Projective modules over subrings of 𝑘[𝑋,𝑌

If Λ \Lambda is an artin algebra there is a partition of ind ⁡ Λ \operatorname {ind} \Lambda , the category of indecomposable finitely generated Λ \Lambda -modules, ind ⁡ Λ = ∪ i ⩾ 0 P _ _ i \operatorname {ind} \Lambda = { \cup _{i \geqslant 0}}{\underline {\underline {P}}_i} , called the preprojective partition. We show that P _ _ i \underline {\underline {P}}_i can be easily constructed for hereditary artin algebras, if P _ _ i − 1 \underline {\underline {P}}_{i - 1} is known: A A is in P _ _ i \underline {\underline {P}}_i if and only if A A is not in P _ _ i − 1 \underline {\underline {P}}_{i - 1} and there is an irreducible map B → A B \to A , where B B is in P _ _ i − 1 \underline {\underline {P}}_{i - 1} .

This paper deals with the unit groups of the endomor- phism rings of projective modules over polynomial rings and further over formal power series rings. A normal subgroup of the unit group is dened and discussed. The local splitting properties of elements of endomorphism rings of projective modules over polynomial rings are given. So, if P is a projective module over a ring A, then (id+ xEndA(x)(P(x))) is constructed. This is a normal subgroup of the unit group of endomorphism rings of a projective module over a poly- nomial ring. If (A,m) is a local ring with dim(A) ≥ µ(m), then we show that (id+ xEndA(x)(P(x))) is a normal subgroup of SLA(x)(P(x)). Also, we will show that under these conditions the similar conclusion can be drawn for the ring of formal power series. In section 3, let P be a projective A-module. Let s1, s2 ∈ A be such that As1 + As2 = A. Then we use the splitting lemma of Quillen to show that every element of (idPs1s2 + xEnd As1s2(x) (Ps1s2 (x))) has two decompositions. Finally, we generalize the result to Theorem 3.4.

Ehresmann semigroups whose categories are EI and their representation theory

Let N be a normal subgroup of a group G. An N-module Q is called G-stable provided that Q is equivalent to the twist Q g of Q by g, for every g ∈ G. If the action of N on Q extends to an action of G on Q, then Q is obviously G-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups G asserts that the (obviously G-stable) projective indecomposable modules (PIMs) Q for the Frobenius kernels G r (r ≥ 1) of G have a G-module structure. It is sometimes just as useful (for a general module Q) to know that a finite direct sum Q ⊕ n of Q has a compatible G-module structure. In this paper, this property is called numerical stability. In recent work (Parshall and Scott, Adv Math 226:2065–2088, 2011), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of k-group schemes and a suitable version of G-stability, called strong G-stability. Among our results here is the determination of necessary and sufficient conditions for the existence of a compatible G-module structure on a strongly G-stable N-module, in the form of a cohomological obstruction which must be trivial precisely when the G-module structure exists. Our main result is achieved by giving an approach to killing the obstruction by tensoring with certain finite dimensional G/N-modules.

A new description of the characters of the protective indecomposable modules for SL(2,pm) in characteristic p is given.From this, one can get the Car tan invariants of these groups by computing the products of certain matrices.This method explains why previous complete descriptions of these invariants are so complicated.We also answer a problem of Upadhyaya concerning the total number of composition factors in one particular projective indecomposable module.

A new canonical induction formula for p-permutation modules

A module M is called dual automorphism invariant if whenever X1 and X2 are small submodules of M, then each epimorphism f : M/X1 → M/X2 lifts to an endomorphism g of M. A module M is said to be d-square free (dual square free) if whenever some factor module of M is isomorphic to N2 for a module N then N = 0. We show that each dual automorphism invariant module over a semiperfect ring which is a small epimorphic image of a projective lifting module is a direct sum of cyclic indecomposable d-square free modules. Moreover, we prove that for each module M over a semiperfect ring which is a small epimorphic image of a projective lifting module (e.g., M is a finitely generated module), M is dual automorphism invariant iff M is pseudoprojective. Also, we give the necessary and sufficient conditions for a dual automorphism invariant module over a right perfect ring to be quasiprojective.

The purpose of this paper is to further study the concept of a very important module theory namely principally supplemented modules. Therefore, our work presents some properties of the relation between lifting property and principally supplemented module. Also we will focus on quasi-discrete module to study principally supplemented in general way. We prove that if R be principal ideal ring and if M be indecomposable R-module such that it has no maximal submodules (Rad(M) = M) then M is principally supplemented module. Also we prove that if M is indecomposable and local module then it is principally supplemented. Every projective module is self-projective therefore we prove that if M projective and supplemented module then M is principally supplemented.   Key words: Principally supplemented, P-hollow module, f-lifting module, quasi-discrete module.

A well-known theorem of Fong states that over large enough fields of any characteristic, the principal indecomposable modules of a soluble finite group are induced from subgroups of order prime to the characteristic.It is shown that this property in fact characterises soluble finite groups.

Blocks with a cyclic defect group

Let R be a left and right artinian ring with identity, and / the Jacobson radical of R. In [4], M. Harada has considered a left serial ring R satisfying a condition (*, 2) that every maximal submodule of a direct sum of any two hollow modules is also a direct sum of hollow modules, and characterized such a ring by the structure of eR for each primitive idempotent e. Further it has been shown that the condition (*, 2) is equivalent to saying that every factor module of ejQ)eR is a direct sum of hollow modules for every primitive idempotent e. Modifying this, we here consider the following condition on a projective indecomposable right module eR over a ring R.

We investigate the categorical behaviour of morphisms between indecomposable projective modules over a special biserial algebra $A$ over an algebraically closed field, which are associated to arrows of the Gabriel quiver of $A$.

This paper presents the socle series of the projective indecomposable modules for the triple cover 3 ^ M 22 \hat 3{M_{22}} in characteristic 2. The results are obtained by computational means; the methods as well as the constructive approach are explained.

Let X = Spec(A) be a smooth affine variety of dimension n ≥ 2 over a field k and P be a projective A-module of rank n. A result of Murthy ([Mu], Theorem 3.8) says that if k is an algebraically closed field and if the top Chern class Cn(P) in CH0(X) is zero, then P splits off a free summand of rank 1 (i.e. P A⊕ Q). We note that over any base field, the vanishing of the top Chern class is a necessary condition for P to split off a free summand of rank 1. However, as is shown by the example of the tangent bundle of an even dimensional real sphere, this condition is not sufficient. In view of these examples, it is of interest to explore whether one can classify examples of projective modules over smooth real varieties which have the property that the top Chern class of the projective module vanishes but the projective module does not split off a free summand of rank one. More precisely we ask the following:

The Loewy and socle structure of the projective indecomposable modules for the group algebras of the Mathieu group M 12 and its automorphism group M 12.2 over an algebraically closed field of characteristic 3 are determined without using computers.