Question regarding isomorphism of localizations of modules

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












Suppose $S$ is a multiplicative subset of a ring $A.$ Let $M$ be an $A$-module. We have the localizations $S^-1A$ and $S^-1M.$ However, my book says that there is a natural isomorphism between $S^-1M cong S^-1A otimes_A M$ defined by the mapping $fracxs mapsto frac1s otimes x.$ I don't exactly see how this is an isomorphism much less a well defined a map. Suppose $fracxs = fracx's'.$ That is, there exists some $s''$ such that $s''(xs' - sx') = 0.$ But why does that imply $frac1s otimes x = frac1s' otimes x'$? I don't know if my following reasoning is correct:



$frac1s' otimes x' = frac1s'ss'' otimes s''sx = frac1s'ss'' otimes s''s'x = frac1s otimes x.$



Injectivity: Suppose $frac1s otimes x = frac1s' otimes x'.$ Then $frac1ss' otimes s'x = frac1ss' otimes sx'.$ Thus, $sx' = s'x$ which implies $fracxs = fracx's'.$



Surjectivity: Consider an arbitrary $fracas otimes x.$ This is simply $frac1s otimes ax$ which is the image of $fracaxs.$



Does this work?




abstract-algebra commutative-algebra




share|cite|improve this question



















  • Your arguments are all correct. Alternatively, to justify that your map is an isomorphism, one can also build its inverse.
    – Suzet
    Jul 30 at 0:41















up vote
0
down vote

favorite












Suppose $S$ is a multiplicative subset of a ring $A.$ Let $M$ be an $A$-module. We have the localizations $S^-1A$ and $S^-1M.$ However, my book says that there is a natural isomorphism between $S^-1M cong S^-1A otimes_A M$ defined by the mapping $fracxs mapsto frac1s otimes x.$ I don't exactly see how this is an isomorphism much less a well defined a map. Suppose $fracxs = fracx's'.$ That is, there exists some $s''$ such that $s''(xs' - sx') = 0.$ But why does that imply $frac1s otimes x = frac1s' otimes x'$? I don't know if my following reasoning is correct:



$frac1s' otimes x' = frac1s'ss'' otimes s''sx = frac1s'ss'' otimes s''s'x = frac1s otimes x.$



Injectivity: Suppose $frac1s otimes x = frac1s' otimes x'.$ Then $frac1ss' otimes s'x = frac1ss' otimes sx'.$ Thus, $sx' = s'x$ which implies $fracxs = fracx's'.$



Surjectivity: Consider an arbitrary $fracas otimes x.$ This is simply $frac1s otimes ax$ which is the image of $fracaxs.$



Does this work?




abstract-algebra commutative-algebra




share|cite|improve this question



















  • Your arguments are all correct. Alternatively, to justify that your map is an isomorphism, one can also build its inverse.
    – Suzet
    Jul 30 at 0:41













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Suppose $S$ is a multiplicative subset of a ring $A.$ Let $M$ be an $A$-module. We have the localizations $S^-1A$ and $S^-1M.$ However, my book says that there is a natural isomorphism between $S^-1M cong S^-1A otimes_A M$ defined by the mapping $fracxs mapsto frac1s otimes x.$ I don't exactly see how this is an isomorphism much less a well defined a map. Suppose $fracxs = fracx's'.$ That is, there exists some $s''$ such that $s''(xs' - sx') = 0.$ But why does that imply $frac1s otimes x = frac1s' otimes x'$? I don't know if my following reasoning is correct:



$frac1s' otimes x' = frac1s'ss'' otimes s''sx = frac1s'ss'' otimes s''s'x = frac1s otimes x.$



Injectivity: Suppose $frac1s otimes x = frac1s' otimes x'.$ Then $frac1ss' otimes s'x = frac1ss' otimes sx'.$ Thus, $sx' = s'x$ which implies $fracxs = fracx's'.$



Surjectivity: Consider an arbitrary $fracas otimes x.$ This is simply $frac1s otimes ax$ which is the image of $fracaxs.$



Does this work?




abstract-algebra commutative-algebra




share|cite|improve this question











Suppose $S$ is a multiplicative subset of a ring $A.$ Let $M$ be an $A$-module. We have the localizations $S^-1A$ and $S^-1M.$ However, my book says that there is a natural isomorphism between $S^-1M cong S^-1A otimes_A M$ defined by the mapping $fracxs mapsto frac1s otimes x.$ I don't exactly see how this is an isomorphism much less a well defined a map. Suppose $fracxs = fracx's'.$ That is, there exists some $s''$ such that $s''(xs' - sx') = 0.$ But why does that imply $frac1s otimes x = frac1s' otimes x'$? I don't know if my following reasoning is correct:



$frac1s' otimes x' = frac1s'ss'' otimes s''sx = frac1s'ss'' otimes s''s'x = frac1s otimes x.$



Injectivity: Suppose $frac1s otimes x = frac1s' otimes x'.$ Then $frac1ss' otimes s'x = frac1ss' otimes sx'.$ Thus, $sx' = s'x$ which implies $fracxs = fracx's'.$



Surjectivity: Consider an arbitrary $fracas otimes x.$ This is simply $frac1s otimes ax$ which is the image of $fracaxs.$



Does this work?





abstract-algebra commutative-algebra





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 30 at 0:38









伽罗瓦

781615




781615











  • Your arguments are all correct. Alternatively, to justify that your map is an isomorphism, one can also build its inverse.
    – Suzet
    Jul 30 at 0:41

















  • Your arguments are all correct. Alternatively, to justify that your map is an isomorphism, one can also build its inverse.
    – Suzet
    Jul 30 at 0:41
















Your arguments are all correct. Alternatively, to justify that your map is an isomorphism, one can also build its inverse.
– Suzet
Jul 30 at 0:41





Your arguments are all correct. Alternatively, to justify that your map is an isomorphism, one can also build its inverse.
– Suzet
Jul 30 at 0:41
















active

oldest

votes











Your Answer




StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);








 

draft saved


draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2866569%2fquestion-regarding-isomorphism-of-localizations-of-modules%23new-answer', 'question_page');

);

Post as a guest






















active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes










 

draft saved


draft discarded


























 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2866569%2fquestion-regarding-isomorphism-of-localizations-of-modules%23new-answer', 'question_page');

);

Post as a guest
































































Comments

Post a Comment

Popular posts from this blog

Color the edges and diagonals of a regular polygon

Image
Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne
Read more

Relationship between determinant of matrix and determinant of adjoint?

Image
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/… – Hector Blandin Nov 17 &
Read more

What is the equation of a 3D cone with generalised tilt?

Image
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an
Read more