Projective summands in tensor products of simple modules of finite dimensional hopf algebras huixiang chen. The functor hom let abe a ring not necessarily commutative. Introduction to the tensor product james c hateley in mathematics, a tensor refers to objects that have multiple indices. Pdf a classical question due to yoneda is, when is the tensor product of any two injective modules injective. Therefore we introduce in section 4 a new tensor product x. The tensor product university of california, berkeley. B, called the hilbert space tensor product of a and b if the vectors a i and b j run through. A tensor product of young modules decomposes as a direct sum of young modules, and in 14 hemmer, nakano and cohen have shown a strong connection between the tensor products of young modules for the symmetric group r and the decomposition matrices of the schur algebra sr. It is also called kronecker product or direct product. Then m is projective if and only if m b is a projective as a bmodule, of course. We say that x is a banach abamodule if x is a banach a.
The tensor product of two projective modules is projec tive. Section6describes the important operation of base extension, which is a process of using tensor products to turn an r module into an s module. Remarks on the cotype of projective tensor products. In this theory, one considers representations of the group algebra a cg of a. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right. Let p and q be projective modules over a commutative ring.
A module over a ring r r is called flat if its satisfies one of many equivalent conditions, the simplest to state of which is maybe. All of this material can be found in grothendiecks ega. In this paper we give some conditions under which the converse is true. We show very generally that projective resolutions for bimodules of two factor algebras can be twisted together to construct a projective resolution for the resulting bimodule for the twisted tensor product given some compatibility conditions. As an application of the tensor product theory under c 2co. The projective tensor product derives its name from the fact that it behaves well with respect to quotient space constructions. Bbe faithfully at, and ma nite amodule anoetherian. Thus f is a z module for which there is defined an injective mapping p. Since any product of injective modules is injective. Cancelling finitely generated projective modules from a tensor product of finitely generated projective modules. Module amenability of the projective module tensor product. The tensor product of gorensteinprojective modules over category.
The origin of our problem the twisted tensor product iterating the twisted tensor products dualities the geometryalgebra dictionary manifolds commutative algebras topological manifolds commutative c. It is easy to see that the hattoris r esult is also true when r is the pro duct of a. However, if c is gptclosed, then the tensor product on gorensteinprojective modules induces the one on kcgproj. As already mentioned in the introduction, from our main result and its kfold generalization, we obtain information on the cotype of the 3fold and kfold projective tensor products. Benson and the second author 6 showed that some hopf algebras constructed from nite groups in positive characteristic have nonprojective modules with projective tensor powers and modules whose tensor products in one order are projective but in the other order are not. In the next section it is shown that if the number of all closed ideals in one of the, calgebras is finite then every closed ideal of ab is a finite sum of product ideals. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Tensor products of young modules connecting repositories. The induced module f,b fg is the direct sum fs, where s is an irreducible and projective fgmodule of dimension the order of u. An unexpected and recurring theme in this work and that of others has been the use of tensor products of irreducible modules.
Unfortunately in fact v 1 v n is not in o has already the wrong central charge n and moreover does not have reasonable niteness properties. Projective summands in tensor products of simple modules. The algebraic tensor product of two hilbert spaces a and b has a natural positive definite sesquilinear form scalar product induced by the sesquilinear forms of a and b. The phrase \set of all must be taken with a grain of logical salt to avoid the well known paradoxes of set theory. Projective summands in tensor products of simple modules of. The tensor product of two projective modules is projective. Namely, the category of artin modules becomes a braided tensor category. Projective modules and tensor products sciencedirect. Furthermore, the tensor products have natural properties of associativity and commutativity. A over z of an rmodule m and an abelian group a is an rmodule via rm.
Recall from 6 that for a finite projective ei category c, the stable category kc gproj modulo projective modules has a natural tensor triangulated. Let lwh denote the minimal n such that the tensor product of n suitably chosen simple hmodules contains the projective. An introduction to morita theory school of mathematics. We study exactness properties of projective tensor product with respect to kernels and countable products. In particular, by taking r z we obtain the tensor product of abelian groups g1 and g2. On the geometry of projective tensor products sciencedirect. Browse other questions tagged homologicalalgebra projectivemodules or ask your own question. Thus f is a zmodule for which there is defined an injective mapping p.
Similarly, the tensor product over z of an rchain complex x and a zchain complex y. If ris a pid, fis a free rmodule of a nite rank, and m f is a submodule then mis a free module and rankm rankf. In this section, we consider properties and in the projective tensor product. For such a group there is a natural choice for the module s. Browse other questions tagged homologicalalgebra projective modules or ask your own question. We give explicit descriptions of projective limits of banach rings as indobjects.
Gilljournalofalgebra36620121234 multiplicities of direct summands in tensor products of young modules are preserved under multiplying partitions by p. Kbaproj, the homotopy category of bounded complexes of nitely generated projective amodules. The tensor product of gorensteinprojective modules over. When x speca is a ne, these categories are t damod, the derived category of amodules, and k dperfa. A good starting point for discussion the tensor product is the notion of direct sums. Module amenability of the projective module tensor product malaysian journal of mathematical sciences 259 and that is called inner derivation. Chinese remainder theorem, endomorphisms of projective modules, projective but not free, tensor product of projective modules 2 so far we have only given a trivial example of projective modules, i. Module amenability of module projective tensor product of banach algebras and semigroup algebras are studied in 4 and 7, respectively.
Since the steinberg module is simple as well as projective, one relies on the very general principle for finite dimensional hopf algebras that tensoring any module with a projective one yields a projective module. Projective summands in tensor products of simple modules of finite dimensional hopf algebras huixiang chen department of mathematics, yangzhou university. Tensor product of a module and a projective module. In section5we will show how the tensor product interacts with some other constructions on modules. We should note that let jl0w 1 w 2 be the l0radical see def.
Let be the projective tensor product of and which is a banach. Let mr be a right r module and rn be a leftr module. As usual, all modules are unital rmodules over the ring r. Projective modules and tensor products mathoverflow. R n,thetensor product of m and n, is an abelian group that is a z. W 1 and w 2, there is a tensor product module w 1 w 2 for each 0 6 c such that w 1 w 2 is an artin v module and is uniquely determined up to isomorphism. If is the conjugate of, then is reflexive by, theorem 4. An introduction to morita theory matt booth october 2015 nov. Then, the tensor product m rnof mand nis an rmodule equipped with a map m n. So in particular it has a natural positive definite quadratic form, and the corresponding completion is a hilbert space a. We begin by noting that there are examples of banach spaces and such that has property.
If ris a pid then every nitely generated projective rmodule is free. In particular, finitely generated faithful multiplication modules are projective. But the projective modules are more elusive and in the general case remain so for small primes. The \only if follows from the same argument as the easy direction of. Roughly speaking this can be thought of as a multidimensional array. Thanks for contributing an answer to mathematics stack exchange.
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