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Degree as a Monoid
Harpreet Bedi
To cite this version:
Harpreet Bedi. Degree as a Monoid. 2020. �hal-02571549�
Degree as a Monoid
1
Harpreet Singh Bedi bedi@alfred.edu
2
May 12, 2020
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Abstract
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In this paper ‘polynomial’ with degree as an ordered monoid say∆are constructed along with corresponding
5
schemes and line bundles O(d), d∈∆. The cohomology of these line bundles is then computed using ˇCech
6
cohomology. The last section of the paper gives new proofs of zero cohomology of affine schemes.
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1 Introduction
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In this paper it is shown that degree can be considered as an ordered monoid (denoted by ∆) for example
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Z≥0,Z[1/p]≥0,Q≥0,R≥0. In section 2 such polynomials (denoted by R[X0, . . . , Xn]∆) are constructed and corre-
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sponding affine schemes are described. The graded rings and Proj construction is given in section 4. The line
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bundles are constructed in section 5 and their cohomology computed in theorems 6.1 and 6.3 given below.
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Theorem(6.1). LetS=R[X0, . . . , Xn]∆andX= ProjS, then for anyn∈∆
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1. There is an isomorphismS' ⊕n∈∆H0(X,OX(n)).
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2. Hn(X,OX(−n−1))is a free module generated by monomials of negative degree.
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Theorem(6.3). LetA=R[X0, . . . , Xn]∆andX= ProjA, m∈∆, thenHi(X,OX(m)) = 0for0< i < n.
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New proofs of vanishing theorems are given in section 7.
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2 Polynomials of degree ∆
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LetBbe a ordered ring with additive identity denoted as0and underlying abelian group(B,+). Since, the ring is
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orderedB≥0forms an ordered monoid with operation+, for exampleZ≥0,Q≥0,R≥0. In this paper monoids of the
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formB≥0 will be considered and denoted by∆.
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2.1 Construction. Let ∆ be a monoid, R a commutative ring and X an indeterminate, then one can define
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polynomials with degree∆as finite sums
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(2.1) X
i∈∆
aiXi where ai∈R and X0:= 1.
The addition is defined as
24
(2.2) f(X) =X
i∈∆
aiXi, g(X) =X
i∈∆
biXi, f(X) +g(X) =X
i∈∆
(ai+bi)Xi.
The multiplication by a monomialcXj, c∈R, j∈∆is given as
25
(2.3) cXjf(X) =cXjX
i∈∆
aiXi =X
i∈∆
caiXi?j wherecai∈Randi ? j∈∆,
where ? is binary operation in the monoid (which is addition by assumption). The multiplication by monomial
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can be extended inductively to multiplication by a polynomial. This ring will be denoted by R[X]∆ and the
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multivariate version will be denoted byR[X0, . . . , Xn]∆. Since,∆ is ordered it is possible to talk about the degree
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of a ‘polynomial’, as the maximum element of∆in the polynomial.
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2.2 Example. The following examples illustrate the construction above.
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1. Z[X]Q≥0 consists of polynomials with coefficients inZand degree inQ≥0 for example1 + 2X−3X1/5.
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2. Z[X]R≥0 consists of polynomials with coefficients inZand degree inR≥0for example1 + 2X−3X1/
√ 5.
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3. Z[X]Z≥0 consists of polynomials with coefficients inZand degree inZ≥0for example1 + 2X−3X5.
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2.1 Zeros of Polynomials
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The ring R[X0, . . . , Xn]∆ is well defined and thus it is possible to define affine schemes and projective schemes
35
directly. In this section zeros of polynomials with degree∆are defined, and the constructions are used for proving
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Nullstellensatz.
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2.3 Definition. LetS be a commutative ring andϕt be a well defined homomorphism given as an evaluation at
38
t∈S (along with the compatibility condition).
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(2.4) ϕt:R[X]∆→S, X7→t∈S
compatibility condition {Xi}i∈∆7→ {ti}i∈∆ The elementt∈Sis called zero of a polynomialf(X)in the ringS iff(t)∈Kerϕt.
40
Notice that√
4 =±2 thus evaluating the polynomial X1/2−2 involves giving a choice (X, X1/2)7→(4,2) or
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(X, X1/2)7→(4,−2), the former map gives zero of the polynomial. The rational degree could give many possible
42
solutions, but that is also true for standard algebraic geometry for example consider infintely many points that
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satisfy the equation of a circleX2+Y2= 1. These evaluation maps are not needed in the general theory of affine
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and projective schemes.
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2.4 Example. The following two examples, illustrate the definition.
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1. Consider the mapZ[X]Z[1/p]≥0→Z, then the evaluation mapX7→n∈Zmakes sense only forn= 0or1,
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sincepth power roots of other elements do not exist in the ring.
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2. The zero ofX
√ 2−5
√
2∈Z[X]R≥0 is5(inRorC), sinceX
√ 2−5
√
2∈Kerϕ5 whereϕ5:Z[X]R≥0→Ror
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ϕ5:Z[X]R≥0→C. Furthermore, the compatibility condition is satisfied since all real powers of5 exist in
50
the ringRorC.
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2.5 Remark. The monoid∆ could as well be a matrix with addition as binary operation. In such a case the user
52
needs to define ringsRandSappropriately. This can be done by carefully considering exponential and logarithm
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maps. For example, for matricesM andN defineMN= exp(NlogM)where
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(2.5) exp(A) =X
n≥0
An
n!, log(I+A) =X
n≥1
(−1)n−1An n
and appropriate assumptions are made for convergence (such as assuming nilpotence, or a bound of one on singular
55
values etc.). This point of view helps make sense of evaluation maps for monomialsaXN as follows
56
(2.6) aXN7→aMN wherea, M, N are Matrices which are ordered in∆.
The above idea can be extended linearly to form polynomials. This gives a ‘new algebraic geometry’ mixing linear
57
algebra and polynomials.
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2.2 Affine Schemes
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LetA:=R[X0, . . . , Xn]∆, sinceAis a ring,SpecAcan be endowed with Zariski topology with closed and open sets
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defined as follows
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(2.7) V(I) :={p∈SpecAsuch thatI⊆p},
D(f) := SpecA\V(f A).
2.6 Construction. Letf ∈Aof degree`∈∆then the elementfn, n∈Z≥0 can be defined as
62
(2.8) fn:=f × · · · ×f
| {z }
ntimes
and degfn= (`+. . .+`)
| {z }
ntimes
.
Hence, {1, f , f2, . . . , fi, . . .}i∈
Z≥0 is a multiplicatively closed, which can be used for localization. Recall that every
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ring is aZmodule, and this is reflected above in the powers.
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The theory of affine schemes can be applied to the ringA. IfMis anAmodule, then for any principal open subset
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D(f)ofSpecA, setM(D(f˜ )) =Mf where Mf is localizationofM with respect to the multiplicatively closed set
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{1, f , f2, . . . , fi, . . .}i∈Z
≥0. This defines a sheaf onSpecAand is called the quasi-coherent sheaf.
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3 Nullstellensatz
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In this section the weak Nullstellensatz is proved by adapting [BSCG09, p 17-18] and [MP07, pp 15]. Letk be an
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uncountable algebraically closed field andA:=k[T1, . . . , Tn]∆ such that the evaluation map
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(3.1) (T1, . . . , Tn)7→(a1, . . . , an), ai∈k
is a well defined homomorphism satisfying definition 2.3.
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3.1 Theorem. LetI(k[T1, . . . , Tn]∆ be a proper ideal withk algebraically closed and uncountable, and∆ countable.
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ThenV(I)defined as
73
V(I) ={y∈kn|f(y) = 0for allf ∈I} is non empty.
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Proof. Using Zorn’s Lemma the ideal I(Ais contained within some maximal ideal mofA. Let B=A/mwhere
75
mis a maximal ideal and thusBis a field. Since, Bis generated overk by the monomials inT1, . . . , Tn and their
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∆powers, the dimension dimkB (as ak vector space) is atmost countable overk. Pickb∈B\k and consider the
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family of elements
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(3.2) 1
a−b
, a∈k withbfixed.
This family is uncountable becausek is uncountable. Since,dimkB is countable, the elements of the family are
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linearly dependent. Hence, there exists a relationship of the form
80
(3.3) λ1
a1−b+. . .+ λj
aj−b = 0for somej∈Z>0
Multiplying throughout byQ
i(ai−b)and settingai =Xgives a polynomialf(X)∈k[X]of degree>0 such that
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f(b) = 0, showing thatb is algebraic overk. Since this holds for any arbitraryb∈B, the fieldBis algebraic over
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k. But,kis algebraically closed, henceB=k.
83
Consider the images of T1, . . . , Tn in the field B =k, given as a1, . . . , an as in 3.1. If P(T1, . . . , Tn)∈ m then
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P(a1, . . . , an) = 0or the point(a1, . . . , an)∈kn is inV(I).
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4 Proj Construction
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4.1 Graded Modules
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Recall that a ring or a module can be graded over a commutative monoid ∆ as in [Bou98, pp 363, Chapter
88
II, §1]. A ring A can be endowed with decomposition A=⊕d≥0Ad of abelian groups such that AdAe ⊆Ad+e
89
for all d, e≥0 where d, e∈∆. Similarly graded Amodules can be defined with AdMe⊆Md+e for all d, e≥0
90
andd, e∈∆.A homogeneous ideal is of the form I =⊕d≥0(I∩Ad) and the quotientA/I has a natural grading
91
(A/I)d =Ad/(I∩Ad). Let ProjAdenote the set of prime ideals of Athat do not contain A+ :=⊕d>0Ad, then
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ProjAcan be endowed with the structure of a scheme. The closed and open sets for homogeneous idealsI are of
93
the form
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(4.1) V+(I) :={p∈Proj Asuch thatI ⊆p}
D+(f) := Proj A\V(f A).
4.2 Localization
95
LetU be a multiplicatively closed set ofR[X]∆, then localization atU gives the ringR[X]∆[U−1]. An important
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example is the multiplicatively closed setU={Xi}i∈
∆, note that1∈U sinceX0:= 1in (2.2).
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4.1 Example. The various avatars of multiplicatively closed setU={Xi}i∈∆in the ringR[X]∆are given as follows
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1. {1, X, X2, . . . , Xi, . . .}i∈
Z≥0 for∆=Z≥0,
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2. {1, X, X2, . . . , Xi, . . .}i∈
Q≥0 for∆=Q≥0,
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3. {1, X, X2, . . . , Xi, . . .}i∈
R≥0 for∆=R≥0.
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4.2.1 Localization of a module
102
LetMbe anAmodule andf ∈A.
103
The localizationMf has a∆grading (or underlying abelian group see remark 4.3 ) where homogeneous elements
104
of degreedare of the formm/fn wherem∈M, f ∈Aare homogeneous and
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(4.2) d= degm−(degf+. . .+ degf)
| {z }
ntimes
, degm,degf ∈∆
wherenis a positive integer and+is the binary operation of the additive monoid. The degree zero elements are
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denoted asM(f)⊂Mf andA(f)⊂Af, furthermore,M(f)is anA(f)module.
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4.2 Example. The polynomial ringA:=R[X0, . . . , Xn]Q≥0 is a gradedRalgebra andAd consists of polynomials
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of degreed∈Q≥0. The localization at the affine plane is given as
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(4.3) A(Xi)=R
"
X0 Xi
!
, . . . , Xn Xi
!#
Q≥0
.
4.3 Remark. The degree d after localization could be <0 due to subtraction as in (4.2), thus it will lie in the
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underlying abelian group of the monoid(B,+).
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5 Twisting Sheaves O (n)
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LetAbe a∆ graded and letn∈∆, define a new gradedAmoduleA(n)d:=An+d for alld∈∆, defineOX(n) :=
113
A(n)e. Theedenotes the localization on the affine open. Thus, on D+(f) an affine open subsetOX(n)|D
+(f)=
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fnOX|D
+(f), furthermore, the usual equality holdsOX(n)⊗O
XOX(m) =OX(n+m).
115
5.1 Example O (1)
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Letkbe a field,∆=Q≥0 and considerProjk[X0, X1]∆, the affine open sets areU0=D(X0) = Spec k[(X1/X0)]∆
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andU1=D(X1) = Spec k[(X0/X1)]∆.
118
For example, consider the section of degree one.
119
(5.1)
Global Section X0+X01/5X14/5+X1
U1 (X0/X1) + (X0/X1)1/5+ 1 U0 1 + (X1/X0)4/5+ (X1/X0)
The transition function from U1 to U0 is given as X1/X0. It is evident that the global sections of O(1) are
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infinitely generated by monomials of the form X0, X1, X0rX1(1−r) where r∈ Q∩(0,1).For example consider the
121
following degree one sections
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(5.2) X0, X1, X01/2X11/2, X01/4X13/4, . . . , X01/2iX11−1/2i, . . .
6 Computing cohomology
123
6.1 Theorem. LetS=R[X0, . . . , Xn]∆andX= ProjS, then for anyn∈∆
124
1. There is an isomorphismS' ⊕n∈∆H0(X,OX(n)).
125
2. Hn(X,OX(−n−1))is a free module generated by monomials of negative degree.
126
Proof. 1. Take the standard cover by affine sets U={Ui}i where each Ui =D(Xi), i = 0, . . . , n. The global
127
sections are given as the kernel of the following map
128
(6.1) Y
SXi
0
−→Y SXi
0Xi1
The element mapping to the Kernel has to lie in all the intersectionsS=∩iSXi, as given on [Har77, pp 118]
129
and is thus the ringS itself.
130
2. Hn(X,OX(−m))is the cokernel of the map
131
(6.2) dn−1:Y
k
SX0···Xˆk···Xn−→SX0···Xn
SX0···Xn is a freeRmodule with basisX0l0· · ·Xnln with eachli∈B(whereBis the underlying group of∆). The
132
image ofdn−1is the free submodule generated by those basis elements with atleast oneli ≥0(that is atleast
133
oneli∈∆). ThusHn is the free module with basis as negative monomials
134
(6.3) {X0l0· · ·Xnln}such thatli<0
135
6.2 Example. LetA=R[X0, . . . , Xn]Q≥0 in the theorem 6.1. In the second part the grading is given byP li and
136
there are infinitely many monomials with degree−n−whereis something very small and∈Q≥0. Recall, that
137
in the standard coherent cohomology there is only one such monomialX0−1· · ·Xn−1 . For example, in case ofP2
138
we haveX0−1X1−1X2−1 but here we also haveX0−1/2X1−1/2X2−2.
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Recall that in coherent cohomology of Pn the dual basis of X0m0· · ·Xnmn is given by X0−m0−1· · ·Xn−mn−1 and the
140
operation of multiplication gives pairing. We do not have this pairing here, but we can pairX0m0 withX0−m0.
141
6.1 Zero cohomology
142
6.3 Theorem. LetA=R[X0, . . . , Xn]∆andX= ProjA, m∈∆, thenHi(X,OX(m)) = 0for0< i < n.
143
The proof is adapted from from [Vak17, pp 474-475]. We will work withn= 2for the sake of clarity, the case for
144
generalnis identical. Another proof for allnis given in section 7.
145
Proof. Start by writing down the ˇCech complex forn= 2.
146
U0 U0∩U1
⊕ ⊕
0 P2 U1 U0∩U2 U0∩U1∩U2 0
⊕ ⊕
U2 U1∩U2 ΓX0 ΓX0,X1
⊕ ⊕
0 Γ ΓX1 ΓX0,X2 ΓX0,X1,X2 0
⊕ ⊕
ΓX2 ΓX1,X2
Figure 1: ˇCech Complex forn= 2
The cohomology groups are zero if it can be shown that the image is equal to kernel. SinceΓX0X1X2 is localization
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of global sections its elements are fractions of the forma/b∈ΓX0X1X2, wherea∈Γ and there are four possibilities
148
forbwhich is formed by monomialsX0a0X1a1X2a2.
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1. All threeX0, X1, X2are inb, that is there are three negative exponents.
150
2. Only two ofX0, X1, X2are inb, that is there are two negative exponents.
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3. Only one ofX0, X1, X2 is inb, that is there is one negative exponent.
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4. None ofX0, X1, X2is inb(orb= 1), that is zero negative exponent.
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3negative exponents The monomialX0a0·X1a1·X2a2 whereai <0. We cannot lift it to any of the coboundaries
154
(that is lift only to0coefficients). IfK012denotes the coefficient of the monomial in the complex (Figure 2),
155
we get zero cohomology except for the spot corresponding toU0∩U1∩U2(as in theorem 6.1).
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0 0
⊕ ⊕
0 0 0 0 K012 0
⊕ ⊕
0 0
Figure 2: 3negative exponents
2negative exponents The monomialX0a0·X1a1·X2a2 where two exponents are negative, saya0, a1<0. Then we
157
can perfectly lift to coboundary coming fromU0∩U1, which gives exactness.
158
0 K01
⊕ ⊕
0 0 0 0 K012 0
⊕ ⊕
0 0
Figure 3: 2negative exponents
1negative exponent The monomial X0a0·X1a1·X2a2 where one exponents is negative, say a0 < 0, we get the
159
complex (Figure 4). Notice thatK0maps injectively giving zero cohomology group.
160
K0 K01
⊕ ⊕
0 0 0 K02 K012 0
⊕ ⊕
0 0
Figure 4: 1negative exponent
Furthermore, the mapping in the Figure 5 gives Kernel whenf =g which is possible for zero only. Again
161
giving us zero cohomology groups.
162
K0 f
⊕ ⊕
0 0 0 g f −g 0
⊕ ⊕
0 0
Figure 5: Mapping for1negative exponent
0negative exponent The monomial X0a0·X1a1·X2a2 where none of the exponents is negative ai >0, gives the
163
complex Figure 6.
164
K0 K01
⊕ ⊕
0 KH0 K1 K02 K012 0
⊕ ⊕
K2 K12
Figure 6: 0negative exponent
Recall that SES of cochain complexesA•→B•→C• gives a LES and if the cohomology groups ofA•and
165
C• are zero, then cohomology groups of B• are also zero. Consider the SES of complex as in Figure 7 .
166
The top and bottom row come from the1negative exponent case, thus giving zero cohomology. The SES
167
of complex gives LES of cohomology groups, since top and bottom row have zero cohomology, so does the
168
middle.
169
A• 0 0 K2 K02⊕K12 K012 0
B• 0 KH0 K0⊕K1⊕K2 K01⊕K02⊕K12 K012 0
C• 0 KH0 K0⊕K1 K01 0 0
Figure 7: SES of Complex
170
6.2 Kunneth Formula
171
We can produce a complex forPn×Pm by taking tensor product of the corresponding ˇCech complex associated
172
with each space, and by the Theorem of Eilenberg-Zilber we get
173
(6.4) Hi(Pn×Pm,O(a, b)) = Xi j=0
Hj(Pn,O(a))⊗Hi−j(Pm,O(b)) a, b∈∆ Furthermore, we can define a cup product following [Liu02, pp 194] to get a homomorphism
174
(6.5) ^:Hp(Pn,O(a))×Hq(Pm,O(b))→Hp+q(Pn×Pm,O(a, b)) a, b∈∆
6.3 Derived functor and ˇCech cohomology
175
Recall the following theorem (see for example, [Vak17, pp 637, 23.5.1 ])
176
Theorem. IfF is a quasicoherent sheaf on a quasicompact separated schemeX, then the ˇCech cohomology agrees with
177
derived functor cohomology.
178
6.4 Proposition. LetS=R[X0, . . . , Xn]∆ setX= Proj S, andF be the sheaf associated with ˇCech complex, then the
179
ˇCech cohomology agrees with derived functor cohomology.
180
Proof. Using the theorem above it needs to be shown that projective space associated with S is quasicompact,
181
quasicoherent and separated.
182
quasicompact : The projective space is covered by affine opens, and each affine open is quasicompact, sinceSpec
183
of a ring is always quasicompact.
184
quasicoherence : The localization definition of quasicoherence is satisfied. At each affine openUi :={Xi}∆ the
185
sheafF (Ui)is obtained via localization atXi.
186
separated : This follows from Proposition 3.6 [Liu02, pp 100] or a direct proof in Proposition 10.1.5 [Vak17, pp
187
281]. The open setsUi×Uj⊂Pn×Pn are of the form
188
(6.6) Ui×Uj= SpecR
"
X0 Xi
, . . . ,Xn Xi
,Y0 Yj
, . . . ,Yn Yj
#
∆
, Xi Xi
−1,Yi Yi
−1
! .
The diagonal morphism corresponds to the mapUi∩Uj→Ui×Ujleading to a surjective mapping of rings
189
(6.7)
R
"
X0 Xi, . . . ,Xn
Xi,Y0 Yj, . . . ,Yn
Yj
#
∆
→R
"
X0 Xi, . . . ,Xn
Xi,Xj Xi
#
∆
Xk Xi
7→ Xk
Xi and Yk
Yj
7→Xk Xi
Xi Xj, which implies separatedness.
190
191
7 Vanishing Cohomology
192
In this section new proofs of vanishing cohomology of affine and projective schemes are given, without using any
193
computations. These proofs use ideas from [Vak17, Chapter 18] and [Sta20, tag01X8] but elimate all computation,
194
and hence, become ideal proofs to present in a classroom.
195
7.1 Remark. Recall that a short exact sequence of complexes0→ A → B → C →0leads to a long exact sequence
196
of cohomology groups.
197
(7.1)
. . . Hi(A) Hi(B) Hi(C) Hi+1(A) Hi+1(B) Hi+1(C)
. . .
δi−1
δi
If the complexAis same asCbut shifted by one, thenHi(C) =Hi+1(A)which immediately implies thatHi(B) = 0
198
for alli >0.
199
7.2 Theorem. LetX= SpecAbe an affine scheme with finite covering by principal open subsetsU =Sn
i=1D(fi)and
200
F a quasicoherent sheaf onX. ThenHp(U,F) = 0forp >0.
201
Proof. LetX be a scheme with open cover U=Sn
i=1D(fi)wheref1, . . . , fn generate the global sections. In other
202
words ifX= SpecA, thenf1, . . . , fn generateA. It is always possible to find a cover ofSpecAsince it is known to
203
be quasi compact for any ringA. Recall ˇCech complex
204
Cp(U,F) = Y
(i0,...,ip)∈Ip+1
F(Ui0...ip) = Y
(i0,...,ip)∈Ip+1
Mfi
0...fip whereM is someAmodule.
The problem is to show that the complex below is exact
205
(7.2) 0→M→Y
i0
Mfi
0
→Y
i0,i1
Mfi
0fi1→ Y
i0,i1,i2
Mfi
0fi1fi2→. . .
Since, localization is an exact functor it is equivalent to check if the complex is exact after localization at a prime
206
idealp.
207
(7.3) 0→Mp→Y
i0
Mfi0p→Y
i0,i1
Mfi0fi1p→ Y
i0,i1,i2
Mfi0fi1fi2p→. . .
Sincef1, . . . , fn generateA, there isfj<por fj lies in the complement of the prime ideal. Since localization at a
208
prime ideal is on the multiplicatively closed set which is a complement of the prime ideal this impliesMfj,p=Mp.
209
It is advisable to see example 7.3 now, since the rest of the proof is simply generalising it.
210
Consider the projection mapQ Mfi
0...fipp→Q
index,jMfi
0...fipp which gives rise to a short exact sequence
211
(7.4) 0→ Y
index=j
Mfi
0...fipp→Y Mfi
0...fipp→ Y
index,j
Mfi
0...fipp→0.
Note thatQ
index=jMfi
0=Mfjp=Mp(consists of only one indexj).The above SES now fits into the complex
212
(7.5)
A 0 Mp Q
index=jMfi
0fi1p Q
index=j
QMfi
0fi1fi2p . . .
B Mp Q
Mfi
0p Q
Mfi
0fi1p Q
Mfi
0fi1fi2p . . .
C Mp Q
index,jMfi
0p Q
index,jMfi
0fi1p Q
index,j
QMfi
0fi1fi2p . . . Notice that the first and the third row are the same, there is just a shifting by 1 in the first row and thus the remark
213
7.1 can be applied to give the result.
214
215
7.3 Example. For example let three elementsf1, f2, f3generateAwithf1<pthen we have
216
(7.6) Mf1p=Mp, Mf1f2p=Mf2p, Mf1f3p=Mf3p and Mf1f2f3p=Mf2f3p
In the diagram below the second row is the ˇCech complex associate with the elements f1, f2, f3, the third row
217
consists of only those elements from second row which do not have f1. For example in the second row fourth
218
column there is an elementMf1f2p⊕Mf2f3p⊕Mf1f3p which projects to Mf2f3p in the third row. The first row is
219
simply the kernel of the projection map and these elements are rewritten according to (7.6). For example, looking
220
at the fourth column again there is a SES
221
(7.7) 0→(Mf2p⊕Mf3p) = (Mf1f2p⊕Mf1f3p)→Mf1f2p⊕Mf2f3p⊕Mf1f3p→Mf2f3p→0.
(7.8)
A 0 Mp Mf2p⊕Mf3p Mf2f3p
B Mp Mf1p⊕Mf2p⊕Mf3p Mf1f2p⊕Mf2f3p⊕Mf1f3p Mf1f2f3p=Mf2f3p
C Mp Mf2p⊕Mf3p Mf2f3p 0
Notice that the first and the third row are the same, there is just a shifting by 1 in the first row and thus the remark
222
7.1 can be applied to give the result.
223
The following example illustrates the zero cohomology groups associated with the projective space.
224
7.4 Example. The augumented ˇCech complex of a given degree associated with the projective space say P2
225
covered by∪n
i=1D(Xi)is given as
226
(7.9) 0→Γ →ΓX1⊕ΓX2⊕ΓX3→ΓX1X2⊕ΓX2X3⊕ΓX1X3→ΓX1X2X3→0.
In order to choose a primeXi<pfor somei= 1,2or3, we have to drop the termΓX1X2X3 (which contains all the
227
Xi and our prime should work for all degrees). The cohomology of the dropped term is computed separately. Let
228
X1<pwe also have to drop the localized avatarΓX1X2X3p=ΓX2X3p. In other words localization atpgives
229
(7.10) 0→Γp→ΓX1p⊕ΓX2p⊕ΓX3p→ΓX1X2p⊕ΓX1X3p→0,
which we want to show is exact. Copying example 7.3 gives the SES of cochain complexes below, and hence the
230
desired result of zero cohomology groups.
231
(7.11)
A 0 Γp ΓX2p⊕ΓX3p 0
B Γp ΓX1p⊕ΓX2p⊕ΓX3p ΓX1X2p⊕ΓX1X3p 0
C Γp ΓX2p⊕ΓX3p 0 0
7.5 Theorem. LetA=R[X0, . . . , Xn]∆andX= ProjA, m∈∆, thenHi(X,OX(m)) = 0for0< i < n.
232
Proof. Let Pn be covered by open sets of the formSn
i=0D(Xi). If Γ denotes the graded global sections, that is
233
Γ =⊕m∈∆Am, then the corresponding ˇCech complex is given as
234
(7.12) 0→Y
i0
ΓXi
0
→Y
i0<i1
ΓXi
0Xi1→ Y
i0<i1<i2
ΓXi
0Xi1Xi2 →. . .→ΓX0···Xn
We want to localize the above complex with respect to a primepsuch that there is atleast someXj <p. Thus we
235
will be forced to drop the termΓX0···Xn since it contains all theXi of all degrees. Furthermore, we will also have to
236
drop any associated localizations. Thus, the localization at the end part is to be computed separately. The ˇCech
237
complex is also augumented at the start to make the cohomology zero.
238
We can now localize at the primeXj<pand follow the process outlined in the proof of theorem 7.2 to get a SES
239
of cochain complexes below which then implies the result.
240
(7.13)
A 0 Γp Q
index=jΓXi
0Xi1p . . .
B Γp Q
ΓXi
0p Q
ΓXi
0Xi1p . . .
C Γp Q
index,jΓXi
0p Q
index,jΓXi
0Xi1p . . .
241
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242
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[BSCG09] M. Beltrametti, F.J. Sullivan, Ettore Carletti, and D. Gallarati. Lectures on Curves, Surfaces and Projective
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Varieties: A Classical View of Algebraic Geometry. EMS textbooks in mathematics. European Mathematical
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[Har77] R. Hartshorne. Algebraic Geometry. Encyclopaedia of mathematical sciences. Springer, 1977. (Cited on
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