Is the tensor product of two torsionfree modules always non. The depth formula, for tensor products of finitely generated modules, is initially due to auslander. 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. Finite rank torsion free modules over dedekind domains e.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. Shyamashree upadhyay april 2015 project supervisor ii. In a 1961 paper, auslander 1 studied torsion in tensor products of nonzero finitely. The tensor product of torsion free groups is a difficult subject. An integral domain is called a prufer domain if every torsionfree module over it. 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. Tensor products of nonfree modules typically will not be torsionfree, even if you assume good depth conditions on the modules. 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. Suppose and are abelian groups possibly equal, possibly distinct. Nothing is known about the algebraic structure of tensor and torsion products of modules in general, except for trivialities and very special cases. 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. 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. The aim of those lectures was to describe my recent work with.
It determines a necessary condition for the depth formula and hence, from another point of view, it complements. Tensor products of free modules sage reference manual v9. Torsionfree extensions of torsionfree abelian groups of. 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. Here we concentrate on the converse and show, under mild conditions, that the tensor product of m and n being torsionfree or satisfying. 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. It is easy to see that we may restrict our attention to extensions of torsion free groups by torsion free ones. 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.
If a a is commutative, then this is a special case of the tensor product in a multicategory. Japan 9 1957, 38851 has shown that over integral domains a. Let rbe a commutative ring with unit, and let m and n be rmodules. Bounds on depth of tensor products of modules sciencedirect.
Let g be a finite dimensional simple lie algebra over the complex numbers c. Pdf torsion in tensor products, and tensor powers, of. Tensor product realizations of simple torsion free modules 2001. The tensor product university of california, berkeley. The main result of this paper is that this group is torsion free except in the case that g is cyclic, quaternion or semidihedral. Is the tensor product of two torsionfree modules always nonzero. R, the tensor product of torsion free r modules is again torsion free if and only if r is pr u fer. Thus if a counter example to my question exists, we must work over domains in which every there are nonprojective finitely generated ideals.
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. 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. In order to clarify these points, we have found two classi. In mathematics, the tensor product of modules is a construction that allows arguments about. Torsion in tensor powers of modules project euclid. A point of interest is that the torsion element we identify comes from a universal example. The tensor product m r nof mand nis a quotient of the free f rm n. Their tensor product as abelian groups, denoted or simply as, is defined as their tensor product as modules over the ring of integers. 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 collection of isomorphism classes of endotrivial modules modulo projectives is an abelian group under tensor product. Torsion in tensor products over noncommutative rings. The tensor product of an algebra and a module can be used for extension of scalars.
Complete reducibility of torsion free c modules of. Unfortunately i dont have a deeper insight into this subject, so i dont know how things behave. Torsion in tensor products, and tensor powers, of modules. Tensor product realizations of simple torsion free modules. Specifically, we look at which rings guarantee nonzero torsion in tensor products of non free modules over the ring. If m is nitely generated, torsion free, and rigid, then m is free.
The conjecture still open reduces to the onedimensional case, where it can be restated as follows. 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. The tensor product r r a 6, where r is the field of rational numbers and.
The dual of a finitely generated module is reflexive, that is, isomorphic to its own double dual. One cannot always expect torsion in tensor powers of non free modules. Unfortunately i dont have a deeper insight into this subject, so i dont know how thi. It is known that every torsion free cn module of finite degree is completely reducible. We conclude that a specific subclass of onedimensional gorenstein rings will have this property. Also, we study torsionfree modules n with the property that its tensor product with any module m has torsion, unless m is very special. Is the tensor product of two torsionfree modules always. Tensor product of torsion free cnmodules of finite. T hese modules were determined by mathieu and using his work we. Commutative algebratorsionfree, flat, projective and free modules. As usual, all modules are unital rmodules over the ring r.
It can be shown that there are many independent torsion tensors with di. 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. Pdf torsion in tensor powers of modules researchgate. The following is an explicit construction of a module satisfying the properties of the tensor product. 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. 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. Here we concentrate on the converse and show, under mild conditions, that the tensor product of m and n being torsion free or satisfying. A module is faithfully flat if taking the tensor product with a sequence.
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. 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. In this thesis we replace the group z by an arbitrary group band aim for a criterion for the torsion freeness of exta,b. Commutative algebratorsionfree, flat, projective and free.
Specifically, we look at which rings guarantee nonzero torsion in tensor products of nonfree modules over the ring. Tensorflow world is the first event of its kind gathering the tensorflow ecosystem and machine learning developers to share best practices, use cases, and a firsthand look at the latest tensorflow product developments. Pdf tensor products of modules and the rigidity of tor. 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. In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps e. Oct 17, 2016 we consider whether the tilting properties of a tilting amodule t and a tilting bmodule t. This is a central theme of auslanders paper modules over unramified regular local rings. Torsion tensor and its geometric interpretation 197 with spinning. It follows from these remarks that a cartesian product of utorsionfree modules is. In particular, in terms of a holonomic cobasis, the components of the torsion tensor are expressed in terms of the christoffel symbols cf. 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. Hence the displayed presentation is a presentation of the tensor product by free sheaves as well.
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