In algebra, a free presentation of a module M over a commutative ring R is an exact sequence of R-modules: Note the image under g of the standard basis generates M. In particular, if J is finite, then M is a finitely generated module. If I and J are finite sets, then the presentation is called a finite presentation; a module is called finitely presented if it admits a finite presentation. Since f is a module homomorphism between free modules, it can be visualized as an (infinite) matrix with entries in R and M as its cokernel. A free presentation always exists: any module is a quotient of a free module: , but then the kernel of g is again a quotient of a free module: . The combination of f and g is a free presentation of M. Now, one can obviously keep "resolving" the kernels in this fashion; the result is called a free resolution. Thus, a free presentation is the early part of the free resolution. A presentation is useful for computation. For example, since tensoring is right-exact, tensoring the above presentation with a module, say N, gives: This says that is the cokernel of . If N is an R-algebra, then this is the presentation of the N-module ; that is, the presentation extends under base extension. For left-exact functors, there is for example Proposition — Let F, G be left-exact contravariant functors from the category of modules over a commutative ring R to abelian groups and θ a natural transformation from F to G. If is an isomorphism for each natural number n, then is an isomorphism for any finitely-presented module M. Proof: Applying F to a finite presentation results in and the same for G. Now apply the snake lemma. (Wikipedia).
Open educational video for the day
In today's video we take a look at an example of a product group https://www.youtube.com/watch?v=w2zo2sA1H2k
From playlist Fun!!!
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From playlist Microsoft Excel
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From playlist Microsoft PowerPoint 2010
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From playlist Microsoft PowerPoint 2007
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From playlist Communicating Effectively
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OpenStax illustrates the impact of a free book.
From playlist Exhibit Playlist
Martin Bridson - Subgroups of direct products of surface groups
After reviewing what is known about subgroups of direct products of surface groups and their significance in the story of which groups are Kähler, I shall describe a new construction that provides infinite families of finitely presented subgroups. These subgroups have varying higher-finite
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Introduction to present value | Interest and debt | Finance & Capital Markets | Khan Academy
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/economics-finance-domain/core-finance/interest-tutorial/present-value/v/introduction-to-present-value A choice between money now and money later. Created by Sal Khan.
From playlist Interest and debt | Finance and Capital Markets | Khan Academy
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Prayagdeep Parija: Random Quotients of Hyperbolic Groups and Property (T)
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Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/economics-finance-domain/core-finance/interest-tutorial/present-value/v/time-value-of-money Why when you get your money matters as much as how much money. Present and
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Martin Bridson - Profinite isomorphism problems.
Martin Bridson (University of Oxford, England)
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Kirk McDermott: Topological Aspects of the Shift Dynamics of the Groups of Fibonacci Type
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From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022