cech cohomology of p1

October 9, 2020

Motivation for classifying vector bundles.

$$0\rightarrow \mathcal{F}\rightarrow \widetilde{S^0}\rightarrow \widetilde{S^1}\rightarrow\cdots$$

On triply overlaping patches you need a compatibility condition (cocycle condition) in order to make things work. . $(x_i) \in \prod \mathcal{F}(U_i)$ being in the kernel of the first morphism means that the difference of the two restriction maps is zero, i.e. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Moreover, if our family was compatible, this is the zero family, by definition of compatible! One way to think about $H^1(A)$ is to use the long exact sequence not as a property of cohomology, but outright as a definition. That way you get the Cech sheaf cohomology idea quite naturally. It only takes a minute to sign up. Princeton: University Press, XIV, 328 p. (1952). Likewise if we want to know how many 2-families are there which were not already there as 1-families (i.e.

There is some proof by using a double cochain complex here.

This whole short exact sequence can be twisted by $(-1)$, noting that twisting a skyscraper sheaf $\mathbb{C}_p$ gives an isomorphic sheaf $\mathbb{C}_p(-1)$ (which I identify with the original sheaf): $$0\rightarrow \mathcal{C}^0(X)\rightarrow \mathcal{C}^1(X)\rightarrow $$ Then there is a surjective map (it is surjective because it is surjective on stalks): A proof without the use of double complexes that Cech (co)homology for triangulable spaces (with respect to an arbitrary abelian group) coincides with the simplicial (co)homology (and so with the singular one) can be found in Chapter IX of. On p. 334 he proves that Cech and Alexander-Spanier cohomology coincide, on p. 340 he proves that Alexander-Spanier and singular cohomology coincide. if we want to determine $2Fam - 1Fam$), it seems a good idea to display 2-families as matching 3-families because the 1-families will be killed in the translation process. So actually we want are talking about $2Fam - (1Fam - 0Fam)$. But what Can someone give an outline of how this proof goes? What is the Zariski topology good/bad for? . CAPPRODUCTAND“CECH” COHOMOLOGY 93 \: Hp(X) H n(X) !H np (X): Noticehowthedimensionswork. What about quadruple intersections? This is $H^1$. Of course, I'd be only too happy to see a more intuitive explanation, and overjoyed to see a geometric one. It only takes a minute to sign up. (I am assuming it is the Algebraic topology book by Spanier). $\alpha:\{\Delta_n\rightarrow U\}\rightarrow A$. Dan Petersen says here that it is in Spanier's book. .

A sheaf $\mathcal{F}$ on a topological space $X$ is the same as a local homeomorphism into $X$: Starting with the sheaf you take the disjoint union of the stalks and put on it the topology generated by the sets $[U,f]:=$ {$f_x \in \mathcal{F}_x \mid x \in U$} where $U$ ranges over all open sets of $X$, $f \in \mathcal{F}(U)$ and $f_x$ denotes the image of $f$ in the stalk. That is, given an exact sequence of sheaves, Notes on Sheaf Cohomology Contents 1 Grothendieck Abelian Categories 1 1.1 The size of an object . On the other hand you can get back a sheaf from a local homeomorphism $E \rightarrow X$ by taking as the value on the open set $U$ the sections $U \rightarrow E$ of the given map. Taking global section functor we have What do you mean by "dimension shift"? Which can be done straight from the definition. ��d�K@��W�~a 0�A�[c�0�Ok��a�P��MwhGa��Z��8��B�v��'��t��=�M�V��?�اI��z�fFO�e_�IQ��cSM�x��Rw��O-wq��|�/�j3�]��y��o�]��Ԥc�I��>�4��>��ϙ��ɻ?��M��r��>��Ep�wf+�k�/�>��c%�qd�P'w�b�)�;Ԛ;$��M��=uM�{e��C��cT��A��]3�ڋ��V�H�'��_q�\6'��c��-�T��v�~��_�^��8��5ܶpq�?Ns�E��V��CS�b��e�]�+��5]4��p�漹��A��X��&��vS�*c��+�پ�XX��z H��GG��Uj1p�xD��j�~'��U��V��u�4�"�4�C�o:��ѳk��d��:��I�HL��7)��s��ó�U�٨ ��Iӵ;Q=��Q�U�ɼr�t�\e��[�oU��Woք�a#�5k��ns��>��a@��a3�c��(��W�5�6�ۿ��^�u _Sq�� s�$x�6��1�e�q@��v�0��(�U MathOverflow is a question and answer site for professional mathematicians. This is an abelian group. On global sections, we then get h�d�� Nice question. Hi, I have to calculate by the definition the first cech group of cohomology the projective line P1 respect the standard covering and the hyperplane bundle , O(1). But how can we characterize the families $x_{ij} \in \prod \mathcal{F}(U_i \cap U_j)$ which come from families on the $U_i$ ? (I haven’t tried this, and have no idea how hard it is.) Making statements based on opinion; back them up with references or personal experience. From a rather reductionist but hands-on perspective, if cohomology of coherent sheaves allows one to recover so many key discrete invariants in geometry, as dimensions of some things that are not a priori dimensions of vector spaces of sections but seem like perfectly good extensions of the idea of Betti number into other interesting fields, I wonder quite where the problem is. . MathJax reference. It is in Spanier's book, but you have to do two steps. Cech Cohomology inˇ Noncommutative Geometry∗ Do Ngoc Diep April 20, 2019 Abstract The Z2-graded Cech cohomology theory is considered in the frame-ˇ work of noncommutative geometry over complex number ﬁeld and in particular the homotopy invariance and Morita invariance are proven. Hence, $H^1(\mathcal{O}(-2))$ must contain at least $\mathbb{C}$ (in fact, it is exactly $\mathbb{C}$, since $H^1(\mathcal{O}(-1))=0$). The higher cohomology groups (for Abelian sheaves) detect obstructions to patching like this. I'll look at it tomorrow. MathOverflow is a question and answer site for professional mathematicians. - this is the same as being in the kernel of the second map in the complex. $\begingroup$ I have seen that book just now.. they do it more generally..they define Singular Cohom of a space with coefficients from not just a group but a sheaf of groups $\mathcal{A}$.. it is more complicated than that of coefficients to be a group...But, as you said, it is only natural to use this notation as in case of. . $j$, resp. . We get a resolution Thanks a lot for the extensive edit. Now map from this whole arrangement into a sheaf of abelian groups (i.e. @ Jan.

$$ 0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ S. Eilenberg, N. Steenrod: Foundations of algebraic topology, (Princeton Mathematical Series No. ӭ��^~��j��N޸�� 9md$��) �NX�sLG@�a ��N�C*��ضZU��n�R�Es�EGkXl]A"��L|ʃ��#� ݈K� endstream endobj 1465 0 obj <>stream It is an error term to our tentative calculation of the difference between global sections and sections on the covering in terms of matching 2-families. November 9, 2014 at 4:35 am. I think looking at Bredon's Sheaf theory book (page $179$ chapter $3$) would be helpful. This result is relavent here because the idea is to produce a resolution of $\underline{A}_X$ that has the above property and we use that resolution to produce sheaf cohomology groups $H^k(X,\underline{A}_X)$.

I hope that was an intelligible first (edit: and second :-) approximation.

\rightarrow \widetilde{S^1}\rightarrow \cdots$$ Show that this functor need not be exact. This is called the "espace etalé" of the sheaf. . 1462 0 obj <>stream (There are other sections that follow things up but they are a bit incomplete so I have not put a copy up there yet.). The same picture is valid when you pass to cohomology over other sites - there you don't have the espace etalé but taking sections is still the same as mapping into the sheaf by the Yoneda lemma. $$0\rightarrow \widetilde{S^0}(X)

Thus, sheaf cohomology groups $H^k(X,\underline{A}_X)$ are isomorphic to singular cohomology groups $H^k(X,A)$. Given an open set $U$ of $X$, consider singular $n$-cochains In other fields that is considered worthy. In particular what do you mean by "if you pass to the next stage" and "were there at the $i−1$ the stage"? The idea being that if one has information about the open sets that make up a space as well as how these sets are glued together one can deduce global properties of the space from local data. �h�[��vm��[�z�� then $H^1(A)$ is measuring the obstruction of global sections to be exact: I do not prove that result but I mentioned just to give some motivation. To learn more, see our tips on writing great answers. . . For every flabby sheaf $\mathcal{G}$ we have $H^i(X,\mathcal{G})=0$ for each $i>0$. h�|X�n��~[{aK�{w�X�Q�Ĉ�+��f��C��Cr�9��G��s��b��*Fa��L��'A E��BKM�Zy��V��I��$a�!! It could be that trying to understand the general sheaf that we are talking about is not the right way. inside $U$ i.e., $\alpha:\{\Delta_n\rightarrow U\}\rightarrow A$. Higher cohomology may be also thought of this way: $H^{i+1}$ measures the failure of $H^i$ to preserve surjective maps. that was directed at Tim. Doing this small rectification recursively in the formula $H^n=m(n+1)Fam - (nFam - (n-1)Fam)$ gives you a bunch of differences in nested parentheses and makes the alternating sums appear which you know from the Euler characteristic... (end of edit). So to calculate Čech cohomology of a contractible space all you have to do is to calculate Čech cohomology of a point. In that case, we have $H^k (\mathcal{U},\underline{A}_X)$ is canonically isomorphic to $H^k (\mathcal{U},\underline{A}_X)$. Dimension shift is the fact that if $0\to M\to P\to N\to 0$ is is an exact sequence in which $P$ is projective, then $Ext^i(M,-)$ and $Ext^{i+1}(N,-)$ are isomorphic. Since the values of $\mathcal{F}$ are abelian groups we can take alternating sums of parallel arrows and thus get a diagram, $$\prod \mathcal{F}(U_i) \rightarrow \prod \mathcal{F}(U_i \cap U_j) \rightarrow \prod \mathcal{F}(U_i \cap U_j) \rightarrow \ldots$$. http://en.wikipedia.org/wiki/Exponential_sheaf_sequence, Responding to the Lavender Letter and commitments moving forward, Intuitive Approach to Sheaf and Cech Cohomology. .

To subscribe to this RSS feed, copy and paste this URL into your RSS reader. ̪�*ɼȍ�B��A��Zp�N0�)F8�9^� ��[��{\�p�k�� 7�`\��W��_�Z(�)��0m 3̑�_|E�JTX�j4h��O��q��5m�h��4[�� v�S��Ѵ��2D�u��۽eĚ��MFl`a��$]�t��ϱЇu��P��fjD��X�t!�4s4ur�NM!�[�aGJ��j��̲"�׭�e�?��%�nՆ�4�ڎ}��-�ۖ��kl�?�ĨHg �f��s�5R,��8Gy�h�Df6��\ܾ�� ۴ȳ�rF��[��@/��|�V5>w]��>�̕A z��2�z�%��Tߏ�M�($��QL�]GF��E�㝴km��c��Q�oblv+m̎��c��J&T��Ȁ��i#�(H��u��i�Q�u1~�pO[#Az(�F�,�:��u�Mg$�(��c site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. It also is instructive to remember that before comparing $2Fam$ and "its subcollection" $1Fam$ we have to map $1Fam$ into $2Fam$ and in the process the subcollection $0Fam$ of $1Fam$ gets killed and $1Fam$ becomes $(1Fam - 0Fam)$. Is the Čech cohomology of an orbifold isomorphic to its singular cohomology? Making statements based on opinion; back them up with references or personal experience. $$ \mathcal{O}\rightarrow\mathbb{C}_p$$ For example I would be very interested in the case of coherent $\mathcal{O}_X$-Modules. . ��e!�V$H�b��6 ZGX҈�f+�L��> ,��P$X!yگmz��BR��*8��d+��d!el҇?f҂�� -6i��pI3r�0ޤ5(��NZ�1��5��\k�9Z3n��( ��-`�iZ�B� @*e� /�! What is sheaf cohomology intuitively? .

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