Finally, a complete answer to the question of yoneda was given by enochs and jenda in 1991 by showing the. In each such case the tensor product is characterized by a similar universal property. Suppose and are abelian groups possibly equal, possibly distinct. The following is an explicit construction of a module satisfying the properties of the tensor product. Torsionless a module is called torsionless if it embeds into its algebraic dual.
It is easy to see that the hattoris r esult is also true when r is the pro duct. W is the complex vector space of states of the twoparticle system. Tensor products of modules and the rigidity of tor article pdf available in mathematische annalen 2991. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way. Graded rings and modules daniel murfet may 16, 2006 in this note we develop the basic theory of graded rings and modules, at least as far as we need it for the theory of projective spaces in algebraic geometry. The third chapter extends tensor theory to spaces other than vector spaces, namely manifolds. Let mr be a right r module and rn be a leftr module. Here are the main results about tensor products summarized in one theorem. A tensor product of rmodules m, n is an rmodule denoted m.
We may generalize the tensor product of two rmodules to finitely many rmodules. More generally yet, if r r is a monoid in any monoidal category a ring being a monoid in ab with its tensor product, we can define the tensor product of a left and a right r r module in an. Let rbe a commutative ring with unit, and let m and n be rmodules. The balanced tensor product of module categories douglas, christopher l. Note that there are two pieces of data in a tensor product.
This note wants to be a proposal for defining the tensor products of hypermodules over a general hyperring. It is easy to see that the hattoris r esult is also true when r is the pro duct of a. Tensor product of lamodules now we would like to show that each or some properties of the usual tensor product hold in the new setting. Shyamashree upadhyay april 2015 project supervisor ii. This theory is being developed in a series of papers including hepth 9309076 and hepth 9309159. In homological algebra and algebraic geometry, a flat module over a ring r is an rmodule m such that taking the tensor product over r with m preserves exact sequences. Tensor products of free modules sage reference manual v9. It is also called kronecker product or direct product. Tensor product of two finitely generated modules stack exchange. The tensor product of an algebra and a module can be used for extension of scalars. More generally yet, if r r is a monoid in any monoidal category a ring being a monoid in ab with its tensor product, we can define the tensor product of a left and a right r rmodule in an. Letu k denote the quantized enveloping algebra corresponding to a finite dimensional simple complex lie algebral. In order to be able to establish this relation, we need to keep in mind that given a copmodule f.
R n,thetensor product of m and n, is an abelian group that is a z. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. The hilbert function of the tensor product of two multigraded modules volume 10 issue 1 d. This tensor product can be generalized to the case when r r is not commutative, as long as a a is a right r r module and b b is a left r r module. Tensor products of quantized tilting modules springerlink. You can see that the spirit of the word tensor is there. B is the free rmodule generated by the cartesian product and g is the rmodule generated by the same relations as above. What links here related changes upload file special pages permanent link page information wikidata item cite this page. One of the applications is simpli cation of the proofs of central results in a paper of c. Consider our action on this expansion we observe that dimv dimv. Hot network questions apex unit test for if null and return statements. Wiegand on the tensor product of modules and the rigidity of tor.
Tensor product bezier patches tensor product of bezier curves. Also assume that this order is odd and not divisible by 3 if typeg 2 occurs. A gentle introduction to tensors washington university in. In mathematics, the tensor product of modules is a construction that allows arguments about. Foundations of module and ring theory a handbook for study and research robert wisbauer university of dus.
The tensor product of two modules a and b over a commutative ring r is defined in exactly the same way as the tensor product of vector spaces over a field. The tensor product university of california, berkeley. We consider whether the tilting properties of a tilting amodule t and a tilting bmodule t. We demonstrate how one can define a reduced tensor product on the. Notesontensorproducts rich schwartz may 3, 2014 1 modules. Let rbe a commutative ring with unit, and let m and n be r modules. As usual, all modules are unital rmodules over the ring r.
I am reading this introduction to tensor products, which is clearly written. Available formats pdf please select a format to send. In this section, we develop the tools needed to describe a system that contains more than one particle. Just as the notion of a monad in a bicategory k k generalizes that of a monoid in a monoidal category, modules over monoids generalize easily to modules over monads. The proof of the tensor product theorem well prove the tensor product theorem, which asserts that if f is a global eld, aits adele ring, vthe places of f, and g is a redactive algebraic group over fthen every irreducible admissible representation of ga decomposes into a restricted tensor product of representations of the groups gfv. A tensor product of rmodules m, n is an r module denoted m. This tensor product can be generalized to the case when r r is not commutative, as long as a a is a right r rmodule and b b is a left r rmodule. The tensor product of functors always exists in the representation theory of small categories. Pdf how to define tensor products of hypermodules researchgate. We give a new, constructionfree proof of the associativity of tensor product for modules for rational vertex operator algebras under certain convergence.
The classification of tilting modules over harada algebras yamaura, kota, journal of the mathematical society of japan, 2012. We demonstrate how one can define a reduced tensor product on the familyf consisting of those finite. Notes on tensor products and the exterior algebra for math 245 k. It is this chapter that provides the foundations for tensor applications in physics. As usual, all modules are unital r modules over the ring r. We introduce the main concepts and announce the main results in a theory of tensor products for module categories for a vertex operator algebra. Pdf tensor products of modules and the rigidity of tor. Keith conrad, tensor products whereas in the world of vector spaces, tensors have a clearly visualizable representations, things become more subtle when we generalize to modules over a ring. The hilbert function of the tensor product of two multigraded.
Flat a module is called flat if taking the tensor product of it with any exact sequence of rmodules preserves exactness. If k is a field, a and b kalgebras, m a faithful left amodule, and n a faithful left bmodule, we recall the proof that the. A conference on vertex algebras and related topics at rims, to appear. More generally, the tensor product can be extended to other categories of mathematical objects in addition to vector spaces, such as to matrices, tensors, algebras, topological vector spaces, and modules. Gus lonergan fourth annual mit primes conference may 17, 2014. Their tensor product as abelian groups, denoted or simply as, is defined as their tensor product as modules over the ring of integers note that in case are abelian groups but are also being thought of as modules over some other ring for instance, as vector spaces over. Simple a simple module s is a module that is not 0 and whose only submodules are. Decomposing tensor products of verma modules shashwat kishore mentor. Mar 18, 2015 the tensor product of two injective rmodules is injective when r is an integral domain. Assume that the quantum parameter is a root of unity ink of order at least the coxeter number forl. Example of a tensor product of modules with nondecomposable elements. Injective modules are defined dually to projective modules. Beware that modules over monads in cat are often called algebras for the monad, since they literally are algebras in the sense of universal algebra, see below. Tensor product of modules metadata this file contains additional information such as exif metadata which may have been added by the digital camera, scanner, or software program used to create or digitize it.