Sheaf Cohomology – II. Sheaf Cohomology

Thanks to Peter Xu, Linus Hamann, Jack Sempliner, Irit Huq-Kuruvilla for helpful (if somewhat orthogonal) discussion related to the subject material, and to Sean Reynolds for showing me how to make the (rather complicated, at least to me) commutative diagram that shows up in this post.

Oh, and Anja Schulz gets a shout out as well!

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When we discuss the construction of a meromorphic function on a Riemann surface with a given principal part at a given point {P}, we have to find holomorphic functions {g_\alpha} on {U_\alpha} such that

\displaystyle f_{\alpha\beta} = g_\beta - g_\alpha

on {U_\alpha \cap U_\beta}. One can formulate this in a general setting by introducing the concept of sheaf cohomology. Suppose {M} is a topological space and {\mathcal{S}} ia sheaf of abelian groups over {M}. Let

\displaystyle \mathcal{U} = \{U_\alpha\}

be a covering of {M} by open subsets. For every nonnegative integer {q}, we define {C^q(\mathcal{U}, \mathcal{S})} as follows. An element of {C^q(\mathcal{U}, \mathcal{S})} is

\displaystyle f = \{f_{\alpha_0\dots\alpha_q}\},

where {f_{\alpha_0\dots\alpha_q}} is a continuous section of {\mathcal{S}} over {U_{\alpha_0} \cap \dots \cap U_{\alpha_q}} and {f_{\alpha_0\dots\alpha_q}} is skew-symmetric in {\alpha_0, \dots, \alpha_q}. We call {C^q(\mathcal{U},\mathcal{S})} the group of alternating {q}-cochains for the covering {\mathcal{U}} with coefficients in the sheaf {\mathcal{S}}. An element of {C^q(\mathcal{U}, \mathcal{S})} is an alternating {q}-cochain for the covering {\mathcal{U}} with coefficients in the sheaf {\mathcal{S}} or simply a {q}-cochain when there is no confusion. For notational convenience, we define {C^q(\mathcal{U}, \mathcal{S})} to be the zero group when {q = -1}.

We define a map

\displaystyle \delta: C^q(\mathcal{U}, \mathcal{S}) \rightarrow C^{q+1}(\mathcal{U}, \mathcal{S})

as follows. The image of

\displaystyle f = \{f_{\alpha_0 \dots \alpha_q}\}

under {\delta} is

\displaystyle g = \{g_{\alpha_0} \dots \alpha_{q+1}\},

where

\displaystyle g_{\alpha_0\dots\alpha_{q+1}} = \sum_{\nu = 0}^{q+1} (-1)^\nu f_{\alpha_0 \dots \widehat{\alpha}_\nu \dots \alpha_{q+1}}

on {U_{\alpha_0} \cap \dots \cap U_{\alpha_{q+1}}} and {\widehat{\alpha}_\nu} means that the index {\alpha_\nu} is omitted. We call the map

\displaystyle \delta: C^q(\mathcal{U}, \mathcal{S}) \rightarrow C^{q+1}(\mathcal{U}, \mathcal{S})

the coboundary map. Denote by {Z^q(\mathcal{U}, \mathcal{S})} the kernel of

\displaystyle \delta: C^q(\mathcal{U}, \mathcal{S}) \rightarrow C^{q+1}(\mathcal{U}, \mathcal{S}),

and denote by {B^q(\mathcal{U}, \mathcal{S})} the image of

\displaystyle \delta: C^{q-1}(\mathcal{U}, \mathcal{S}) \rightarrow C^q(\mathcal{U}, \mathcal{S}).

The group {Z^q(\mathcal{U}, \mathcal{S})} (respectively {B^q(\mathcal{U}, \mathcal{S})}) is respectively called the group of alternating {q}-cocycles (respectively {q}-coboundaries) for the covering {\mathcal{U}} with coefficients in the sheaf {\mathcal{S}}.

The composite of

\displaystyle \delta: C^{q-1}(\mathcal{U}, \mathcal{S}) \rightarrow C^q(\mathcal{U}, \mathcal{S}),\text{ }\delta: C^q(\mathcal{U}, \mathcal{S}) \rightarrow C^{q+1}(\mathcal{U}, \mathcal{S})

is zero, because if the image of the element

\displaystyle h = \{h_{\alpha_0 \dots \alpha_{q-1}}\}

of {C^{q-1}(\mathcal{U}, \mathcal{S})} under

\displaystyle \delta: C^{q-1}(\mathcal{U}, \mathcal{S}) \rightarrow C^q(\mathcal{U}, \mathcal{S})

is

\displaystyle f = \{f_{\alpha_0 \dots \alpha_q}\},

then

\displaystyle g_{\alpha_0 \dots \alpha_{q+1}} = \sum_{\nu = 0}^{q+1} (-1)^\nu f_{\alpha_0 \dots \widehat{\alpha}_\nu \dots \alpha_{q+1}}

\displaystyle = \sum_{\nu = 0}^{q+1} (-1)^\nu \left( \sum_{\mu < \nu} (-1)^\mu h_{\alpha_0 \dots \widehat{\alpha}_\mu \dots \widehat{\alpha}_\nu \dots \alpha_{q+1}} + \sum_{\mu > \nu} (-1)^{\mu - 1} h_{\alpha_0 \dots \widehat{\alpha}_\mu \dots \widehat{\alpha}_\nu \dots \alpha_{q+1}}\right)

\displaystyle = \sum_{0 \le \mu < \nu \le q+1} (-1)^{\mu + \nu} h_{\alpha_0 \dots \widehat{\alpha}_\mu \dots \widehat{\alpha}_\nu \dots \alpha{q+1}} + \sum_{0 \le \nu < \mu \le q + 1} (-1)^{\mu + \nu - 1} h_{\alpha_0 \dots \widehat{\alpha}_\mu \dots \widehat{\alpha}_\nu \dots \alpha_{q+1}}

\displaystyle =0.

Hence, {B^q(\mathcal{U}, \mathcal{S})} is contained in {Z^q(\mathcal{U}, \mathcal{S})}. Denote by {H^q(\mathcal{U}, \mathcal{S})} the quotient {Z^q(\mathcal{U}, \mathcal{S})/B^q(\mathcal{U}, \mathcal{S})}. The group {H^q(\mathcal{U}, \mathcal{S})} is called the cohomology group of dimension {q} for the covering {\mathcal{U}} with coefficients in the sheaf {\mathcal{S}}. It is clear from the definition that {H^0(\mathcal{U}, \mathcal{S})} is simply the set of all global continuous sections of the sheaf {\mathcal{S}} over {M} and is usually denoted also by {\Gamma(M, \mathcal{S})}.

In our construction of a meromorphic function on a Riemann surface with a given principal part at a given point {P}, the collection {\{f_{\alpha\beta}\}} is a {1}-cocycle with coefficients in the sheaf of germs of holomorphic functions on {M}, and the existence of {\{g_\alpha\}} is equivalent to {\{f_{\alpha\beta}\}} being a {1}-cboundary. In this formulation, the obstruction to the solution of the problem is the cohomology group of dimension {1}. When we try to piece together local meromorphic functions to form a global meromorphic function, we want to get rid of the discrepancies of the local meromorphic functions, and it would serve the same purpose if we can get rid of the discrepancies by going to a refinement of the covering. The problem is solved if the cohomology group of dimension {1} vanishes when one goes to a refinement of the covering. This suggests that one should take all coverings of the space and consider the direct limit of the cohomology groups for all the coverings.

In the general case when we have a refinement {\mathcal{V}} of the covering {\mathcal{U}}, we have

\displaystyle C^q(\mathcal{U}, \mathcal{S}) \rightarrow C^q(\mathcal{V}, \mathcal{S}),\text{ }Z^q(\mathcal{U}, \mathcal{S}) \rightarrow Z^q(\mathcal{V}, \mathcal{S}),

\displaystyle B^q(\mathcal{U}, \mathcal{S}) \rightarrow B^q(\mathcal{V}, \mathcal{S}),\text{ }H^q(\mathcal{U}, \mathcal{S}) \rightarrow H^q(\mathcal{V}, \mathcal{S}).

We define {H^q(M, \mathcal{S})} as the direct limit of {H^q(\mathcal{U}, \mathcal{S})} as {\mathcal{U}} runs through the directed set of all open coverings of {M}. The group {H^q(M, \mathcal{S})} is called the cohomology group of dimension {q} of {M} with coefficients in the sheaf {\mathcal{S}}. We denote respectively by {C^q(M, \mathcal{S})}, {Z^q(M, \mathcal{S})}, and {B^q(M, \mathcal{S})} the direct limits of {C^q(\mathcal{U}, \mathcal{S})}, {Z^q(\mathcal{U}, \mathcal{S})}, and {B^q(\mathcal{U}, \mathcal{S})}. Then we have an induced coboundary map

\displaystyle \delta: C^q(M, \mathcal{S}) \rightarrow C^{q+1}(M, \mathcal{S})

whose kernel is {Z^q(M, \mathcal{S})} and whose image is {B^{q+1}(M, \mathcal{S})}. Moreover, {H^q(M, \mathcal{S})} is the quotient of {Z^q(M, \mathcal{S})} by {B^q(M, \mathcal{S})}.

Among all cohomology groups of positive dimension, the most important cohomology group is the one of dimension {1}. Its vanishing enables one to piece together local continuous sections of a sheaf to form a global continuous section. Why are the cohomology groups of higher dimensions introduced? They are introduced the help us compute the cohomology group of dimension {1}, because when we have a short exact sequence of three sheaves, we have a long exact sequece of cohomology groups. We are going to discuss this long exact sequence of cohomology groups.

Suppose

\displaystyle 0 \rightarrow \mathcal{S}' \rightarrow \mathcal{S} \rightarrow \mathcal{S}'' \rightarrow 0

is an exact sequence of sheaves and sheaf-homomorphisms. For every {q}, we have the short exact sequence

\displaystyle 0 \rightarrow C^q(M, \mathcal{S}') \rightarrow C^q(M, \mathcal{S}) \rightarrow C^q(M, \mathcal{S}'') \rightarrow 0.

This is clear except the surjectivity of {C^q(M, \mathcal{S}) \rightarrow C^q(M, \mathcal{S}'')}. Suppose we have an element

\displaystyle f'' = \{f_{\alpha_0 \dots \alpha_q}\}

of {C^q(\mathcal{U}, \mathcal{S}'')}. We have to show that its restriction to some refinement {\mathcal{W}} of {\mathcal{U}} is the image of some element {C^q(\mathcal{W}, \mathcal{S})}. First, let us make a trivial observation. Suppose we have a continuous section {g''} of {\mathcal{S}''} over an open subset {G} of {M}. We give {M} a metric {d(\,\cdot\,,\, \cdot\,)}. For every point {P} in {G}, there exists a maximum positive number

\displaystyle \eta = \eta(g'', P)

such that the ball {B_\eta(P)} of radius {\eta} centered at {P} is contained in {G}, and for every {r < \eta}, the restriction {g''|B_r(P)} of {g''} to {B_r(P)} is the image of some continuous section {\mathcal{S}} over {B_r(P)}. The function {\eta(g'', P)} is clearly a lower semi-continuous function of {P} so that if

\displaystyle \eta(g'', P_0) > \epsilon,

then

\displaystyle \eta(g'', P) > \epsilon

for all {P} in some open neighborhood of {P_0}. We can assume, without loss of generality, that the covering {\mathcal{U}} is locally finite. We choose {U_\alpha'} relatively compact in {U_\alpha} so that

\displaystyle \mathcal{U}' = \{U_\alpha'\}

still covers {M}. For every point {P} in {M}, we let {\epsilon(P)} be the minimum of {\eta(f_{\alpha_0\dots\alpha_q}, P)} for all {U_{\alpha_0}'} containing {P}. Clearly, every {P} admits an open neighborhood on which the function {\epsilon(\,\cdot\,)} has a positive lower bound. Now, for every point {P} in {M}, let {W_P} be an open metric ball centered at {P} of radius {r(P)} contained in {U_\alpha'} for some

\displaystyle \alpha = \alpha(P)

so that the function

\displaystyle \epsilon(\,\cdot\,)is > 2r(P)

on {W_P}. Let

\displaystyle \mathcal{W} = \{W_P\}_{P \in M}.

Then {\mathcal{W}} is a refinement of {\mathcal{U}'}. Suppose {W_{P_0}, \dots, W_{P_q}} have a common point {P}. We can choose a number {r} such that

\displaystyle 2r(P_\nu) < r < \epsilon(P)

for {0 \le \nu \le q}. Let {B} be the metric ball centered at {P} whose radius is {r}. Then {B} contains {W_{P_\nu}} for {0 \le \nu \le q}, and {B} is contained in {U_{\alpha(P_0)} \cap \dots \cap U_{\alpha(P_q)}}. Moreover, {f'_{P_0\dots P_q}} be the restriction of {g} to {W_{P_0} \cap \dots \cap W_{P_q}}. We skew-symmetrize {f'_{P_0 \dots P_q}} with respect ot {P_0, \dots, P_q}. Then the restriction of {f''} to {\mathcal{W}} is the image of the element {\{f'_{P_0 \dots P_q}\}} of {C^q(\mathcal{W}, \mathcal{S})}.

From the commutative diagram with exact rows

we get a long exact sequence

\displaystyle 0 \rightarrow \Gamma(M, \mathcal{S}') \rightarrow \Gamma(M, \mathcal{S}) \rightarrow \Gamma(M, \mathcal{S}'') \rightarrow H^1(M, \mathcal{S}') \rightarrow \dots \rightarrow

\displaystyle H^q(M, \mathcal{S}') \rightarrow H^q(M, \mathcal{S}) \rightarrow H^q(M, \mathcal{S}'') \overset{\varphi}{\rightarrow} H^{q+1}(M, \mathcal{S}') \rightarrow \dots.

The only map in the long exact sequence that needs some explanation is the so-called connecting homomorphism {\varphi}. It is defined as follows. Take an element {f''} of {Z^q(M, \mathcal{S}'')}. We can find an element {f} of {C^q(M, \mathcal{S})} whose image under {\mathcal{S} \rightarrow \mathcal{S}''} is {f''}. Let {g \in C^{q+1}(M, \mathcal{S})} be the image of {f} under the coboundary map {C^q(M, \mathcal{S}) \rightarrow C^{q+1}(M, \mathcal{S})}. From the above commutative diagram, it follows that {g} is the image of some element under {\varphi} of the element of {H^q(M, \mathcal{S}'')} defined by {f''}. The exactness of the sequence is a consequence of straightforward diagram chasing.