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To link to an nLab page but show a different link text, do the following: [[ name of page | link text to show ]].</li> <li style="font-size: 0.8em">LaTeX can be used inside single dollar signs (inline) or double dollar signs or \[ and \], as usual. </li> <li style="font-size: 0.8em">To create a table of contents, add \tableofcontents on its own line.</li> <li style="font-size: 0.8em">For a theorem or proof, use \begin{theorem} \end{theorem} as you would in LaTeX. Labelling and referencing is exactly as in LaTeX, with use of \label and \ref. The full list of supported environments can be found in the <a href="/nlab/show/HowTo#DefinitionTheoremProofEnvironments">HowTo</a>. </li> <li style="font-size: 0.8em">Tikz can be used for figures almost exactly as in LaTeX. Similarly, tikz-cd and xymatrix can be used for commutative diagrams. See the <a href="/nlab/show/HowTo#diagrams">HowTo</a>.</li> <li style="font-size: 0.8em">As an alternative to the Markdown syntax for sections (headings), one can use the usual LaTeX syntax \section, \subsection, etc.</li> <li style="font-size: 0.8em">For further help, see the <a href="/nlab/show/HowTo">HowTo</a>, or you are very welcome to ask at the <a href="https://nforum.ncatlab.org/">nForum</a>.</li> </ol> </div> <form accept-charset="utf-8" action="/nlab/save/satellite" id="editForm" method="post"> <div style="display: none;"> <input name="see_if_human" id="see_if_human" style="tabindex: -1; autocomplete: off"/> </div> <div> <textarea name="content" id="content" style="height: 45em; width: 70%;"> +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Homological algebra +--{: .hide} [[!include homological algebra - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Definition If $T \colon X\to Y$ is an [[additive functor]] between [[abelian category|abelian categories]] with sufficiently many [[projective object|projectives]] and [[injective object|injectives]], then one defines its __right satellite__ $S^1 T$ and __left satellite__ $S^{-1}T=S_1 T:C\to D$ via the formulas $$ S^1 T (A) = ker(T(M)\to T(P)) $$ $$ S_1 T (A) = coker(T(Q)\to T(N)) $$ where $0\to M\to P\to A\to 0$ and $0\to A\to Q\to N\to 0$ are short [[exact sequence]]s where $P$ is projective and $Q$ is injective in $X$. This definition does not depend on the choice of these exact sequences, and moreover $S^1$ and $S_1$ extend to functors. Higher satellites are defined by $S^n T = (S^1)^n T$ and $S^{-n} T = S_n T = (S_1)^n T$ for $n\geq 0$. For every exact sequence $0\to A\to A'\to A''\to 0$ there are natural connecting morphisms such that $S^n T$ (with $-\infty\lt n\lt\infty$) evaluated at $A,A',A''$ compose a long [[exact sequence]]. ## Properties If $T$ is [[exact functor|right exact]] then $S^n T=0$ for all $n\gt 0$ and if $T$ is left exact then $S^n T =0$ for all $n\lt 0$. If $T$ is covariant and $A$ projective then $(S^n T)(A)=0$ for $n\lt 0$, for contravariant $T$ or injective $A$ replace $n\lt 0$ with $n\gt 0$ in the conclusion. ## References There is also an axiomatic definition of satellites and their relation to [[derived functor]]s in the case when $T$ is half exact. See * [[Henri Cartan]], [[Samuel Eilenberg]], _Homological algebra_, Princeton UP 1953, chapter 3. There are generalizations to non-[[additive category|additive categories]]. See * [[George Janelidze|G. Z. Janelidze]], On satellites in arbitrary categories, Bull. Georgian Acad. Sci. __82__ (1976), no. 3, 529-532, in Russian, with a reprint translated in English at [arXiv:0809.1504](http://front.math.ucdavis.edu/0809.1504). for early work; and also in a bit different setup recent * [[Alexander Rosenberg|A. Rosenberg]], [Homological algebra of noncommutative 'spaces' I](http://www.mpim-bonn.mpg.de/preprints/send?bid=3623), 199 pages, preprint Max Planck, Bonn: MPIM2008-91. or for a shorter presentation, Ch. 3 or so of the survey * [[Alexander Rosenberg|A. Rosenberg]], _Topics in noncommutative algebraic geometry, homological algebra and K-theory_, preprint MPIM Bonn 2008-57 [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3589) [[!redirects satellites]]</textarea> <p> <input id="alter_title" name="alter_title" onchange="toggleVisibility();" type="checkbox" value="1" /> <label for="alter_title">Change page name.</label><br/> <span id="title_change" style="display:none"><label for="new_name">New name:</label> <input id="new_name" name="new_name" onblur="addRedirect();" type="text" value="satellite" /></span> </p> <div> <p style="font-size: 0.8em; width: 70%;"> For non-trivial edits, please briefly describe your changes below. A discussion thread for this page does not yet exist at the <a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">nForum</a>, but will be created upon submission of your changes, and your comments will be added to it. This discussion thread can also be used for further discussion related to this page. 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