Japan 9 1957, 38851 has shown that over integral domains a. Pdf torsion in tensor powers of modules researchgate. The depth formula, for tensor products of finitely generated modules, is initially due to auslander. The collection of isomorphism classes of endotrivial modules modulo projectives is an abelian group under tensor product. Here we concentrate on the converse and show, under mild conditions, that the tensor product of m and n being torsion free or satisfying. One cannot always expect torsion in tensor powers of non free modules. Is the tensor product of two torsionfree modules always nonzero. The tensor product of an algebra and a module can be used for extension of scalars. The tensor product r r a 6, where r is the field of rational numbers and.
The tensor product of torsion free groups is a difficult subject. In this thesis we replace the group z by an arbitrary group band aim for a criterion for the torsion freeness of exta,b. Shyamashree upadhyay april 2015 project supervisor ii. Here we concentrate on the converse and show, under mild conditions, that the tensor product of m and n being torsion free or satisfying higher serre conditions forces vanishing of tor. Torsion in tensor products over noncommutative rings. It is known that every torsion free cn module of finite degree is completely reducible. Suppose and are abelian groups possibly equal, possibly distinct. Torsion in tensor powers of modules project euclid.
Bounds on depth of tensor products of modules sciencedirect. The last section contains some miscellaneous results related to the. Tensor products of free modules sage reference manual v9. Hence the displayed presentation is a presentation of the tensor product by free sheaves as well. The tensor product m r nof mand nis a quotient of the free f rm n. The ring will be denoted f and the nite free modules v and v0, but this is just psychologically suggestive of the case of vector spaces.
C,\otimes with a right and left action, respectively, of some monoid a a, their tensor product over a a is the quotient of their tensor product in c c by this action. Fernando reduced the classification of infinite dimensional simple g modules with a finite dimensional weight space to determining the simple torsion free g modules for g of type a or c. Finite rank torsion free modules over dedekind domains e. The conjecture still open reduces to the onedimensional case, where it can be restated as follows. 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. 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.
The tensor product university of california, berkeley. Dec 03, 2019 tensor product plural tensor products mathematics the most general bilinear operation in various contexts as with vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, modules, and so on, denoted by. Pdf torsion in tensor products, and tensor powers, of. 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. R, the tensor product of torsion free r modules is again torsion free if and only if r is pr u fer. A tensor of type that is skewsymmetric with respect to its indices, obtained by decomposing the torsion form of a connection in terms of a local cobasis on a manifold. In particular, in terms of a holonomic cobasis, the components of the torsion tensor are expressed in terms of the christoffel symbols cf. An integral domain is called a prufer domain if every torsionfree module over it. As usual, all modules are unital rmodules over the ring r.
Commutative algebratorsionfree, flat, projective and free modules. We conclude that a specific subclass of onedimensional gorenstein rings will have this property. Their tensor product as abelian groups, denoted or simply as, is defined as their tensor product as modules over the ring of integers. Nothing is known about the algebraic structure of tensor and torsion products of modules in general, except for trivialities and very special cases. If i was working over a semiheriditary domain, then modules are flat if and only if they are torsion free, and so i believe the result should always hold in this case. Various facts concerning groups that were proved originally in an ad hoc fashion may be verified more clearly, and perhaps more elegantly, by using homological methods, in particular, the long exact sequences connecting the tensor and torsion products as well as hom and ext. Torsion in tensor products, and tensor powers, of modules. Specifically, we look at which rings guarantee nonzero torsion in tensor products of non free modules over the ring. Motivation in this handout, we work out some interesting isomorphisms involving tensor products of modules. Commutative algebratorsionfree, flat, projective and free. The dual of a finitely generated module is reflexive, that is, isomorphic to its own double dual.
Oct 17, 2016 we consider whether the tilting properties of a tilting amodule t and a tilting bmodule t. Tensor product of torsion free cnmodules of finite. The aim of those lectures was to describe my recent work with. 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. Complete reducibility of torsion free c modules of. The following is an explicit construction of a module satisfying the properties of the tensor product. In a 1961 paper, auslander 1 studied torsion in tensor products of nonzero finitely. In this thesis, we provide a formula for the decomposition of the tensor product of any simple torsion free cn module of finite degree with any simple finite dimensional cn module. It can be shown that there are many independent torsion tensors with di. Tensor product realizations of simple torsion free modules. Let g be a finite dimensional simple lie algebra over the complex numbers c. In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps e.
Pdf torsion in tensor products, and tensor powers, of modules. I know that the problem of torsion in tensor products, even of torsion free modules, is a very delicate thing. The main result of this paper is that this group is torsion free except in the case that g is cyclic, quaternion or semidihedral. Here we concentrate on the converse and show, under mild conditions, that the tensor product of m and n being torsionfree or satisfying. If a a is commutative, then this is a special case of the tensor product in a multicategory. In mathematics, the tensor product of modules is a construction that allows arguments about. Torsionfree extensions of torsionfree abelian groups of.
A point of interest is that the torsion element we identify comes from a universal example. Thus if a counter example to my question exists, we must work over domains in which every there are nonprojective finitely generated ideals. It follows from these remarks that a cartesian product of utorsionfree modules is. Also, we study torsionfree modules n with the property that its tensor product with any module m has torsion, unless m is very special. It determines a necessary condition for the depth formula and hence, from another point of view, it complements. Let m be a manifold with an affine connection on the tangent bundle aka covariant derivative the torsion tensor sometimes called the cartan torsion tensor of. Let rbe a commutative ring with unit, and let m and n be rmodules. T hese modules were determined by mathieu and using his work we. Is the tensor product of two torsionfree modules always non. Unfortunately i dont have a deeper insight into this subject, so i dont know how things behave.
Pdf tensor products of modules and the rigidity of tor. A module is faithfully flat if taking the tensor product with a sequence. Iyengar this is a fairly literal transcript of my lectures at the 7th japanvietnam joint seminar on commutative algebra, held in quy nhon, vietnam, from 12th16th december 2011. If m is nitely generated, torsion free, and rigid, then m is free. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Tensor products defn and a ton of examples duration. The main problem is that tensor product can create torsion and co torsion and reflexive modules have neither. Unfortunately i dont have a deeper insight into this subject, so i dont know how thi. Specifically, we look at which rings guarantee nonzero torsion in tensor products of nonfree modules over the ring. 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. Is the tensor product of two torsionfree modules always. Torsion tensor and its geometric interpretation 197 with spinning. In order to clarify these points, we have found two classi.
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