τ-tilting modules over right 2-Nakayama algebras

We consider a Krull–Schmidt, Hom-finite, 2-Calabi–Yau triangulated category with a basic rigid object T, and show a bijection between the set of isomorphism classes of basic rigid objects in the finite presented category pr T of T and the set of isomorphism classes of basic τ-rigid pairs in the module category of the endomorphism algebra Endc(T) op . As a consequence, basic maximal objects in pr T are one-to-one correspondence to basic support τ-tilting modules over End c (T) op . This is a generalization of correspondences established by Adachi–Iyama–Reiten.

Let H be the upper half plane and T a discrete subgroup of AutH.Suppose that H mod Y is of finite measure.This work stems from the question whether there is an algebraic interpretation for the moduli of unitary representations of T similar to the case whenWe show that this is indeed the case via the moduli of vector bundles on the compactification of H mod r, provided with some additional structures which we propose to call parabolic structures.The idea of parabolic structures is inspired from A. Weil's work [6,§2, Chapter I, p. 56].Let Xbe a smooth, irreducible, projective curve defined, say, over an algebraically closed field k.By vector bundles on X we understand algebraic vector bundles.DEFINITION 1.Let V be a vector bundle on X and Q G X. Then a quasiparabolic structure of Vat Q is giving a flag on the fibre VQ of Fat Ô, i.e., giving linear subspaces F*VQ of VQ,We call I = (l l9 . . .9 l r ) the type (or flag type) of the quasi-parabolic structure.Let k x = l x -/ 2 , k 2 = l 2 -/ 3 , . . ., fc fWl = /,._! -l r , k r = l r \ then k t are called the multiplicities of the quasi-parabolic structure.DEFINITION 2. Let V be a vector bundle on X and Q G X. Then a parabolic structure of V at Q is giving (i) a quasi-parabolic structure of V at Q; say I = (l t , . . ., l r ) is its type and {k.} its multiplicities, and(ii) constants a = (a 1 , . . ., a n ) called the weights of the parabolic structure such that 0 < a x < a 2 < • • • < ot n < 1 and there are r distinct elements among a, say a' = (o^, . . ., Û£), 0 < a\ < a 2 < • • • < a r < 1, such that OL\ occurs k x times, a 2 occurs k 2 times, . . ., a r occurs k r times among a.We call a. the weight ofF l V Q .Note that l x = « = rkV.Let F, H> be vector bundles on X with quasi-parabolic structures at Q.An isomorphism ƒ: F -• W of vector bundles is said to be a quasi-parabloic isomorphism if the types of V, W at Q are the same and fçiF'Vç) = F 1 WQ (/^:

ϕ-Coordinated modules for quantum vertex algebras and associative algebras

Congruence classes of presentations for the complex reflection groups G( m, 1, n) and G( m, m, n)

Introduction. In '[12] Thrall introdcuced three generalizations of the quasiFrobenius (=QF) algebras of Nakayama [9], [10]. In this paper we shall be concerned with ring theoretic generalizations of two of these Thrall algebras-namely QF-2 algebras and QF-3 algebras. If R is a ring then a one-sided ideal of R is primitive in case it is generated by a primitive idempotent, and an R-module is minimalfaithful in case it is faithful and has no proper faithful direct summand. Extending Thrall's original definitions to (two-sided) artinian rings we have: QF-2 rings: An artinian ring is QF-2 in case each of its primitive one-sided ideals has a simple socle. QF-3 rings: An artinian ring is QF-3 in case it has (to within isomorphism) a unique minimal faithful module. It is not difficult to show that QF rings are both QF-2 and QF-3 (see [2, ??58-59]). Moreover, Thrall [12] has shown that QF-2 algebras are QF-3 but not necessarily QF. Most of the information about QF-2 and QF-3 rings is limited to finite dimensional algebras (see [8], [12], [13]). Two notable exceptions generalize to QF-3 rings results known to hold for QF-3 algebras almost from their inception. Specifically, Jans [7] has characterized QF-3 rings as those artinian rings whose injective hulls are projective and Harada [5] has shown that the QF-3 property is actually two-sided (i.e., a QF-3 ring has a unique minimal faithful right module). In ?2 of this paper we obtain ideal theoretic characterizations of the injective projective modules (and hence of the unique minimal faithful module) over left QF-3 rings. Our main results appear in ?3. With the aid of Morita's duality theorems [8] we obtain characterizations of QF-3 rings that are analogous to Nakayama's original definition of QF rings in terms of socles of primitive one-sided ideals [10], his characterization of QF-rings in terms of the double annihilator property for one-sided ideals [10], and the fact (see [8, ?14]) that QF rings are precisely those artinian rings for which the functor HomR ( , R) provides a duality between the categories of finitely generated and finitely generated right R-modules.

Here we give an upper bound for the number of irreducible components of the moduli scheme of stable rank r torsion-free sheaves of fixed degree on the integral curve X. This bound depends only on r, Sing(X),pa(X) and the corresponding number for the rank 1 case. Let X be an integral complete curve of arithmetic genus Pa and geometric genus g. For all integers r, d with r > 1 let M(X, r, d) (or just M(r, d)) be the moduli scheme of stable rank r torsion-free sheaves on X with degree d. For general background on stable vector bundles and sheaves on curves and their moduli schemes, see [N] or [S]. The number of irreducible components of M (r, d) does not depend on d, but only on X and r (see Theorem 1.2). Let n(r, X) (or just n(r)) be the number of irreducible components of M(r, d). Here we want to give upper bounds on n(r) depending only on the singularities of X, the integer r and n(l). This is given by Theorem 0.1 (see the discussion in the last part of Remark 1.4). To state Theorem 0.1 (our main result) we need to introduce a few concepts (e.g. measures for the singularities of X) and fix some notation. Let 7r: C -* X be the normalization (hence g = pa(C)). Set 6 := Pa g. Set o:= Ox. For any x C Sing(X), set Ox := Ox,x and let mx be the maximal ideal of Ox; set kx := Ox/mx. Set fx := dimK(Ext1(kx,Ox)) and f := max{fx}xEsirlg(x). For example, we have fx = 1 if and only if Ox is Gorenstein (see e.g. [Co], Lemma 2.1.4). Let ex be the multiplicity of X at x; set := max(ex) and e := Ex ex. Let F be a rank r torsion-free sheaf on X; the set of all formal isomorphism classes of the germs {FX}xEsirlg(x) will be called the formal isomorphism class of F. The set W of all possible isomorphism classes for rank r torsion-free sheaves around Sing(X) has in a natural way a scheme structure; for any reduced subscheme Q of W, let M(r, d, Q) be the reduced subscheme of M(r, d)r.eci parametrizing sheaves with formal isomorphism type in Q. Let n(r, Q) (or n(r, d, Q)) be the number of irreducible components of M(r, d, Q). By Theorem 1.2 this integer does not depend on d. Our main interest is when Q is an irreducible component of Wr.ec. Here is our main result. Received by the editors November 28, 1994. 1991 Mathematics Subject Classification. Primary 14H60, 14D20, 14B99. This research was partially supported by MURST and GNSAGA of CNR (Italy). The author is a member of Europroj (and its group Vector bundles on curves). ?)1997 Arnericari Mathernatical Society

Let F ( G ) \mathcal {F}(G) denote the set of isomorphism classes of finite homomorphic images of a group G. We say that groups G and H have isomorphic finite quotients if F ( G ) = F ( H ) \mathcal {F}(G) = \mathcal {F}(H) . Let H \mathcal {H} denote the class of finite extensions of finitely generated nilpotent groups. In this paper we show that if G is in H \mathcal {H} , then the groups H in H \mathcal {H} for which F ( G ) = F ( H ) \mathcal {F}(G) = \mathcal {F}(H) lie in only finitely many isomorphism classes.

Semi-formal transformation of secure business processes into analysis class and use case models: An MDA approach

Let G be a finitely generated nilpotent group and let %(G) denote the set of isomorphism classes of finite homomorphic images of G. If H is another finitely generated nilpotent group, we will say that G and H have isomorphic finite quotients if %(G) —%(H). The finite quotients of a finitely generated nilpotent group provide much information about the structure of the group [3], [5], although they do not determine the group up to isomorphism [8, and G. Higman, unpublished]. The following result shows, however, that a finitely generated nilpotent group is determined to a large extent by its finite quotients.

ABSTRACTLet M be a semiprime Goldie module which is projective in σ[M]. We give several decompositions of the M-injective hull of M in terms of its minimal prime submodules. Then, we use this to give a decomposition of the endomorphism ring of . We also show that the set of minimal primes of M is in bijective correspondence with the set of isomorphism classes of simple modules over . We also investigate the relationships between semiprime Goldie modules, QI-modules and co-semisimple modules. As an application we extend certain results on left QI-rings and V-rings.

Let X1 be a curve of genus g, projective and smooth over Fq. Let S1 X1 be a reduced divisor consisting of N1 closed points of X1. Let (X;S) be obtained from (X1;S1) by extension of scalars to an algebraic closure F of Fq. Fix a primel not dividingq. The pullback by the Frobenius endomorphism Fr ofX induces a permutation Fr of the set of isomorphism classes of rank n irreducible Ql-local systems on X S. It maps to itself the subset of those classes for which the local monodromy at each s2 S is unipotent, with a single Jordan block. Let T (X1;S1;n;m) be the number of xed points of Fr m acting on this subset. Under the assumption that N1 2, we show that T (X1;S1;n;m) is given by a formula reminiscent of a Lefschetz xed point formula: the function m 7! T (X1;S1;n;m) is of the form P ni m for suitable integers ni and \eigenvalues i. We use Laorgue

In his 1944 paper Veranderliche Riemannsche Flachen , Teichmuller defined a structure of complex manifold on the set of isomorphism classes of marked closed Riemann surfaces of genus g. The complex manifold he obtained is the space called today Teichmuller space. In the same paper, Teichmuller introduced the so-called universal Teichmuller curve -- a space over Teichmuller space where the fiber above each point is a Riemann surface representing that point. In fact, Teichmuller proved the existence of the Teichmuller curve as a space of Riemann surfaces parametrized by an analytic space, with an existence and uniqueness theorem establishing this analytic structure. This result was later reformulated and proved by Grothendieck in a series of ten lectures he gave at Cartan's seminar in 1960-1961. In his approach , Grothendieck replaced Teichmuller's explicit parameters by a general construction of fiber bundles whose base is an arbitrary analytic space. This work on Teichmuller space led him to recast the bases of analytic geometry using the language of categories and functors. In Grothendieck's words, the Teichmuller curve becomes a space representing a functor from the category of analytic spaces into the category of sets. In this survey, we comment on Grothendieck's series of lectures. The survey is primarily addressed to low-dimensional topologists and geometers. In presenting Grothendieck's results, we tried to explain or rephrase in more simple terms some notions that are usually expressed in the language of algebraic geometry. However, it is not possible to short-circuit the language of categories and functors. The survey is also addressed to those algebraic geometers who wish to know how the notion of moduli space evolved in connection with Teichmuller theory. Explaining the origins of mathematical ideas contributes in dispensing justice to their authors and it usually renders the theory that is surveyed more attractive. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichmuller theory. This volume is dedicated to the memory of Alexander Grothendieck.

Combinatorics of quasi-hereditary structures

In this paper, we study the Nisnevich sheafification \mathcal{H}^1_{\text{ét}}(G) of the presheaf associating to a smooth scheme the set of isomorphism classes of G -torsors, for a reductive group G . We show that if G -torsors on affine lines are extended, then \mathcal{H}^1_{\text{ét}}(G) is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for G -torsors. We also identify the sheaf \mathcal{H}^1_{\text{ét}}(G) with the sheaf of \mathbb{A}^1 -connected components of the classifying space \mathrm{B}_{\text{ét}}G . This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of \mathbb{A}^1 -connected components in terms of unramified G -torsors over function fields whenever Nisnevich-local purity holds for G -torsors.

We extend the Horrocks correspondence between vector bundles and cohomology modules on the projective plane to the product of two projective lines. We introduce a set of invariants for a vector bundle on the product of two projective lines, which includes the first cohomology module of the bundle, and prove that there is a one to one correspondence between these sets of invariants and isomorphism classes of vector bundles without line bundle summands.