Скачать или смотреть Spanning Sets of Tensor Products (Algebra 2: Lecture 21 Video 1)

Lecture 21: We started this lecture by proving a result about spanning sets of tensor products of modules.  We then saw that m \otimes 0 = 0 \otimes n = 0.  We proved that for any finite abelian group A we have Q \otimes_Z A = 0, and in fact, saw something a little more general than that.  We discussed how to understand R-module homomorphisms from a tensor product and emphasized that these maps come from bilinear functions from M x N.  We discussed some basic questions about tensor products, such as, "What does it mean to say that m \otimes n = 0?” and, "What does it mean to say that M \otimes_R N = 0?”  We discussed the tensor product of two finite cyclic groups and saw how to give a basis for the tensor product of two free R-modules.  We gave an example of a tensor in the tensor product of a free R-module with itself that we could prove was not an elementary tensor.  At the end of the lecture we stated some results that you should be aware of, but we did not give full proofs.

Reading: In this lecture we also closely followed Conrad's Tensor Products notes: https://kconrad.math.uconn.edu/blurbs... We started by proving Theorem 3.3.  We then proved Theorem 3.5 and gave the examples following it.  We gave a fairly detailed discussion of the 'Beginner Questions about the Tensor Product' on pages 12-13.  We proved Theorems 4.1 and 4.9, and went over Example 4.11 in detail.  At the end of the lecture we mentioned some important results that you should know about, but did not give full proofs.  We highlighted Theorems 4.3, 4.5, 4.21, 5.1, 5.2, 5.3, and the first part of Theorem 5.7.  I recommend that you read over the proofs of these results, but do not worry too much about the details.  You would not be asked to prove these results on an exam, but you may want to use these results when solving problems.  This concludes our section on tensor products.  I encourage you to read all of Section 10.4 of Dummit and Foote to see how this material is presented differently there.