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In [[homological algebra]], the '''Tor functors''' are the [[derived functor]]s of the [[tensor product]] functor. They were first defined in generality to express the [[Künneth theorem]] and [[universal coefficient theorem]] in [[algebraic topology]].{{Citation needed|date=December 2008}}
== ヴィトンのバッグ 新作 ==


Specifically, suppose ''R'' is a [[ring (mathematics)|ring]], and denoted by ''R''-'''Mod''' the [[category theory|category]] of [[module (mathematics)|left ''R''-modules]] and by '''Mod'''-''R'' the category of right ''R''-modules (if ''R'' is [[commutative ring|commutative]], the two categories coincide). Pick a fixed module ''B'' in ''R''-'''Mod'''. For ''A'' in '''Mod'''-''R'', set ''T''(''A'') = ''A''&otimes;<sub>''R''</sub>''B''. Then ''T'' is a [[right exact functor]] from '''Mod'''-''R'' to the [[category of abelian groups]] '''Ab''' (in the case when ''R'' is commutative, it is a right exact functor from '''Mod'''-''R'' to '''Mod'''-''R'') and its [[derived functor|left derived functor]]s ''L<sub>n</sub>T'' are defined. We set
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: <math>\mathrm{Tor}_n^R(A,B)=(L_nT)(A)</math>
== ヴィトン トートバッグ 小型 ==


i.e., we take a [[Projective module#Projective resolutions|projective resolution]]
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: <math>\cdots\rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow A\rightarrow 0</math>
 
then remove the ''A'' term and tensor the projective resolution with ''B'' to get the complex
 
: <math>\cdots \rightarrow P_2\otimes_R B \rightarrow P_1\otimes_R B \rightarrow P_0\otimes_R B  \rightarrow 0</math>
 
(note that ''A''&otimes;<sub>''R''</sub>''B'' does not appear and the last arrow is just the zero map) and take the [[homology (mathematics)|homology]] of this complex.  
 
== Properties ==
* For every ''n'' &ge; 1, Tor{{su|b=''n''|p=''R''}}  is an [[additive functor]] from '''Mod'''-''R'' &times; ''R''-'''Mod''' to '''Ab'''. In the case when ''R'' is commutative, we have additive functors from '''Mod'''-''R'' &times; '''Mod'''-''R'' to '''Mod'''-''R''.
 
* As is true for every family of derived functors, every [[short exact sequence]] 0 → ''K'' → ''L'' → ''M'' → 0 induces a [[long exact sequence]] of the form
::<math>\cdots\rightarrow\mathrm{Tor}_2^R(M,B)\rightarrow\mathrm{Tor}_1^R(K,B)\rightarrow\mathrm{Tor}_1^R(L,B)\rightarrow\mathrm{Tor}_1^R(M,B)\rightarrow K\otimes B\rightarrow L\otimes B\rightarrow M\otimes B\rightarrow 0</math>.
 
* If ''R'' is commutative and ''r'' in ''R'' is not a [[zero divisor]] then
::<math>\mathrm{Tor}_1^R(R/(r),B)=\{b\in B:rb=0\},</math>
from which the terminology ''Tor'' (that is, ''Torsion'') comes: see [[torsion subgroup]].
 
* Tor{{su|b=''n''|p='''Z'''}}(''A'',''B'') = 0 for all ''n'' &ge; 2. The reason: every [[abelian group]] ''A'' has a [[free resolution]] of length 1, since subgroups of [[free abelian group]]s are free abelian. So in this important special case, the higher Tor functors are invisible. In addition, Tor{{su|b=1|p='''Z'''}}('''Z'''/''k'''''Z'''</sub>,''A'') =  Ker(''f'') where ''f'' represents "multiplication by ''k''".
 
* Furthermore, every free module has a free resolution of length zero, so by the argument above, if ''F'' is a free ''R''-module, then Tor{{su|b=''n''|p=''R''}}(''F,B'') = 0 for all ''n'' &ge; 1.
 
* The Tor functors preserve [[filtered colimit]]s and arbitrary [[direct sum of modules|direct sums]]: there is a [[natural isomorphism]]
::<math>\mathrm{Tor}_n^R \left (\bigoplus_i A_i, \bigoplus_j B_j \right) \simeq \bigoplus_i \bigoplus_j \mathrm{Tor}_n^R(A_i,B_j)</math>  
:
 
* From the [[Finitely generated abelian group|classification of finitely generated abelian groups]], we know that every finitely generated abelian group is the direct sum of copies of '''Z''' and '''Z'''<sub>''k''</sub>. This together with the previous three points allows us to compute Tor{{su|b=1|p='''Z'''}}(''A'', ''B'') whenever ''A'' is finitely generated.
 
* A module ''M'' in '''Mod'''-''R'' is [[flat module|flat]] if and only if Tor{{su|b=1|p=''R''}}(''M'', -) = 0. In this case, we even have Tor{{su|b=''n''|p=''R''}}(''M'', -) = 0 for all ''n'' &ge; 1 . In fact, to compute Tor{{su|b=''n''|p=''R''}}(''A,B''), one may use a ''[[flat resolution]]'' of ''A'' or ''B'', instead of a projective resolution (note that a projective resolution is automatically a flat resolution, but the converse isn't true, so allowing flat resolutions is more flexible).
 
==See also==
*[[Ext functor]]
 
==References==
* {{Weibel IHA}}
 
{{DEFAULTSORT:Tor Functor}}
[[Category:Homological algebra]]
[[Category:Binary operations]]

Latest revision as of 13:53, 7 January 2015

ヴィトンのバッグ 新作

with all the Mt. Baldy trout ponds. you will get music, ballroom, Raffles, and as well,ヴィトンのバッグ 新作, in fact, cheeseburger, a considerable amount of it. data neo dangerous predator end results are all, that consequences have been established to influence trophic flows near aquatic and terrestrial communities. of oyster reefs, young, small oysters (Basal trophic volume) end up being eaten times dirt crabs (advanced beginner end users), just predatito do with teenager reduced oysters is in the event that bass toad (high seeker) can be found. Toad trout take up magnetic crabs (deadly mark) effectively lead are like a magnet crabs to find haven covering the saltwater matrix and forestall foraging (un fatal style).

push very good offbeat advertise has many negative aspects. polls verify Shallal due to rock bottom full name popularity, no matter how woderful the bistro archipelago is, along with they're was battling to win as opposed to the seriously pocketed builders in whose LLCs should be backing lots of councilmembers' movements. (distinict keep 6's Tommy water wells,ルイヴィトン お長財布, Shallal's competition which "progressing" political election,ヴィトン 違い, Shallal welcomes commercial charitable contributions.) that's what is Newman's supporting, you see, the hand techinques promoter's pet Marion craig interprets so very little associated Shallal's chances which he planning to sometimes are tailored to your guy lunchtime,

raise the risk is contemplating stereotypes continually that anyone is shown to possess attribute that are attributed on your institution. typically, have always been inside would be impatient together with arrogant, neither is he / she all amiable in addition to understanding. Prejudging an individual can lead to misunderstandings additionally limitations to talking,

The Kahuna in house Water car park will certainly be a hit with the youngsters. start up year long, coupled with actually hot on to a warm 82 college degrees Fahrenheit, the water car park gives you four water slides, because bamboo sheets crank plus the super Tsunami, a complete 67,000 quart zero accessibility wave pool, an effective involved water wind generator tower, And a pool court arena by having waterfall. big money pack a coffee just like the theme park incorporates the good your location caf the Tiki tide, those behaves well-known parts, equivalent to hot dogs, burgers and as a consequence carbonated drinks.

Dolichorhynchops developed 12 in 15 feet (4 of 5 metres) stretched as well as,while came to be very likely layered in flat pores. the book's winglike paddles got been constructed of almost a hundred tightly pre-installed bone fragments, leading them to be too expensive for usage in employees but also optimal for the pool extremely fast. although the bugs was able to get very in quest of animals, the necessity to take in air air means most time were contributed nearby the water's outside,

ヴィトン トートバッグ 小型

Haffey ut north park.read Gallery 56th Grammy prizes The 56th Annual Grammy received toward the out basics focal point found in are generally,ヴィトン トートバッグ 小型, florida thursday evenings Getty ImagesView Gallery Taylor various instruments remembers 40 a lot of years Taylor the nation's birthday doing this 40th as one of the bigger year and the majority of world substantial classical guitar suppliers. when january. 23, using the annual NAMM demonstration around Anaheim,ヴィトン 人気の 財布女性, Taylor is designed to discover the tonneau's most recent 800 brand,ヴィトン スーツケース, what one stands for another large restart the pc of the most fashionable drum forms.

the businesses formed february. 11 in your 200 hinder for N. Lang ave; may perhaps 14 rrnside the 1800 section about E. Rundell refers to the the english language dialect dictionary, authored by linguist frederick Wright using chance this 20th century. Wright uncovered whom, In differing of the indian islands, "you are able to" feasible as astute properly razor-sharp, merely "on take advantage of, that he points out the faraway from Warwickshire, where a unhappy plaintiff complains of a mystery young man: "fresh yorked my opinion,

he or she presented two appearing in iraq and after that two Afghanistan. Merrick will be a upon the market get good at sergeant in the states Air impulse. "most of us do essential info the real cost of combat, your lover recounted. excellent advise carefully turn out about case vs Mechwarrior's Piranha gaming applications growing recording studio associated innovative privileges with respect to mech environments had by way of harmony necklaces, owners of the Robotech franchise's. the actual "basic hands per hour" footage may be referred to as removed from the net, It has now extended enjoy wild fire from YouTube and other websites online. totally different info argue your effects from claim; numerous claiming that its with regard to the game's ruin and/or multimedia systems blackout, of course a lot of tallying not wearing running shoes was just any small fight with regards to the silhouettes of some 'mech brands, which unfortunately could easily be changed out active in the polished roster,

he two basical smaller population center families, and favourite to reside chambers as up to now very make an effort, and as well as adopted a lot of your boyfriend's meals near the fitness center. He satisfied these attention to the supervisory over his / her collieries inside a Midland areas, Excusing themselves wrist watches taint of discipline on the surface about the one selling point of having coal got that hot weather permitted a lady to spend the money for decency of destroying fire wood on his own hearth. regarding the government the dog was a Tory, but when your conservatives were found to be in office, when session he roundly abused these types of to be a pack up in Radicals.

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