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# Class Note for MATH 518 at UA

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j u quot ii z xi iiimn Jiii iY g Victor Piercey University of Arizona Math 518 May 7 2008 Introduction In algebraic geometry one encounters invertible sheaves or equivalently line bundles over an algebraic variety in roui In algebraic geometry one encounters invertible sheaves or equivalently line bundles over an algebraic variety lnvertible sheaves over a fixed algebraic variety form a group under the tensor product called the Picard group mum Cum in rzoui In algebraic geometry one encounters invertible sheaves or equivalently line bundles over an algebraic variety lnvertible sheaves over a fixed algebraic variety form a group under the tensor product called the Picard group We will develop a purely algebraic definition for the Picard group of a commutative ring with identity in rzoui In algebraic geometry one encounters invertible sheaves or equivalently line bundles over an algebraic variety lnvertible sheaves over a fixed algebraic variety form a group under the tensor product called the Picard group We will develop a purely algebraic definition for the Picard group of a commutative ring with identity If we think of the ring as the coordinate ring of an affine variety the Picard group of the affine variety is the same as the Picard group of the ring In this talk we will 0 define invertible modules and the Picard group of a commutative ring with identity In this talk we will 0 define invertible modules and the Picard group of a commutative ring with identity 9 identify a method for computing Picard groups of nonsingular affine curves which will require a little bit of geometry mml Gr In this talk we will 0 define invertible modules and the Picard group of a commutative ring with identity 9 identify a method for computing Picard groups of nonsingular affine curves which will require a little bit of geometry 3 explicitly describe the isomorphism class of the Picard groups of nonsingular curves over C In this talk we will 0 define invertible modules and the Picard group of a commutative ring with identity 9 identify a method for computing Picard groups of nonsingular affine curves which will require a little bit of geometry 3 explicitly describe the isomorphism class of the Picard groups of nonsingular curves over C 9 determine a method for computing Picard groups of singular affine curves and In this talk we will 0 define invertible modules and the Picard group of a commutative ring with identity 9 identify a method for computing Picard groups of nonsingular affine curves which will require a little bit of geometry 3 explicitly describe the isomorphism class of the Picard groups of nonsingular curves over C 9 determine a method for computing Picard groups of singular affine curves and Q compute the Picard group of a curve with a cusp and a curve with a node For a ring A an Amodule l is invertible if I is finitely generated and for every prime ideal p e SpecA lF E AF as Apmodules W Eml wiv iatm mxm 39Dei39ti on For a ring A an Amodule l is invertible if I is finitely generated and for every prime ideal p e SpecA lF E AF as Apmodules Rgma k The condition that lF E AF is often described by saying that l is locally free of rank 1 W st wi itm mm Ill For a ring A an Amodule l is invertible if I is finitely generated and for every prime ideal p e SpecA lF E AF as Apmodules Ramark The condition that lF E AF is often described by saying that l is locally free of rank 1 It I is an ideal in A and is invertible as an Amodule we say that is an invertible ideal The notation for the modules is chosen because we will see that every invertible module is isomorphic to an invertible ideal Examples 9 If is afree module of rank 1 then is invertible Examples 9 It is atree module of rank 1 then is invertible 0 Any principal ideal generated by a nonzero divisor is invertible Examp o It is atree module of rank 1 then is invertible 0 Any principal ideal generated by a nonzero divisor is invertible 0 Let A Zx75 and let I 21 75 Then is invertible but not principle Wl Grout Wm Carma Exam 0 It is atree module of rank 1 then is invertible 0 Any principal ideal generated by a nonzero divisor is invertible 0 Let A Zx75 and let I 21 75 Then is invertible but not principle 0 Let k be an algebraically closed field and let A kXyy2 7 X3 X Then every maximal ideal m e MaXA is invertible but not principal mml Gloria39s aiming IW Fractional Ideals Let A be a commutative ring with identity and let K KA be the total ring of fractions Fractional Ideals Let A be a commutative ring with identity and let K KA be the total ring of fractions The Asubmodules of KA are called fractional ideals of A Let A be a commutative ring with identity and let K KA be the total ring of fractions The Asubmodules of KA are called fractional ideals of A It I is a finitely generated fractional ideal of A we write all the generators over a common denominator This shows that l is isomorphic as an Amodule to an ordinary ideal of A tea u r mm r 2 Ewe imam rm Let A be a commutative ring with identity and let K KA be the total ring of fractions The Asubmodules of KA are called fractional ideals of A If I is a finitely generated fractional ideal of A we write all the generators over a common denominator This shows that l is isomorphic as an Amodule to an ordinary ideal of A For any set I C K A but especially fractional ideals define l 1 s e KA l le A Wm 11qu Duality Pairings and Natural Maps Given an Amodule I the dual module is defined to be 1 HomAu A Duality Pairings and Natural Maps Given an Amodule I the dual module is defined to be 1 HomAu A The duality pairing on I x gives rise to a unique homomorphism u I X l a A Duality Pairings and Natural Maps Given an Amodule I the dual module is defined to be I HomAI A The duality pairing on I x gives rise to a unique homomorphism u I X l a A This map is given by g X a l gt Ma Let A be a Noetherian domain Q If is an Amodule then I is invertible if and only if the natural map In l X l a A is an isomorphism W lml wiw lmcnxw Let A be a Noetherian domain Q If is an Amodule then I is invertible if and only if the natural map In l X l a A is an isomorphism 9 Every invertible module is isomorphic to a fractional ideal of A Every invertible fractional ideal contains a nonzerodivisor of A mango wimrmwm ne War was Let A be a Noetherian domain 0 If is an Amodule then I is invertible if and only if the natural map u l X l a A is an isomorphism 9 Every invertible module is isomorphic to a fractional ideal of A Every invertible fractional ideal contains a nonzerodivisor of A Q If I J C K A are invertible modules then the natural map l X J a J given by s X t gt gt st is an isomorphism as is the natural map I lJ a HomAl J given by t gt gt Apt Where ta ta In particular I 1 g I Emmi 39r aquot mum Clam Let A be a Noetherian domain Q If is an Amodule then I is invertible if and only if the natural map u l X l a A is an isomorphism 9 Every invertible module is isomorphic to a fractional ideal of A Every invertible fractional ideal contains a nonzerodivisor of A Q If I J C K A are invertible modules then the natural map l X J a J given by s X t gt gt st is an isomorphism as is the natural map I lJ a HomAl J given by t gt gt Apt Where ta ta In particular I 1 g I G If I C K A is any Asubmodule then I is invertible if and only if I ll A i F me mmm mum Clams The Picard Gr up As an immediate corollary the collection of isomorphism classes of invertible modules forms a set As an immediate corollary the collection of isomorphism classes of invertible modules forms a set Moreover the theorem states that the set of isomorphism classes of invertible modules forms a group under the tensor product Wm 11qu As an immediate corollary the collection of isomorphism classes of invertible modules forms a set Moreover the theorem states that the set of isomorphism classes of invertible modules forms a group under the tensor product The identity element is the isomorphism class of A as an Amodule The inverse of an invertible module is the dual l Wm 11qu As an immediate corollary the collection of isomorphism classes of invertible modules forms a set Moreover the theorem states that the set of isomorphism classes of invertible modules forms a group under the tensor product The identity element is the isomorphism class of A as an Amodule The inverse of an invertible module is the dual l The Picard group of A denoted by PicA is the group of all isomorphism classes of invertible Amodules W39le e smurmmw Cartier Divisors The collection of all invertible submodules of K forms a group under ideal multiplication with identity element given by A as a module over itself and where the inverse of l is given by Iquot The collection of all invertible submodules of K forms a group under ideal multiplication with identity element given by A as a module over itself and where the inverse of l is given by Iquot The group of invertible submodules of K is called the group of Cartier divisors and denoted by CA The collection of all invertible submodules of K forms a group under ideal multiplication with identity element given by A as a module over itself and where the inverse of l is given by Iquot The group of invertible submodules of K is called the group of Cartier divisors and denoted by CA A principal divisor is an element of CA of the form Au for u e KX They form a subgroup of CA denoted by PCA Wl C I39WIQ E warm 1qu law l llrrquot 11 w rm l mlm The collection of all invertible submodules of K forms a group under ideal multiplication with identity element given by A as a module over itself and where the inverse of l is given by Iquot The group of invertible submodules of K is called the group of Cartier divisors and denoted by CA A principal divisor is an element of CA of the form Au for u e KX They form a subgroup of CA denoted by PCA W mlaxe wr llmtimw 1m v mum m Wm 153tth Lefcp CA a PicA be given by Ml I Where I is the isomorphism class of the invertible module I 0 The map o is a surjeotive group homomorphism and ker o7 K X AX mommy simmwa law v mm m We 1th Lefcp CA a PicA be given by Ml I Where I is the isomorphism class of the invertible module I D The map u is a surjeotive group homomorphism and keer K X AX Q The group CA is generated by the set of invertible ideals of A mangle wi nkmmm law v mm m We 1th Lefcp CA a PicA be given by Ml I Where I is the isomorphism class of the invertible module I D The map u is a surjeotive group homomorphism and keer K X AX Q The group CA is generated by the set of invertible ideals of A mangle wi nkmmm law v mm m We 1th Lefcp CA a PicA be given by Ml I Where I is the isomorphism class of the invertible module I D The map u is a surjeotive group homomorphism and keer K X AX Q The group CA is generated by the set of invertible ideals of A Combined with the lemma above we have mangle wi nkmmm law v mm m We 1th Lefcp CA a PicA be given by Ml I Where I is the isomorphism class of the invertible module I D The map u is a surjeotive group homomorphism and keer K X AX Q The group CA is generated by the set of invertible ideals of A Combined with the lemma above we have PicA g CAPCA mangle wi nkmmm Picard Gr ups of N nSingularCurves Let k be an algebraically closed field Picard Groups of NonSingular Curves Let k be an algebraically closed field If X is a nonsingular affine curve its Picard group PicX is the Picard group PicA where A is the affine coordinate ring kX1 Xl of the curve Let k be an algebraically closed field If X is a nonsingular affine curve its Picard group PicX is the Picard group PicA where A is the affine coordinate ring kX1 Xl of the curve We will be able to describe these Picard groups explicitly when the ground field is C Wl C I39WIQ E warm 1qu Let k be an algebraically closed field If X is a nonsingular affine curve its Picard group PicX is the Picard group PicA where A is the affine coordinate ring kX1 Xl of the curve We will be able to describe these Picard groups explicitly when the ground field is C To lay the ground work for this discussion we begin with objects called Weil divisors Emmi rmly mm 1qu m1 Qui39yi nn The group of Weil divisors of a ring A denoted DivA is the free abelian group on the set of codimension one prime ideals of A A Weil divisor is an element of DiVA memes Watm mxm rm Quilting The group of Weil divisors of a ring A denoted DivA is the free abelian group on the set of codimension one prime ideals of A A Weil divisor is an element of DiVA Weil divisors and Cartier divisors are generally speaking very different W Ewul thm mxm The group of Weil divisors of a ring A denoted DivA is the free abelian group on the set of codimension one prime ideals of A A Weil divisor is an element of DiVA Weil divisors and Cartier divisors are generally speaking very different However if A is a Dedekind domain CA DiVA momma armrewa The Class G oup If A is a Dedekind domain and CA DiVA then PCA lt DivA The Class Group It A is a Dedekind domain and CA DiVA then PCA lt DivA We can therefore form the class group C1A DivAPCA The Class Group It A is a Dedekind domain and CA DiVA then PCA lt DivA We can therefore form the class group C1A DivAPCA Accordingly it A is a Dedekind domain PicA C1A It A is a Dedekind domain and CA DiVA then PCA lt DivA We can therefore form the class group C1A DivAPCA Accordingly it A is a Dedekind domain PicA C1A Since PCA can be identified with principal ideals of A and a Dedekind domain is a PD it and only it it is a UFD it follows that a Dedekind domain A is a UFD it and only it PicA is trivial wr image arming IW nsingular C rves An affine curve X is nonsingular if and only if its coordinate ring A is a Dedekind domain Nonsingular Curves An affine curve X is nonsingular if and only if its coordinate ring A is a Dedekind domain Accordingly PicX C1A Nonsingular Curves An affine curve X is nonsingular it and only it its coordinate ring A is a Dedekind domain Accordingly PicX C1A The method to compute these groups is to embed the curve into projective space LPN for some N r Normal An affine curve X is nonsingular it and only it its coordinate ring A is a Dedekind domain Accordingly PicX C1A The method to compute these groups is to embed the curve into projective space LPN for some N We will make sense of the Picard group and the class group of a projective curve mmi Gr WWW A An affine curve X is nonsingular if and only if its coordinate ring A is a Dedekind domain Accordingly PicX C1A The method to compute these groups is to embed the curve into projective space LPN for some N We will make sense of the Picard group and the class group of a projective curve Then we will determine how to undo the projectivization to obtain the Picard group of the affine curve The Picard Group of a Projective Curve Suppose X is a nonsingular curve of genus g embedded into LPN for some N The Fquot 39 p of a Projective Curve Suppose X is a nonsingular curve of genus g embedded into LPN for some N Such a curve does not correspond to a single coordinate ring Instead it has affine patches that have corresponding coordinate rings which may not all be the same P I p of a Projective Curve Suppose X is a nonsingular curve of genus g embedded into LPN for some N Such a curve does not correspond to a single coordinate ring Instead it has affine patches that have corresponding coordinate rings which may not all be the same Taking a collection of invertible modules over the affine coordinate rings and gluing them together in a consistent manner forms a sheaf of modules that is locally free of rank one Such a sheaf is called an invertible sheaf Wrrm Cum P I p of a Projective Curve Suppose X is a nonsingular curve of genus g embedded into LPN for some N Such a curve does not correspond to a single coordinate ring Instead it has affine patches that have corresponding coordinate rings which may not all be the same Taking a collection of invertible modules over the affine coordinate rings and gluing them together in a consistent manner forms a sheaf of modules that is locally free of rank one Such a sheaf is called an invertible sheaf The Picard group PicX is the group of isomorphism classes of invertible sheaves The operation inverses and identity as in Wrrm Cum Weil Divisors on Projective Curves The group of Weil divisors on the curve X is the tree abelian group on the closed points of the curve Weil Divisors on Projective Curves The group of Weil divisors on the curve X is the tree abelian group on the closed points of the curve On any affine patch the closed points are precisely the codimension 1 prime ideals which are maximal ideals in a curve Weill v39 The group of Weil divisors on the curve X is the free abelian group on the closed points of the curve On any affine patch the closed points are precisely the codimension 1 prime ideals which are maximal ideals in a curve A principal divisor is the formal sum of zeros and poles of a certain class of functions on the curve where the coefficients are the order of the corresponding zero or pole Weill v39 The group of Weil divisors on the curve X is the free abelian group on the closed points of the curve On any affine patch the closed points are precisely the codimension 1 prime ideals which are maximal ideals in a curve A principal divisor is the formal sum of zeros and poles of a certain class of functions on the curve where the coefficients are the order of the corresponding zero or pole Over C one considers the meromorphic functions with the usual definition of the order of a zero or a pole The Degree Map One can now form the class group on a projective curve as the Weil divisors modulo principal divisors The Degree Map One can now form the class group on a projective curve as the Weil divisors modulo principal divisors There is a surjective homomorphism D DiVX a Z given by D Z niPi 2 Hi One can now form the class group on a projective curve as the Weil divisors modulo principal divisors There is a surjective homomorphism D DiVX a Z given by D Z niPi 2 Hi This map descends to a well defined map d C1X a Z called the degree map One can now form the class group on a projective curve as the Weil divisors modulo principal divisors There is a surjective homomorphism D DiVX a Z given by D Z niPi 2 Hi This map descends to a well defined map d C1X a Z called the degree map For a nonsingular projective curve PicX E C1X so we can define the degree map on PicX mml Gr A Short Exact Sequence The subgroup of invertible sheaves of degree zero is denoted Pic0X The subgroup of invertible sheaves of degree zero is denoted Pic0X This is precisely the kernel of the degree map A Short Exact Sequence The subgroup of invertible sheaves of degree zero is denoted Pic0X This is precisely the kernel of the degree map We now have the following short exact sequence 0 a Pic0X a PicX a Z a o The subgroup of invertible sheaves of degree zero is denoted Pic0X This is precisely the kernel of the degree map We now have the following short exact sequence 0 a Pic0X a PicX a Z a 0 Since Z is a projective Zmodule this exact sequence splits and we obtain PicX e Z ea Pic0X mml Gr A Brief Deto r The result above shows that it suffices to study Pic X A Brief Detour The result above shows that it suffices to study Pic X There is a particularly nice description of Pic X when the ground field is C an writingwatt WWW m The result above shows that it suffices to study Pic X There is a particularly nice description of Pic X when the ground field is C This requires a brief detour where we discuss the Jacobian of a compact Riemann surface mums 61mm The Jacobian of a Riemann Surface Let X be a compact Riemann surface of genus g The Jacobian of a Riemann Surface Let X be a compact Riemann surface of genus g Recall the first singular homology group is H1XZ 229 Let X be a compact Riemann surface of genus g Recall the first singular homology group is H1XZ 229 If w is a holomorphic 1form then w is closed Hence if c 8A is a boundary in H1 X Z then by Stokes39 w timith am 13 t W Let X be a compact Riemann surface of genus g Recall the first singular homology group is H1XZ 229 It to is a holomorphic 1form then to is closed Hence it c 8A is a boundary in H1 X Z then by Stokes39 Let Q X be the vector space of holomorphic 1forms on X Then for each c e H1XZ we have a well defined linear functional 2 X a c 0 Wl Group warm 1qu The Jacobian of a Riemann S fface We now have an embedding H1XZ gt Q1 X The Jacobian of a Riemann Surface We now have an embedding H1XZ gt Q1 X The Jacobian of X is defined to be JX Q1 XH1 X Z The Jacobian of a Riemann Surface We now have an embedding H1XZ gt Q1 X The Jacobian of X is defined to be JX Q1 XH1 X Z The vector space Q1 X is isomorphic to C9 The Jacobian of a Riemann Surface We now have an embedding H1XZ gt Q1 X The Jacobian of X is defined to be JX Q1 XH1 X Z The vector space Q1 X is isomorphic to C9 Since H1XZ E 229 we have the following isomorphism of abelian groups JX g 19229 g RZQZZQ T29 PicX over C A deep theorem of Abel and Jacobi shows that Pic X JX Pic X over C A deep theorem of Abel and Jacobi shows that Pic X JX Therefore for a projective curve over C PicX Z ea T29 Pic over C A deep theorem of Abel and Jacobi shows that Pic X JX Therefore for a projective curve over C PicX Z ea T29 One interesting consequence is that if X is an elliptic curve PicX e Z ea X F nitenesm mm m A deep theorem of Abel and Jacobi shows that Pic X JX Therefore for a projective curve over C PicX Z ea T29 One interesting consequence is that if X is an elliptic curve PicX e Z ea X If X is a rational curve genus zero over C PicX Z We can describe the generator concretely Since X is rational it is biholomorphic to it The generator of PicX is the socalled tautological bundle the bundle over it whose fibers over a point 6 is the line 6 mml Gr N SinguarAffine Curves Suppose A kX1 Xl is the coordinate ring of a nonsingular affine curve X NonSingular Affine Curves Suppose A kX1 Xl is the coordinate ring of a nonsingular affine curve X To compute PicA we will complete the curve to a projective curve X obtaining the Picard group PicX as above Suppose A kX1 Xl is the coordinate ring of a nonsingular affine curve X To compute PicA we will complete the curve to a projective curve X obtaining the Picard group PicX as above We will then determine how the Picard group responds when restricting to the affine part that we started with mml Gr Points at Infinity The coordinate ring gives us a closed embedding of our curve X SpecA into Aquot SpeckX17 7Xn Points at Infinity The coordinate ring gives us a closed embedding of our curve X SpecA into Aquot SpeckX17 7Xn Complete Aquot to LPquot by adding in the hyperplane at infinityquot which can be chosen to be 2 O The coordinate ring gives us a closed embedding of our curve X SpecA into Aquot SpeckX17 7Xn Complete Aquot to LPquot by adding in the hyperplane at infinityquot which can be chosen to be 2 O The curve X is completed by adding finitely many points in this hyperplane These are the points at infinity on the curve mml GT Realms The coordinate ring gives us a closed embedding of our curve X SpecA into Aquot SpeckX17 7Xn Complete Aquot to LPquot by adding in the hyperplane at infinityquot which can be chosen to be 2 O The curve X is completed by adding finitely many points in this hyperplane These are the points at infinity on the curve The question therefore becomes how the Picard group of X changes when finitely many points are removed mml 1139 mm CLAW Let X be a nonsingular curve embedded in projective space 1Pquot Let ZP177PkCX be a finite subset of the points of X and let U X Z Then there is a short exact sequence 2 a C1X a C1U a o Mamba whittm mxm Let X be a nonsingular curve embedded in projective space 1quot Let ZP177PkCX be a finite subset of the points of X and let U X Z Then there is a short exact sequence 2 a C1X a C1U a o Letting U be the original affine curve with coordinate ring A and let p1 pk be the points at infinity Then PicA g Pic0X Zk 1 mamas wi tm uxw PicA over C When the ground field is C we have PicA g ngZkquot PicA over C When the ground field is C we have PicA g ngZkquot Recall that if A is a Dedekind domain A is a UFD if and only if PicA is trivial When the ground field is C we have PicA g ngZkquot Recall that it A is a Dedekind domain A is a UFD it and only it PicA is trivial As an interesting consequence the coordinate ring of a nonsingular affine curve is a UFD it and only it the curve is rational genus O Wi Grout Wm Cum Singular Affine Curves The method used for nonsingular curves no longer works in the singular case The method used for nonsingular curves no longer works in the singular case In the nonsingular case the ring A is a Dedekind domain and we have isomorphisms with the divisor class group Affine coordinate rings for singular curves are Noetherian domains but they are not Dedekind domains Emmi rm Wm 11qu The method used for nonsingular curves no longer works in the singular case In the nonsingular case the ring A is a Dedekind domain and we have isomorphisms with the divisor class group Affine coordinate rings for singular curves are Noetherian domains but they are not Dedekind domains Instead we will compute the Picard group by constructing the beginning of a MeyerVietorisquot sequence that will be applied to a special fiberproduct diagram called the conductor squarequot Projective Modules over Fiber Products To derive the MeyerVietorisquot sequence we will need to construct projective modules over fiber products To derive the MeyerVietorisquot sequence we will need to construct projective modules over fiber products Suppose the following is a fiber product square 0 2 191 Azas 192 To derive the MeyerVietorisquot sequence we will need to construct projective modules over fiber products Suppose the following is a fiber product square 0 2 191 Azas 92 Suppose in addition that 31 is surjective and a1 is injective Projective Modules over Fiber Products II The ring A can be explicitly described as Aa17a2 6 A1 X A2 W191 5292 Projective Modules over Fiber Products II The ring A can be explicitly described as A a17a2 6 A1 X A2 W191 5292 Suppose P is an Aimodule for i 12 and we have an S module isomorphism o7 P1 A1 S a P2 A2 S Projective Modules over Fiber Products II The ring A can be explicitly described as Aa17a2 6 A1 X A2 W191 5292 Suppose P is an Aimodule for i 12 and we have an S module isomorphism o7 P1 A1 S a P2 A2 8 Define MP17P27w1P17P2 P1X P2 W001 1 P2 1 Prey 39 gui es over Fiber Products The ring A can be explicitly described as Aa17a2 6 A1 X A2 i 51a1 5292 Suppose P is an Aimodule for i 12 and we have an S module isomorphism o7 P1 A1 S a P2 A2 8 Define MP1P27w1P17P2 P1X P2 Mp1 1 P2 lt81 MP1 P2 o7 is an Amodule under the action a 11402 a1a P17a2a P2 Wi Group wi tiirm 1qu Projective Modules over Fiber Products lll The following theorems are proved in Milnor39s book on Ktheory and in my paper posted on my website mmwwmm m les im iEilidules over Fiber Products I The following theorems are proved in Milnor39s book on Ktheory and in my paper posted on my website 7 rem If P are finitelygenerated projective A modules then MP1 P2 4p is a finitelygenerated projective Amodule W mlae awnimrimmee wmlk mt39 iasi 1W ME WWW it The following theorems are proved in Milnor39s book on Ktheory and in my paper posted on my website If P are finitelygenerated projective A modules then MP1 P2 4p is a nitelygenerated projective Amodule Theorem Every projective Amodule is isomorphic to M P1 P2 4p for some P1 P2 and 4p mama wmfm em Modules over Fiber Produts III The following theorems are proved in Milnor39s book on Ktheory and in my paper posted on my website If P are finitelygenerated projective A modules then MP1 P2 4p is a finitelygenerated projective Amodule Theorem Every projective Amodule is isomorphic to M P1 P2 4p for some P1 P2 and 4p Theorem If P are projective Amodules and 4p P1 A1 S a P2 A2 S is an isomorphism then P E MP1 P2 w A Ai iglww i mlpa i tim gmm MeyerVietoris To derive the MeyerVietoris sequence we need a couple more lemmata To derive the MeyerVietoris sequence we need a couple more lemmata Suppose P and P are invertible A modules fori 12 Then MP1 P247 MP P y ifand only if there are isomorphisms 1p P H P fori 12 such that w wg 1ds o w ow1 lds W Eml e wiv tittm mm If P Q are projective Aimodules fori 12 and 72P1 ASHP2 ASziJQ1 ASH02 ASare isomorphisms then MP17 P2W A MQ17 02711 MP1 A1Q17P2 A2 027 1M Wimpy summatiwa Algebraic K Theory We also need a bit of Algebraic K Theory z tlglfa g 39 t We also need a bit of Algebraic K Theory For a commutative ring A with identity K0A is the quotient of the tree abelian group on isomorphism classes of finitely generated projective modules over A modulo the relation P O P69 0 mml Gr it t a 39 miean 6mm z llglfa Q t We also need a bit of Algebraic K Theory For a commutative ring A with identity K0A is the quotient of the tree abelian group on isomorphism classes of finitely generated projective modules over A modulo the relation P O P69 0 K0 A can be made into a ring under the tensor product The multiplicative identity is the isomorphism class of A as an Amodule Alg lfa Q We also need a bit of Algebraic K Theory For a commutative ring A with identity K0A is the quotient of the tree abelian group on isomorphism classes of finitely generated projective modules over A modulo the relation P O P69 0 K0 A can be made into a ring under the tensor product The multiplicative identity is the isomorphism class of A as an Amodule PicA embeds into K0A it is the group of units of K0 A as a ring Wm 11qu MeyerVietoris I We will now define the maps that will be used in the MeyerVietorisquot sequence MeyerVietoris We will now define the maps that will be used in the MeyerVietorisquot sequence In general a ring homomorphism f A a Btakes units to units and therefore induces a group homomorphism f AX a BX We will now define the maps that will be used in the MeyerVietorisquot sequence In general a ring homomorphism f A a Btakes units to units and therefore induces a group homomorphism f AX a BX Define L AX a Af 69A by La a1aa2a Wm 2mm We will now define the maps that will be used in the MeyerVietorisquot sequence In general a ring homomorphism f A a Btakes units to units and therefore induces a group homomorphism f AX a BX Define L AX a Af 69A by La a1aa2a ell391lt9393 lX69AH 3X bYa17a2 513152a271 Wl CHM Wm Cum MeyerVietoris IV For each i define 6 PicA a PicA by 6l A A l MeyerVietoris IV For each i define 6 PicA a PicA by 6l A A I If p e SpecA and q 6 SpecA with p0 q then AI A 0n AIM Aq In g AIM Aq Aq g AIL Hence 6l e PicA For each i define 6 PicA a PicA by 6l A A I If p e SpecA and q 6 SpecA with p0 q then AI A 0n AIM Aq In g AIM Aq Aq g AIL Hence 6l e PicA The maps 6 are group homomorphisms by associativity of the tensor product Wt Gnglgg nmmm 2mm For each i define 6 PicA a PicA by 6l A A I If p e SpecA and q 6 SpecA with p0 q then AI A 0n AIM Aq In g AIM Aq Aq g AIL Hence 6l e PicA The maps 6 are group homomorphisms by associativity of the tensor product Define 6 PicA a PicA1 ea PicA2 by 6 6162 Wt mung mum 2mm There is a homomorphism 8 8X a PicA such that following sequence is exact o a AX A Af eaAZX L 3X 3 PicA i PicA1 PicA2 W Eml whittm mm Proof of MeyerVietoris We will prove this theorem in the following steps 0 Ker hm Proof of MeyerVietoris We will prove this theorem in the following steps 0 Ker km 9 Define 8 and show that it lands in PicA Proof of MeyerVietoris We will prove this theorem in the following steps 0 Ker km 9 Define 8 and show that it lands in PicA O Ker Proof of MeyerVietoris We will prove this theorem in the following steps 0 Ker km 9 Define 8 and show that it lands in PicA O Ker 9 Ker6 III18 Proof of MeyerVietoris We will prove this theorem in the following steps 0 Ker km 9 Define 8 and show that it lands in PicA O Ker 9 Ker6 III18 Woof We will prove this theorem in the following steps 0 Ker km 9 Define 8 and show that it lands in PicA Q Ker 9 Ker6 Im8 Note that exactness at the left end is obvious by the assumption that m is injective Wl CHM Wm CMW Proof of MeyerVietoris Since 31a1a 32a2a it is obvious that Imj C Ken Proof of MeyerVietoris Since 1a1a 32a2a it is obvious that Imj C Ken Suppose511512 1 This meansthat 31 i1 za2 1 1 and therefore 31a1 3292 Since 31a1a 32a2a it is obvious that Imj C Kerr Suppose511512 1 This meansthat 31 i1 za2 1 1 and therefore 31a1 3292 Since A is the fiber product of A1 and A2 over 8 there is a e A such that aa a which precisely says that La 511512 mmi C I39WIQE MWhh IW Since 31a1a 32a2a it is obvious that Imj C Kerr Suppose511512 1 This meansthat 31 i1 za2 1 1 and therefore 31a1 3292 Since A is the fiber product of A1 and A2 over 8 there is a e A such that aa a which precisely says that La 511512 This establishes the other containment mmi C I39WIQE MWhm IW Proof of MeyerVietoris Next we define 8 Proof of MeyerVietoris Next we define 8 Consider P A viewed as Aimodules Proof of MeyerVietoris Next we define 8 Consider P A viewed as Aimodules Observe that A Ar Sis generated by 1A 2 15 as S modules Proof of MeyerVietoris Next we define 8 Consider P A viewed as Amodues Observe that A Af Sis generated by 1A 2 15 as S modules Therefore 3 e 8 defines an S module homomorphism LpSIA1 A1SgtA2 8A2 Sgiven by 1A1 X 15 gt gt 31A2 215 Next we define 8 Consider P A viewed as Amodues Observe that A Af Sis generated by 1A 2 15 as S modules Therefore 3 e 8 defines an S module homomorphism LpSIA1 A1SgtA2 8A2 Sgiven by1A1 1s gt gt 31A2 215 If s 6 8X dag is an isomorphism and therefore we can form the Amodule M5 MA1A2 dag mmi 1139 Win CLAW Proof of MeyerVietoris IV I claim that the isomorphism class of M5 denoted by M5 is in PicA Th Proof of MeyerVietoris IV I claim that the isomorphism class of M5 denoted by M5 is in PicA Since A are finitely generated projective Aimodules M5 6 K0A Proof of MeyerVietoris IV I claim that the isomorphism class of M5 denoted by M5 is in PicA Since A are finitely generated projective Amodues M5 6 K0A It therefore suffices to show that M5 is a unit in the ring K0A Proof 2 I claim that the isomorphism class of M5 denoted by M5 is in PicA Since A are finitely generated projective Amodules M5 6 K0A It therefore suffices to show that M5 is a unit in the ring K0A Observe the following for s t 6 8X Ms AMt MA1 A1A17A2 A2A27lps 4pt MA17A27 ws w g MA17A27 wst Mst mml Gloria39s aiming th Maw men I claim that the isomorphism class of M5 denoted by M5 is in PicA Since A are finitely generated projective Amodules M5 6 K0A It therefore suffices to show that M5 is a unit in the ring K0A Observe the following for s t 6 8X Ms AMt MA1 A1A17A2 A2A27lps 4pt MA17A27 ws w g MA17A27 wst Mst Observe in addition that M1S A as A modules Wl C I39WIQ E warm 1qu Proof of MeyerVietoris V Therefore in Ko A MSHMY A Proof of MeyerVietoris V Therefore in KO A MSHMV A It follows that M5 6 PicA Proof of MeyerVietoris V Therefore in KO A MsMsew A It follows that M5 6 PicA Define a sx a PicA by 83 M5 Proof of MeyerVietoris V Therefore in K0A MSHMY A It follows that M5 6 PicA Define 8 SX a PicA by 83 M5 Note that 831 Mst Ms A Mt Ms A Mt WSW Proof of MeyerVietoris V Therefore in K0A MSHMV A It follows that M5 6 PicA Define 8 SX a PicA by 83 M5 Note that 831 Mst Ms w M Ms A Mr 8s8t Thus 8 is a group homomorphism Proof of MeyerVietoris VI Suppose s ja1a2 31 5103292quot Then Ms Mmaomazw Mm a1 A M 25239 Proof of MeyerVietoris VI Suppose s ja1a2 31 5103292quot Then Ms Mmaomazw Mm a1 A Mme By a lemma used to prove our results for MP1 P2 w N A 2ag M191 51 g A and M daz 1 M Proof of MeyerVietoris VI Suppose s ja1a2 31 5103292quot Then Ms Mmaomazw Mm a1 A Mme By a lemma used to prove our results for MP1 P2 4p M191 51 g A and M daz 1 M N A39 2ag Accordingly if s 6 may then 83 Ms A A A A which implies s e ker8 Proof of MeyerVietoris VI Now suppose 83 M5 A Proof of MeyerVietoris VI Now suppose 83 M5 A Then MA1A2w5 z MA1A2 w Proof of MeyerVietoris VI Now suppose 83 M5 A The MA17A2Ws g MA17A27 1 By one of the lemmata we have isomorphisms 1p A a A such that w wg 1ds o ws 0 1 1ds Proof of MeyerVietoris V Plug 141 2 15 into the equation On the right hand side we have 11A1 1S1A2 18 Proof of MeyerVietoris V Plug 141 2 15 into the equation On the right hand side we have 11A1 1S1A2 18 A computation shows that on the left side we have 511111A262 E11A21A2 318 Proof of MeyerVietoris V Plug 141 2 15 into the equation On the right hand side we have 11A1 1S1A2 18 A computation shows that on the left side we have 511111A262 E11A21A2 318 It follows that 51 11A252 11A2 1s Proof of MeyerVietoris V Since 1 is an isomorphism for each i1J391A must be a unit Proof of MeyerVietoris V Since 1 is an isomorphism for each i1J391A must be a unit Let a1 mumi1 and let a2 151mg Proof of MeyerVietoris V Since 1 is an isomorphism for each i1J391A must be a unit Leta1 1J11A1 1 andletag 151mg Then our equation reads 351a152a21 Proof of MeyerVietoris V Since 1 is an isomorphism for each i1J391A must be a unit Leta1 1J11A1 1 andletag 151mg Then our equation reads 351a152a21 Therefore 3 51K 111 1 1523271 51a1523271a17a2 Proof of MeyerVietoris IX Suppose I e Im8 Then I M5 for some 3 6 8X Proof of MeyerVietoris IX Suppose I e Im8 Then I M5 for some 3 6 8X Therefore lg MA1A2 75 Th Proof of MeyerVietoris IX Suppose I e Im8 Then I M5 for some 3 6 8X Therefore lg MA1A2 75 ThUS I Af A g A Proof of MeyerVietoris IX Suppose I e Im8 Then I M5 for some 3 6 8X Therefore IE MA1A2 p5 ThUS I Ar A g A It follows that 6I A1 A2 which implies I e ker6 Proof of MeyerVietoris X Suppose 5WD A1L A2 Proof of MeyerVietoris X Suppose 5WD A1L A2 There are projective Aimodules P and an S module isomorphism w P1 A S a P2 gm 8 such that I MP17 Paw Proof of MeyerVietoris X Suppose 5WD A1L A2 There are projective Aimodules P and an S module isomorphism w P1 A S a P2 gm 8 such that I MP17 Paw Then we have Pigl AAigAlz Suppose 5WD A1i7 A2 There are projective Aimodules P and an S module isomorphism w P1 A S a P2 gm 8 such that I MP17 Paw Then we have Pigl AAigAlz Therefore lg MA1A2 w for some S module isomorphism 1A1 A3HA2 AS mmi C I39WIQE Mum IW Proof of MeyerVietoris XI Since A A Sis generated by 1A 2 15 there is some 3 6 8X such that 71A1 15 s1A2 215 Proof of MeyerVietoris XI Since A A Sis generated by 1A 2 15 there is some 3 6 8X such that 71A1 15 s1A2 215 Therefore w e75 and I MA1A2 s M5 83 Proof of MeyerVietoris XI Since A A Sis generated by 1A 2 15 there is some 3 6 8X such that 71A1 15 31A2 215 Therefore o7 cps and I MA1A2 s M5 83 This completes the proof 1 N rmalizations and Cond ctors Let A be the coordinate ring of an affine singular curve Let A1 be its normalization Normalizations and Conductors Let A be the coordinate ring of an affine singular curve Let A1 be its normalization Then A gt A1 and A1 is an Amodule containing A In particular AA1 is an Amodule quotm 4 Mmkt msmri39w vmmmm n m llltl rl f sew Em 39 ns and Conductors Let A be the coordinate ring of an affine singular curve Let A1 be its normalization Then A gt A1 and A1 is an Amodule containing A In particular AA1 is an Amodule The annIhIator in A of the Amodue A1 A is called the conductor of A and denoted c momenta mr ttm uxw Let L A gt A1 be the natural inclusion Then ce c J W39 mwg I Mthme 31ng irigcmmmma39 2 by MM Let L A gt A1 be the natural inclusion Then ce c J Proof By definition cA1 C A W39 mm attituthng Let L A gt A1 be the natural inclusion Then ce c J Proof By definition cA1 C A If f e c6 LC c then there are f e c and g 6 A1 such that f 2 fg Wimpy ai ttmmm Let L A gt A1 be the natural inclusion Then ce c Proof By definition cA1 C A If f e c6 LC c then there are f e c and g 6 A1 such that f 2 fg For each i fig 6 A so fig 6 A m c6 c for each i Wimpy ai ttmmm Let L A gt A1 be the natural inclusion Then ce c J Proof By definition cA1 C A If f e c6 LC c then there are f e c and g 6 A1 such that f 2 fg For each i fig 6 A so fig 6 A m c6 c for each i Hence f e c 1 Wimpy ai ttmmm The Con uctor Square The following is a fiber product diagram ALA1 w w Ac Am c The Conductor Square The following is a fiber product diagram ALA1 r2 r Ac Am c The maps a1 A gt A1 and 32 A2 gt S are the natural inclusions and a2 A a A2 and B1 A1 a S are the natural projections The following is a fiber product diagram ALA1 lw lm Ac Am c The maps a1 A gt A1 and 32 A2 gt S are the natural inclusions and a2 A a A2 and B1 A1 a S are the natural projections This diagram is called the conductor square The Conductor Square Note that 31 is surjective and 041 is injective so this square satisfies the hypotheses for the MeyerVietoris sequence The Conductor Square Note that 31 is surjective and 041 is injective so this square satisfies the hypotheses for the MeyerVietoris sequence Therefore the following is an exact sequence 0 a AX a Amway a A1 cX a PicA a PicA1e PicAc Note that 31 is surjective and 041 is injective so this square satisfies the hypotheses for the MeyerVietoris sequence Therefore the following is an exact sequence 0 a AX a Amway a A1 cX a PicA a PicA1e PicAc This is the tool used to compute Picard groups for singular affine curves mmi Gr if t a 39 miean 6mm Note that 31 is surjective and 041 is injective so this square satisfies the hypotheses for the MeyerVietoris sequence Therefore the following is an exact sequence 0 a AX a A1XeaAcX a A1cX a PicA a PicA1PicAc This is the tool used to compute Picard groups for singular affine curves This tool does not work for nonsingular curves since the affine coordinate rings are their own normalizations Projective Modules over Fiber Products MeyerVietoris and The Conductor Square The Cusp and the Node The Cusp Let k be a field and consider the ring A kttz t3 kiwiy2 x3 Victor Piercey University of Arizona Math 518 Picard Groups of Affine Curves Projective Modules over Fiber Products MeyerVietoris and The Conductor Square Singular Affine Curves The Cusp and the Node The Cusp Let k be a field and consider the ring A kitZ t3 klxyylyz x3 This is a singular curve in Al with a cusp at the origin Victor l Piercey University of Arizona Math 518 Picard Groups of Affine Curves Projective Modules over Fiber Products MeyerVietoris and The Conductor Square Singular Affine Curves The Cusp and the Node The Cusp Let k be a field and consider the ring A kitZ t3 klxyylyz x3 This is a singular curve in Al with a cusp at the origin When k R the picture as follows y Victor l Piercey University of Arizona Math 518 Picard Groups of Affine Curves The Normalization of A I claim the normalization of A is A1 kt The Normalization of A I claim the normalization of A is A1 kt Denote A kWW Since Y2 73 in the field of fractions KA we have my2 y The Normalization of A l I claim the normalization of A is A1 kt Denote A kWW Since Y2 73 in the field of fractions KA we have my2 y Therefore Y is integral over A It follows that A is not integrally closed The Normalization of A II Considerthe ring C k jj The Normalization of A II Considerthe ring C k jj Since y y my C kWY The Normalization of A ll Considerthe ring C k jj Since y y my C kWY Since YY2 7 C kY7 kt which is integrally closed Eb Emil illi39 131ml Considerthe ring C k jj Since Y 7 YY C kWY Since YY2 7 C kY7 kt which is integrally closed Thus C is a minimal integrally closed subring of KA that contains A and therefore is the integral closure of A mmi image aiming IW The Conductor I claim the conductor is c 1213 C A The Con uctor I claim the conductor is c 1213 C A Since71Y A712 c The Con uctor I claim the conductor is c 1213 C A Since71Y A712 e c Moreover WWW feA The Conductor I claim the conductor is c 1213 C A Since71Y A712 e c Moreover WWW feA Therefore Z 1213 C c The Conductor I claim the conductor is c 1213 C A Since71Y A712 e c Moreover WWW feA Therefore Z 1213 C c Since this is a maximal ideal in A and the entire ring cannot be the conductor the claim follows MeyerVietoris fo A It is now immediate that Af kX AcX kX PicA1 O and PicAc O MeyerVietoris for A It is now immediate that Af kX AcX kX PicA1 O and PicAc 0 It remains to compute A1cX MeyerVietoris for A It is now immediate that Af kX AcX kX PicA1 O and PicAc 0 It remains to compute A1cX In A1 M1113 6 12 therefore A1c kititz MeyerVietoris for A It is now immediate that Af kX AcX kX PicA1 O and PicAc 0 It remains to compute A1cX In A1 kt 13 e 12 therefore A1c kt12 Suppose ft a bt e A1cX with inverse c dt Then 1 51 btc dt ac bc dat It is now immediate that Af kX AcX kX PicA1 O and PicAc 0 It remains to compute A1cX In A1 kt 13 e 12 therefore A1c kt12 Suppose ft a bt e A1cX with inverse c dt Then 1 51 btc dt ac bc dat Therefore a c e kX and be ida Wi Group39s azrmrm 2mm Amy e W m Rewrite fas ft a1 bat If we let u ba we have 1 a1 ut Amy e W m Rewrite fas ft a1 bat If we let u ba we have 1 a1 ut If we denote the additive group of the field by lg we have A1cX a1 ut i a e kXu 6 lg Amy e W m Rewrite fas ft a1 bat If we let u ba we have 1 a1 ut If we denote the additive group of the field by lg we have A1cX a1 ut i a e kXu 6 lg Therefore A1cX kX 69h Rewrite fas ft a1 bat If we let u ba we have 1 a1 ut If we denote the additive group of the field by lg we have A1cX a1 ut l a e kXu 6 lg Therefore A1cX kX 69h In particular k embeds into A1cX as 1 ut Wl Grudge azr lmrm 2mm now claim that PicA lg now claim that PicA lg Collecting all the computations together our MeyerVietoris sequence is as follows OHAXHKXXKXHKXGBKgtP10AgtO mm arm illvlg l now claim that PicA lg Collecting all the computations together our MeyerVietoris sequence is as follows OHAXHKXXKXHKXGBKgtP10AgtO Since the conductor square commutes and the units in A1 and Ac are the images of the units in A it follows that both factors of kX have the same image namely the factor of W Wi CHM Wm Cum Fi ll g l now claim that PicA for Collecting all the computations together our MeyerVietoris sequence is as follows OHAXHKXXKXHKXGBKgtP10AgtO Since the conductor square commutes and the units in A1 and Ac are the images of the units in A it follows that both factors of kX have the same image namely the factor of W Therefore PicA 2 kX ea kkx 2 m mmi Gr The Node Now suppose chark 7 2 and consider the ring B kt2 7113 7 t kxyy27X2X 1 The Node Now suppose chark 7 2 and consider the ring B kt2 7113 7 t kxyy27X2X 1 This is a singular curve in Ai with a node at the origin The Node Now suppose chark 7 2 and consider the ring B 7 kttz 71x3 7 r 7 kiwioz 7 W 1 This is a singular curve in Ai with a node at the origin When k R the picture as follows The Normalization of B I claim the normalization of B is B1 kt The Normalization of B I claim the normalization of B is B1 kt Denote B kWW In the field of fractions KB we have W Y 1 The Normalization of B l I claim the normalization of B is B1 kt Denote B kWW In the field of fractions KB we have W Y 1 Therefore YR is integral over B It follows that B is not integrally closed The Normalization of B H Considerthe ring C kY77Y The Normalization of B H Considerthe ring C kY77Y Since Kym C HEY The Normalization of B ll Considerthe ring C kY77Y Since Kym C HEY Since 72 7 1 Y C k7Y kt which is integrally closed Eb arm ill 131ml Considerthe ring C kY77Y Since Kym C HEY Since 72 7 1 Y C k7Y kt which is integrally closed Thus C is a minimal integrally closed subring of KB that contains B and therefore is the integral closure of B ywl 21mg alrmrm IW The Conductor I claim the conductor is c 12 7113 7 t C B The Con uctor I claim the conductor is c 12 7113 7 t C B Sincet7 AYtzi16c The Conductor I claim the conductor is c 12 7113 7 t C B Sincet7 AY 1271 e c Moreover Yt2YY2Y 1YYY 1 e B The Con uctor I claim the conductor is c 12 7113 7 t C B Sincet7 AY 1271 e c Moreover Yt2YY2Y 1YYY 1 e B Therefore Z 12 7 113 7 t C c mm arm illvlg l I claim the conductor is c 12 7113 7 t C B Sincet7 AY 1271 e c Moreover Yt2YY2Y 1YYY 1 e B Therefore Z 12 7 113 7 t C c Since this is a maximal ideal in B and the entire ring cannot be the conductor the claim follows Wl CHM Wm Carma MeyerVietoris for B It is now immediatethat Bf kX BcX kX PicB1 O and PicBc O MeyerVietoris for B It is now immediatethat Bf kX BcX kX PicB1 O and PicBc 0 It remains to compute B1cX MeyerVietoris for B It is now immediatethat Bf kX BcX kX PicB1 O and PicBc 0 It remains to compute B1cX In B1 kt 13 716 12 7 1 therefore B1c ktt2 71 It is now immediatethat Bf kX BcX kX PicB1 O and PicBc 0 It remains to compute B1cX In B1 kt 13 716 12 7 1 therefore B1c ktt2 71 Since I assumed chark 7 2 1 1 and t 7 1 are coprime in B1 hence ktt271 ktt71 t1 ktt71xktt1 kxk wx mung WWW 2mm It is now immediatethat Bf kX BcX kX PicB1 O and PicBc 0 It remains to compute B1cX In B1 kt 13 716 12 7 1 therefore B1c ktt2 71 Since I assumed chark 7 2 1 1 and t 7 1 are coprime in B1 hence ktt271 ktt71 t1 ktt71xktt1 kxk Therefore B1cX kX eakx Wwi1mravs WNW IW now claim that PicB kX PicB N W now claim that PicB kX Collecting all the computations together our MeyerVietoris sequence is as follows OgtBgtlt HKXGBKX HKXGBKX HPiCBgtO mm arm illvlg l ragga now claim that PicB kX Collecting all the computations together our MeyerVietoris sequence is as follows OgtBgtlt HKXGBKX HKXGBKX HPiCBgtO Since the conductor square commutes and the units in B1 and 3 are the images of the units in B it follows that both factors of kX have the same image namely a single factor of W Wm 2mm Wl Grout mm arm illvlg l ragga now claim that PicB kX Collecting all the computations together our MeyerVietoris sequence is as follows OgtBgtlt HKXGBKX HKXGBKX HPiCBgtO Since the conductor square commutes and the units in B1 and 3 are the images of the units in B it follows that both factors of kX have the same image namely a single factor of W Therefore P143 2 kX ea kXkx g m Wl CHM Wm Cum A Geometric Interpretation The normalization of the curve Y SpecB is Y1 z A Speckt and PicY1 0 implies that the only line bundle on Y1 istrivial A Geometric Interpretation The normalization of the curve Y SpecB is Y1 z A Speckt and PicY1 0 implies that the only line bundle on Y1 istrivial The map Y a Y1 can be considered as separating the node into two points p1p2 6 Y1 The normalization of the curve Y SpecB is Y1 z A Speckt and PicY1 0 implies that the only line bundle on Y1 istrivial The map Y a Y1 can be considered as separating the node into two points p1p2 6 Y1 Every line bundle on Y is obtained by some identification of the fibers over p1 and p2 of the trivial line bundle on Y1 The normalization of the curve Y SpecB is Y1 z A Speckt and PicY1 0 implies that the only line bundle on Y1 istrivial The map Y a Y1 can be considered as separating the node into two points p1p2 6 Y1 Every line bundle on Y is obtained by some identification of the fibers over p1 and p2 of the trivial line bundle on Y1 The possible transition functions are given by multiplication by an element a e W hence it makes geometric sense that PicY kX

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