 3, 4, 5, 6, 8, 9 or 12. In view of Baeth's
results, it is likely that the list can be shortened to the following: 1, 2, 3,
4 or 6, though this has not been worked out in full generality. The
construction, in Section 5 of the Wiegand/Wiegand paper, of indecomposable
modules (in the non-local case) of ranks 5, 7, 8, 9 and 12, is invalid, since it
depends on the incorrect assertion in Example 3.12.

In the paper "Noetherian rings of bounded representation type", MR 1015536, Lemma
2.4 makes the ludicrous claim that the functor F (reduction modulo an ideal)
preserves indecomposability of modules over Artinian pairs. (Consider, for
example, the Artinian pair (k[t^3,t^7], k[t]), with t^12 = 0. This is the bottom
line of the conductor square for the ring k[[T^3,T^7]], which has infinite
representation type. Therefore the Artinian pair has indecomposable modules of
arbitrarily large rank. If we take I = tk[t] in the statement of Lemma 2.4, the
resulting Artinian pair is (k, k), which has only one indecomposable module.
Letting (V,W) be an indecomposable module over (k[t^3,t^7], k[t]), with W free
of rank >1, we see that F(V,W) must decompose.) Fortunately this claim is
never used in the paper.