On Residually Reducible Representations on Local Rings

In the first two sections of this chapter, we focus our attention on two special classes of rings, namely, local rings and semilocal rings. By definition, a ring R is local if R/rad R is a division ring, and R is semilocal if R/rad R is a semisimple ring. Thus, local rings include all division rings, and semilocal rings include all left or right artinian rings. The basic properties of local and semilocal rings are developed, respectively, in §19 and §20. We shall see, for instance, that local rings are connected with the problem of the uniqueness of Krull-Schmidt decompositions, and that semilocal rings are connected with the problem of “cancellation” of modules.

Local rings whose modules are almost serial

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a proper nonnilpotent ideal in a local (Noetherian) ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R comma upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(R,M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a reduction of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; that is, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="subset-of-or-equal-to"> <mml:semantics> <mml:mo> ⊆ </mml:mo> <mml:annotation encoding="application/x-tex">\subseteq</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J upper I Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:msup> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">JI^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="equals"> <mml:semantics> <mml:mo>=</mml:mo> <mml:annotation encoding="application/x-tex">=</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Superscript n plus 1"> <mml:semantics> <mml:msup> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">I^{n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some nonnegative integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove that there exists a finite free local unramified extension ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the ideal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I upper S"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">IS</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a minimal reduction <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="subset-of-or-equal-to"> <mml:semantics> <mml:mo> ⊆ </mml:mo> <mml:annotation encoding="application/x-tex">\subseteq</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J upper S"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">JS</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the property that the number of elements in a minimal generating set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-te

Let (R, m) be a local (Noetherian) ring. The main result of this paper asserts the existence of a local extension ring S of R such that (i) S dominates R, (ii) the residue field of S is a finite purely transcendental extension of R/m, (iii) every associated prime of (0) in S contracts in R to an associated prime of (0), and (iv) dim(S) < 1. In addition, it is shown that S can be obtained so that either mS is the maximal ideal of S or S is a localization of a finitely generated R-algebra.

The development of the theory of local rings has been greatly stimulated by the importance of the applications to algebraic geometry, but it is none the less true that this stimulus has produced a theory which, on aesthetic grounds, is somewhat unsatisfactory. In the first place, if a local ring Q arises in the ordinary way from a geometric problem, then Qwill have the same characteristic as its residue field. It is partly for this reason that our knowledge of equicharacteristic local rings is much more extensive than it is of those local rings which present the case of unequal characteristics. Again, in the geometric case, the integral closure of Q in its quotient field will be a finite Q-module. Here, once more, we have a special situation which it would be desirable to abandon from the point of view of a general abstract theory.

A criterion is established, in terms of the Massey products structure carried by the homology of partial resolutions, for the Yoneda cohomology algebra ExtA(k, k) to be a free module over the universal envelope of a free graded Lie subalgebra.It is shown that several conjectures on the (co)homology of local rings, in particular on the asymptotic behaviour of the Betti numbers, follow from such a structure.For all rings with edim A -depths * 3, and for Gorenstein rings with edim A -depth A = 4, the following dichotomy is proved: Either A is a complete intersection, or xiA(k, k) contains a nonabelian free graded Lie subalgebra.0. Ten conjectures on the homology of local rings (in place of Introduction).In this paper (A, rrt, k) will denote a noetherian local ring A with maximal ideal m and residue field k.A well-known homological invariant of A is its Poincar series 00 ; = 0 where b (= b(A)) is the /'th Betti number of A, i.e. the rank of the /'th free module in the unique, up to noncanonical isomorphism, yl-free resolution X of k.Much of the work on the homology of local rings has been motivated by the following famous conjecture:C0. PA(t) represents a rational function.Recently this has been settled in the negative by D. Anick,1 who constructed for each k a A>algebra A requiring 5 generators for m and having m3 = 0, with transcendental Poincar series.On the other hand, more than 20 years of research have produced different kinds of conditions, which ensure the rationality of PA(t).Before collecting for further reference several of them in a theorem, we introduce some notation, to be kept for the rest of the paper.edim A = embedding dimension of A := dimt m/m2 (= bx(A)); dim A -Krull dimension of A ;depth A = maximal length of an ^-regular sequence in m; d(A) = complete intersection defect of A '-dim^ a/no.-(dim R -dim A), where (R, n) is a regular local ring such that R/a is isomorphic to the m-adic

&lt;p&gt;The goal of this thesis is to study non-commutative deformation rings of representations of algebras. The main motivation is to provide a generalization of the deformation theory over commutative local rings studied by B. Mazur, M. Schlessinger and others. The latter deformation theory has played an important role in number theory, and in particular in the proof of Fermat's Last Theorem.&lt;/p&gt; &lt;p&gt;The thesis is divided into two parts.&lt;/p&gt; &lt;p&gt;In the first part, &lt;em&gt;A&lt;/em&gt; is an arbitrary λ-algebra for a complete local commutative Noetherian ring λ with residue field &lt;em&gt;k&lt;/em&gt;. A category Ĉ is defined whose objects are complete local λ-algebras &lt;em&gt;R&lt;/em&gt; with residue field &lt;em&gt;k&lt;/em&gt; such that &lt;em&gt;R&lt;/em&gt; is a quotient ring of a power series algebra over λ in finitely many non-commuting variables. If &lt;em&gt;V&lt;/em&gt; is a finite dimensional &lt;em&gt;k&lt;/em&gt;-vector space that is also a left &lt;em&gt;A&lt;/em&gt;-module and that satisfies a natural finiteness condition, it is proved that &lt;em&gt;V&lt;/em&gt; has a so-called versal deformation ring &lt;em&gt;R(A,V)&lt;/em&gt;. More precisely, &lt;em&gt;R(A,V)&lt;/em&gt; is an object in Ĉ such that the isomorphism class of every lift of &lt;em&gt;V&lt;/em&gt; over an object &lt;em&gt;R&lt;/em&gt; in Ĉ arises from a morphism α : &lt;em&gt;R(A,V)&lt;/em&gt;→ &lt;em&gt;R&lt;/em&gt; in Ĉ and α is unique if &lt;em&gt;R&lt;/em&gt; is the ring of dual numbers &lt;em&gt;k&lt;/em&gt;[ϵ].&lt;/p&gt; &lt;p&gt;In the second part, two particular examples of λ, &lt;em&gt;A&lt;/em&gt; and &lt;em&gt;V&lt;/em&gt; are studied and the versal deformation ring &lt;em&gt;R(A,V)&lt;/em&gt; is determined in each of these cases. In the first example, λ=&lt;em&gt;k&lt;/em&gt;, &lt;em&gt;A&lt;/em&gt; is a series of non-commutative &lt;em&gt;k&lt;/em&gt;-algebras depending on a parameter &lt;em&gt;r≥2&lt;/em&gt;, and &lt;em&gt;V&lt;/em&gt; is a particular quotient module of &lt;em&gt;A&lt;/em&gt;; it is shown that &lt;em&gt;R(A,V)&lt;/em&gt; is isomorphic to &lt;em&gt;A&lt;/em&gt;. The second example builds on the first example when &lt;em&gt;r=2&lt;/em&gt; and uses that, if additionally the characteristic of &lt;em&gt;k&lt;/em&gt; is 2, then &lt;em&gt;A&lt;/em&gt; is isomorphic to the group ring &lt;em&gt;k[D&lt;sub&gt;8&lt;/sub&gt;]&lt;/em&gt; of a dihedral group &lt;em&gt;D&lt;sub&gt;8&lt;/sub&gt;&lt;/em&gt; of order 8.&lt;/p&gt; &lt;p&gt;It is shown that if &lt;em&gt;k&lt;/em&gt; is perfect and &lt;em&gt;W&lt;/em&gt; is the ring of infinite Witt vectors over &lt;em&gt;k&lt;/em&gt;, then &lt;em&gt;R(W[D&lt;sub&gt;8&lt;/sub&gt;],V)&lt;/em&gt; is isomorphic to &lt;em&gt;W[D&lt;sub&gt;8&lt;/sub&gt;]&lt;/em&gt;.&lt;/p&gt;

Let Q be a complete local ring which has the same characteristic as its residue field P, and, for the present, let us denote by A the image of a subset A of Q under the natural homomorphism of Q onto P. Then a subfield F of Q is called a coefficient field if = P. It has been shown in [2] and in [3] that a complete equicharacteristic local ring, such as the above, always possesses at least one coefficient field; this is the embedding theorem for the equicharacteristic case.

A characterization of blocks with vertices

Let R be a commutative noetherian local ring with residue field k and assume that it is not Gorenstein. In the minimal injective resolution of R, the injective envelope E of the residue field appears as a summand in every degree starting from the depth of R. The number of copies of E in degree i equals the k-vector space dimension of the cohomology module ExtiR(k, R). These dimensions, known as Bass numbers, form an infinite sequence of invariants of R about which little is known. We prove that it is non-decreasing and grows exponentially if R is Golod, a non-trivial fiber product, or Teter, or if it has radical cube zero.

Introduction. Let T be a simply connected CW complex of finite type. Its homotopy Lie is the rational Lie 7r*(9 T) 0 Q, equipped with the Samelson product. Let A be a local (noetherian) ring with residue field k (of arbitrary characteristic). Its homotopy Lie algebra, 7r*(A), is the naturally defined Lie whose enveloping is ExtA(k, k)-cf. [Av]. These Lie algebras have finite type (i.e. are finite dimensional in each degree) and are concentrated in strictly positive degrees. Thus, for the sake of simplicity, we shall use graded Lie algebra to mean graded Lie of finite type concentrated in strictly positive degrees throughout this paper. On the other hand these Lie algebras are usually nonzero in infinitely many degrees. Thus, while we may (and do) form the sum, R, of all the solvable ideals in such a Lie and call it the radical, it may well happen that R, itself, is not solvable. Indeed, this does happen in the topological context, since by Quillen's result [Q] every rational Lie of finite type occurs as the homotopy Lie of a space. By contrast, in the case of local rings, or of CW complexes with finite Lusternik-Schnirelmann (L.S.) category, serious restrictions apply to the homotopy Lie algebra. Indeed, our first main result is the remarkable

A cohomological study of local rings of embedding codepth 3

Extended plus closure in complete local rings

Almost slender rings and compact rings

About a year ago Angus Macintyre raised the following question. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be complete local noetherian rings with maximal ideals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German m"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A slash German m Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">m</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">A/\mathfrak {m}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B slash German n Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">n</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">B/\mathfrak {n}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Does it follow that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are isomorphic? We show that the answer is <italic>yes</italic> if the residue field is algebraic over its prime field. The proof uses a strong approximation theorem of Pfister and Popescu, or rather a variant of it, which we obtain by a method due to Denef and Lipshitz. Examples by Gabber show that the answer is <italic>no</italic> in general.