Supersingular isogeny graphs and endomorphism rings. Part of the progress in mathematics book series pm, volume 12 log in to check access. The theory of rings, algebras and their representations has evolved into a well defined subdiscipline of general algebra. The global dimension of the endomorphism ring 91 said to be a predecessor of z provided there is a path from z.
An \ algebra is a ring with some additional structure. It is shown that a projective module has a local endomorphism ring if and only if it is a cyclic module with a unique maximal ideal. Like all rings, it has a multiplication that is compatible with its structure as a zmodule, making it a z algebra. See langs book 7, chapter for a modern exposition.
The endomorphism ring of an artinian uniform module is a local ring. Here an auslander algebra is by definition the endomorphism algebra of an amodule m such that add m add ma, where a is an algebra of finite representation type and ma is the direct sum of all nonisomorphic indecomposable amodules. Cenrbe a ring homomorphism from kinto the center of r. Unless the ring is of finite representation type we must limit attempts at. No part of this book may be reproduced in any form by photostat, microfilm, or any other.
On some classes of modules and their endomorphism ring. The centralizer of an element in an endomorphism ring. Endomorphism algebras of abelian varieties are important objects for studying abelian. B endj falg that commutes with the action of galfalg jf. On the structure of the endomorphism ring of a certain local. The global dimension of the endomorphism ring of a generator. Pdf the centralizer of an element in an endomorphism. A homomorphism of algebras is a ring homomorphism which restricts to.
The purpose of this book is to provide an introduction to the theory of abstract algebra. Endomorphism rings of abelian groups algebra and applications. Endomorphism algebra project gutenberg selfpublishing. The relevance of the global dimension d of the endomorphism ring endm of. We take this opportunity to recall the definition of an algebra. The endomorphism rings of modules which are both generators and cogenerators have attracted much interest morita, tachikawa, and many others, see, for example, 6. On the socle of an endomorphism algebra researchgate. We additionally provide variant types and in addition to type of the books to browse. The theory of endomorphism rings can also be useful for studies of the structure of additive groups of rings, emodules, and homological properties of abelian groups.
We also describe how to identify the finite symplectic automorphism groups in these r. The centralizer of an element in an endomorphism ring arxiv. The endomorphism ring of a nonzero right uniserial module has either one or two maximal right ideals. Constructing homomorphism spaces and endomorphism rings. Right here, we have countless ebook rings, modules, algebras, and abelian. Constructing supersingular elliptic curves with a given endomorphism ring. The results are applied to the modular hecke algebra of a finite group with a. On the one hand this book intends to provide an introduction to module theory and the related part of ring theory. The books of baer 52 and kaplansky 245 have played an important role in the early development of the theory of endomorphism rings of abelian groups and modules.
A algebra alinear amodule addition admits algebra apply associative automorphism basis belongs bijective called canonical canonical mapping commutative commutative ring consider consisting contained conversely corollary corresponding deduce defined definition denoted derived dimension direct sum distinct dual elements endomorphism equal. Another important example of endomorphism algebras in the above theorem is given by auslander algebras. We are going to describe the possibilities for the global dimension of endm in terms of. All in all, i highly recommend the book to everyone interested in cotorsion pairs. The modernday proof of this theorem involves an area of algebra called galois theory, named after its main discoverer. In this note we provide a general method to study the endomorphism rings of hcomodule algebras over subalgebras. Thetermalgebra is supposed to distinguish it from the endomorphism ring end kx. Taking a di erent approach from several books, it helps the reader.
The proof of the main theorem in this section theorem 3. Endomorphism rings and tensor products of linearly compact modules. By induction, we claim that any indecomposable direct. The endomorphism ring enda of an abelian variety a is an order in a. Kcenrbe a ring homomorphism from kinto the center of r. The global dimension of the endomorphism ring of a generatorcogenerator for a hereditary artin algebra vlastimil dlab and claus michael ringel abstract.
M which has a generator with right perfect endomorphism ring. Now it follows from the definition of ring homomorphism that kerf is an ideal of r. Moreover, we are working in characteristic zero, so a theorem of vaserstein applies see the paper referenced in my original question. We prove that its left endomorphism algebra is a left sgalois extension of aop. Under this addition, and with multiplication being defined as function composition, the endomorphisms of an abelian group form a ring the endomorphism ring. In addition, new sections of this theory app eared after the pub lication of this bo ok.
It contains 11 chapters 318 pages and an index of terms 9 pages. We note the converse that a frobenius extension satisfying a generator condition is d2 if its endomorphism algebra extension is d2. Endomorphism rings of abelian varieties and their representations. Starting from a basic understanding of linear algebra the theory is presented with complete proofs. We also give constructive versions of deurings correspondence, which associates to a maximal order in a certain quaternion algebra an isomorphism class of supersingular elliptic curves. I dont know the book by curtis and reiner, but a wellestablished convention in ring and module theory is to write morphisms on the opposite side to the scalars and probably the book uses it. Matrix ring from wikipedia, the free encyclopedia in abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication.
Let o be a maximal order in the quaternion algebra bp over q ramified at p and. Actually, this result can be generalised a little bit to all modules over commutative rings which are certainly required to be finitely continue reading. Let d be the endomorphism of v that is given by formal differentiation, d dx. This avoids awkward references to the opposite ring that pop out when morphisms are written on the same side. Numerical computation of endomorphism rings of jacobians msp. Endomorphism ring project gutenberg selfpublishing. For any commutative ring r, an associative unital r algebra a is a not necessarily commutative ring equipped with a ring homomorphism r a that maps r into the center of a.
Rigorous computation of the endomorphism algebra of a. On the other hand the presentation includes most recent results and includes new ones. Endomorphism rings of abelian groups piotr krylov springer. More generally, a projective module has a semiperfect endomorphism ring if and only if it is a finite direct sum of modules each of which has a local endomorphism ring. On some classes of modules and their endomorphism ring ebook. The character is a ring homomorphism from rg to the class functions on. Rigorous computation of the endomorphism algebra of a jacobian. These are notes of two talks with the aim of giving some basic properties of the endomorphism ring of an abelian variety a and its representations on certain linear objects associated to a. Endomorphism rings of leavitt path algebras sciencedirect. Determinant trick, cayleyhamilton theorem and nakayamas. The matrix ring mnr can be identified with the ring of endomorphisms of the free rmodule of rank n, mnr. Statements appear in several papers that heuristically, these two problems should be equivalent gps16, klpt14, section.
For a commutative local ring r, consider noncommutative ralgebras. Constructing supersingular elliptic curves with a given. The present book is entirely devoted to endomorphism rings of abelian groups. If there is no risk of confusion one simply refers to end0 kx as the endomorphism algebra of x. In the sequel we shall often suppress the subscript k. Some of the terminology is coming from the fact that lie algebras are algebras over fields, albeit strange nonassociative ones, so for instance an ideal of a lie algebra is an exactly what it is for any other algebra or ring. Endomorphism rings of finite global dimension graham j. On finiteness of endomorphism rings of abelian varieties. The theory of rings, algebras and their representations has evolved into a welldefined subdiscipline of general algebra. Computing the structure of modules is a fundamental task in the representation theory of finitedimensional algebras. The ring mnd over a division ring d is an artinian simple ring, a special type of semisimple ring. With this method we may derive the endomorphism rings of crossed products and some. The procedure for matrix multiplication can be traced back to compositions of endomorphisms in this endomorphism ring. If the endomorphism ring is of some object r m in the category of left rmodules, it might be natural to extend the definition of the endomorphism ring to a left r algebra does this make sense.
The socle of an endomorphism algebra of a finite dimensional module of a finite dimensional algebra is described. As a corollary, endomorphism ring theorems for d2 and galois extensions are derived from the d2 characterization of galois extension. Smith, introduction to abstract algebra is a careful treatment of the principal topics of abstract algebra in 327 pages. Recall from lecture 7 that the endomorphism ring ende of an elliptic curve ek consists of the isogenies from e to itself, together with the zero morphism. In this case, obviously a m is a generatorcogenerator. First basic results about the endomorphism algebra. Pdf endomorphism rings of abelian groups researchgate. On the one hand this book intends to provide an introduction to module theory and. For a positive integer e let er denote the ring r viewed as an r algebra via fe. Let kbe a commutative ring, let rbe a ring, and let.
This identi es these last two problems as one possible way to attack some systems, such. The global dimension of the endomorphism ring of a. If the module is artinian, noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring. Cayleyhamilton theorem is usually presented in a linear algebra context, which says that an vector space endomorphism and its matrix representation satisfies its own characteristic polynomial. We know from linear algebra that endomorphism rings of kdimensional. In this article, to compute the geometric endomorphism ring of jmeans to compute an abstractly presented z algebra bassociative with 1 and free of nite rank as a zmodule, equipped with an action of galfalg jf, and a computable ring isomorphism 1. An algebra is called gendosymmetric in case it is isomorphic to the endomorphism ring of a generator over a. In abstract algebra, the endomorphism ring of an abelian group x, denoted by endx, is the set of all homomorphisms of x into itself. Quaternion algebras dartmouth mathematics dartmouth college. For the rational endomorphism ring end f q e, only the following two options are possible.
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