The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

The map shows both the Angeln and the Schwansen peninsulas

The base does not appear on public U. S. government maps ; the USGS topographic map for the area only shows the long-disused Groom Mine.

A map of Cyprus in the later Bronze Age ( such as is given by J. L. Myres and M. O. Richter in Catalogue of the Cyprus Museum ) shows more than 25 settlements in and about the Mesaorea district alone, of which one, that at Enkomi, near the site of Salamis, has yielded the richest Aegean treasure in precious metal found outside Mycenae.

The latest Ordnance Survey Ireland " Discovery Series " ( Third Edition 2005 ) 1: 50, 000 map of the Dublin Region, Sheet 50, shows the boundaries of the city and three surrounding counties of the region.

This map shows their relative distribution in various parts of Europe.

Information richness or a map that is multivariate shows relationships within the map.

This issue assumes more importance as the scale of the map gets smaller ( i. e. the map shows a larger area ) because the information shown on the map takes up more space on the ground.

This map of the world in 1914 shows the large colonial empires that powerful nations established across the globe

* ESP shows textual information about the enemy ; such as health, name and distance ; also information about weapons lying around the map, which could be missed without the hack.

The Sebastian Cabot map ( Antwerp 1544 ) shows a number of islands to the south which may be the Mascarene Islands.

This electrostatic potential map shows how the oxygen atom has a more negative charge ( red ) than the positive ( blue ) hydrogen atoms of a water molecule.

The map also shows Hong Kong's maritime boundaries.

The map at the beginning of this article shows the 32 traditional counties that were in place in Ireland by early 17th century.

A 1905 map shows the Khazar empire in 814.

This 1856 map shows slave states ( gray ), free states ( pink ), U. S. territories ( green ), and Kansas in center ( white ).

A map of the world in the years 1920 – 1945, which shows the League of Nations members during its history

As the map shows, the firestorm jumped natural firebreaks ( river channels ), as well as prepared firebreaks.

* Macbeth's Scotland on Google Maps This map shows some of the primary locations in Macbeth.

The national Railroad Development Corporation map shows a proposed extension across the border to Chipata in Zambia.

The map also shows the 75th parallel north and 60th parallel north.

The set of regions of a map can be represented more abstractly as an undirected graph that has a vertex for each region and an edge for every pair of regions that share a boundary segment.

This graph is planar ( it is important to note that we are talking about the graphs that have some limitations according to the map they are transformed from only ): it can be drawn in the plane without crossings by placing each vertex at an arbitrarily chosen location within the region to which it corresponds, and by drawing the edges as curves that lead without crossing within each region from the vertex location to each shared boundary point of the region.

Because the western boundary was contested at the time of the Purchase, President Jefferson immediately began to organize three missions to explore and map the new territory.

Pre-Simla British map published in 1909 shows the so called " Outer Line " as India's northern boundary.

" A 1908 map of The Province of Eastern Bengal and Assam prepared for the Foreign Department of the Government of India, showed the international boundary from Bhutan continuing to the Baroi River, following the Himalayas foothill alignment.

* Yahoo map showing jog from west to east for northerly traffic and indicating the approximate location of the original boundary line

There is a red line marking the boundary between New York and Quebec, and the upper third of the map shows the Richelieu River extending north to meet the Saint Lawrence River at Sorel, with Montreal southwest of that point.

On 29 December 1724, Peter I of Russia asked Bering to command a voyage east, probably to map the lands ( and possibly seas ) between Russia's eastern boundary and the North America continent.

Once this has been established the features on the lot can be drawn, dimensions can be shown from these features to the boundary line and a map or plat is prepared for the client.

The district boundary map is provided by the Michigan Department of Information Technology.

See this link for a more accurate map of the Hazard district boundary.

The notification includes a description of the interest ( citation ), a map of the boundary, and a list of activities requiring consultation ( see below ).

* Remington-Whitewater school district boundary map, USD 206

* Remington-Whitewater school district boundary map, USD 206

* Remington-Whitewater school district boundary map, USD 206

" This boundary dispute, which was officially caused by a map error, was resolved in 1853, with the Gadsden Purchase.

Given two complexes M < sub >*</ sub > and N < sub >*</ sub >, a chain map between the two is a series of homomorphisms from M < sub > i </ sub > to N < sub > i </ sub > such that the entire diagram involving the boundary maps of M and N commutes.

Define C < sub > n </ sub >( X ) for natural n to be the free abelian group formally generated by singular n-simplices in X, and define the boundary map

Under Roman domination, the map of the city changed, starting from the modification of the boundary walls.

" The 1784 John Stockdale map shows the River Mersey extending to Mottram, and forming the boundary between Cheshire and Derbyshire.

map sheet showing boundary of former Ifni Province

A method of delineating biotic communities is to map the food network to identify which species feed upon which others and then determine the system boundary as the one that can be drawn through the fewest consumption links relative to the number of species within the boundary.

The domain invariance theorem may be generalized to manifolds: if M and N are topological n-manifolds without boundary and f: M → N is a continuous map which is locally one-to-one ( meaning that every point in M has a neighborhood such that f restricted to this neighborhood is injective ), then f is an open map ( meaning that f ( U ) is open in N whenever U is an open subset of M ).