
	Where results make sense

Topic: Short exact sequence

	Exact sequence - Wikipedia, the free encyclopedia
		In mathematics, especially in homological algebra and other applications of abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next.
		When dealing with exact sequences of groups, it is common to write 1 instead of 0 for the trivial group with a single element.
		The five lemma gives conditions under which the middle map in a commutative diagram with exact rows of length 5 is an isomorphism; the short five lemma is a special case thereof applying to short exact sequences.
	en.wikipedia.org /wiki/Exact_sequence (1010 words)

	Exact Sequences (Site not responding. Last check: 2007-11-03)
		If an exact sequence begins 0 → a → b, the image of 0 is 0, which becomes the kernel of a, hence a embeds in b.
		If an exact sequence ends b → c → 0, c is the kernel of the second homomorphism, and the image of the first, hence b maps onto c.
		If a short exact sequence is split exact, then b is the direct product of a and c, at least up to isomorphism.
	www.mathreference.com /mod-hom,exact.html (300 words)

	Springer Online Reference Works
		Exact sequences often occur and are often used in (co)homological considerations.
		Analogous long exact sequences occur in a variety of other homology and cohomology theories.
		Homology theory; Cohomology; Cohomology sequence; Homology sequence, and various articles on the (co)homology of various kinds of objects, such as Cohomology of algebras; Cohomology of groups; Cohomology of Lie algebras.
	eom.springer.de /e/e036750.htm (120 words)

	Group theory terms
		An exact sequence is a chain of groups connected by homomorphisms such that the image of any one homomorphism in the chain is the kernel of the next homomorphism.
		A short exact sequence is one with only five groups in it, the first and last of which are both the trivial group.
		A short exact sequence is related to the quotient operation on groups.
	groupexplorer.sourceforge.net /help/rf-groupterms.html (3947 words)

	PlanetMath: split short exact sequence
		In an abelian category, a short exact sequence
		"split short exact sequence" is owned by antizeus.
		This is version 4 of split short exact sequence, born on 2002-01-05, modified 2003-09-20.
	planetmath.org /encyclopedia/SplittingBackmap.html (70 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		>the value of exactness is as a means of calculating >unknown groups' structure.
		if you have a short exact >sequence in which you know two of the three groups, >then the corresponding long-exact sequence enables >you to calculate the 3rd, unknown group.
		then the corresponding long exact seq enables..." the whole point of the nudge i was trying to give, was to get quickly to the sense, and let the guy's textbooks take care of the rigor.
	www.math.niu.edu /~rusin/known-math/99/exact (438 words)

	[No title]
		A homology theory on a triangulated category S is an exact functor to an Abelian category which preserves the coproducts that exist in S. Unless we state otherwise, the target category will always be taken to be the category Ab of Abelian groups.
		Y to the short exact sequence 0 -!
		A linear extension of B by a bimodule D is a category C with the same objects as B, together with short exact sequences D(A, B) -j!C(A, B) -p!B(A, B) such that p is a functor and j(p(f)p(g)u) = g O j(u) O f.
	jdc.math.uwo.ca /papers/phantoms.txt (11288 words)

	[No title]
		The short exact sequence * *is Theorem 3.1.1, and it is the central result of the analysis of Part I. Theorem 1.1.3.
		It follows that a cofibre sequence of dg A-objects also induces a lo* ng exact* sequence in homology, but cofibre sequences have the advantage that they are pr* eserved by application of Hom(.; Y) and Hom(X;.), and hence they also induce long exact* * *sequences on applying [.; Y ] and [X;.].
		A, exact in the sense that triangles are taken to lo* ng exact* sequences; in the applications H* is some form of homology.
	www.math.purdue.edu /research/atopology/Greenlees/s1qPartI.txt (19630 words)

	Tensoring a Short Exact Sequence
		Recall that a short exact sequence is an embedding of a into b, with quotient module c, and is denoted as follows.
		The proof is the same, and the following exact sequence appears.
		Let the inclusion of the even integers into Z produce a short exact sequence as follows.
	www.mathreference.com /mod-pit,texact.html (815 words)

	Springer Online Reference Works
		(a particular case of this is the spectral sequence of a Serre fibration).
		The general constructions referred to above also give the spectral sequence of a mapping and take into account (along with their cohomological structure) the degree of disconnectedness of pre-images of points; this is especially effective for zero-dimensional or finite-to-one mappings (in the case of coverings it becomes the Cartan spectral sequence).
		Leray (1945 and later) in connection with the study of homological properties of continuous mappings of locally compact spaces, and he also gave the definition of cohomology (with compact support) with coefficients in a sheaf.
	eom.springer.de /s/s084840.htm (2238 words)

	Math 55a: A preview of ``abstract nonsense'' (Site not responding. Last check: 2007-11-03)
		The sequence is said to be ``exact at V
		An exact sequence 0 --> U --> V --> V/U --> 0 is called a ``short exact sequence''; an exact sequence involving four or more vector spaces between the initial and final zero is called a ``long exact sequence''.
		In any exact sequence of finite-dimensional vector spaces with an initial and final zero, the dimensions of the even- and odd-numbered vector spaces in the sequence have the same sum; in other words, the alternating sum of the dimensions (a.k.a.
	www.math.harvard.edu /~elkies/M55a.05/nonsense.html (424 words)

	Ulrich Oertel Page
		In work related to the classification of automorphisms of 3-manifolds, I obtain a short exact sequence relating various mapping class groups (diffeotopy groups) associated to a compression body of arbitrary dimension.
		This is applied to obtain a short exact sequence for the mapping class group of a reducible 3-manifold M. The sequence gives the mapping class group of the disjoint union of irreducible summands of M as a quotient of the entire mapping class group of M. There are further applications.
		For example, a short exact sequence which gives information about the mapping class group of a 3-manifold which has a "rigid characteristic decomposition."
	www.andromeda.rutgers.edu /~oertel (724 words)

	Math 720
		Use part a) to show that any exact sequence can be obtained by “splicing” together short exact sequences.
		(The snake lemma) Consider the commutative diagram of modules with exact rows:
		Show that if the columns are exact and the bottom two rows are exact, then so is the top.
	www.ndsu.nodak.edu /ndsu/coykenda/M721.2.S2001.htm (96 words)

	102a
		in this short exact sequence is given as follows.
		-algebra of Bost-Connes we must split the above short exact sequence.
		defines a splitting of the short exact sequence.
	www.aimath.org /WWN/rh/articles/html/102a (167 words)

	Derived functor - Wikipedia, the free encyclopedia
		But we can compute its homology at the i-th spot (the kernel of the map from F(I
		Note that left exactness means that 0 →F(X) → F(I
		If X is itself injective, then we can choose the injective resolution 0 → X → X → 0, and we obtain that R
	en.wikipedia.org /wiki/Derived_functor (1209 words)

Try your search on: Qwika (all wikis)