More General Algebraic Contexts
Let R be a commutative ring and let M be a finite free module over R. Then contraction operates on the full (mixed) tensor algebra of M in exactly the same way as it does in the case of vector spaces over a field. (The key fact is that the natural pairing is still perfect in this case.)
More generally, let OX be a sheaf of commutative rings over a topological space X, e.g. OX could be the structure sheaf of a complex manifold, analytic space, or scheme. Let M be a locally free sheaf of modules over OX of finite rank. Then the dual of M is still well-behaved and contraction operations make sense in this context.
Read more about this topic: Tensor Contraction
Famous quotes containing the words general, algebraic and/or contexts:
“The general review of the past tends to satisfy me with my political life. No man, I suppose, ever came up to his ideal. The first half [of] my political life was first to resist the increase of slavery and secondly to destroy it.... The second half of my political life has been to rebuild, and to get rid of the despotic and corrupting tendencies and the animosities of the war, and other legacies of slavery.”
—Rutherford Birchard Hayes (1822–1893)
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“The “text” is merely one of the contexts of a piece of literature, its lexical or verbal one, no more or less important than the sociological, psychological, historical, anthropological or generic.”
—Leslie Fiedler (b. 1917)