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Cohen–Macaulay ring

In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.

They are named for, who proved the unmixedness theorem for polynomial rings, and for , who proved the unmixedness theorem for formal power series rings. All Cohen–Macaulay rings have the unmixedness property.

For Noetherian local rings, there is the following chain of inclusions.

Definition
For a commutative Noetherian local ring R, a finite (i.e. finitely generated) R-module $$M\neq 0$$ is a Cohen-Macaulay module if $$\mathrm{depth}(M) = \mathrm{dim}(M)$$ (in general we have: $$\mathrm{depth}(M) \leq \mathrm{dim}(M)$$, see Auslander–Buchsbaum formula for the relation between depth and dim of a certain kind of modules). On the other hand, $$R$$ is a module on itself, so we call $$R$$ a Cohen-Macaulay ring if it is a Cohen-Macaulay module as an $$R$$-module. A maximal Cohen-Macaulay module is a Cohen-Macaulay module M such that $$\mathrm{dim}(M)=\mathrm{dim}(R)$$.

The above definition was for a Noetherian local rings. But we can expand the definition for a more general Noetherian ring: If $$R$$ is a commutative Noetherian ring, then an R-module M is called Cohen–Macaulay module if $$M_\mathrm{m}$$ is a Cohen-Macaulay module for all maximal ideals $$\mathrm{m}\in \mathrm{Supp}(M)$$. (This is a kind of circular definition unless we define zero modules as Cohen-Macaulay. So we define zero modules as Cohen-Macaulay modules in this definition.) Now, to define maximal Cohen-Macaulay modules for these rings, we require that $$M_\mathrm{m}$$ to be such an $$R_\mathrm{m}$$-module for each maximal ideal $$\mathrm{m}$$ of R. As in the local case, R is a Cohen-Macaulay ring if it is a Cohen-Macaulay module (as an $$R$$-module on itself).

Examples
Noetherian rings of the following types are Cohen–Macaulay.
 * Any regular local ring. This leads to various examples of Cohen–Macaulay rings, such as the integers $$\Z$$, or a polynomial ring $$K[x_1,\ldots ,x_n]$$ over a field K, or a power series ring $$Kx_1,\ldots ,x_n$$ . In geometric terms, every regular scheme, for example a smooth variety over a field, is Cohen–Macaulay.
 * Any 0-dimensional ring (or equivalently, any Artinian ring).
 * Any 1-dimensional reduced ring, for example any 1-dimensional domain.
 * Any 2-dimensional normal ring.
 * Any Gorenstein ring. In particular, any complete intersection ring.
 * The ring of invariants $$R^G$$ when R is a Cohen–Macaulay algebra over a field of characteristic zero and G is a finite group (or more generally, a linear algebraic group whose identity component is reductive). This is the Hochster–Roberts theorem.
 * Any determinantal ring. That is, let R be the quotient of a regular local ring S by the ideal I generated by the r × r minors of some p × q matrix of elements of S. If the codimension (or height) of I is equal to the "expected" codimension (p−r+1)(q−r+1), R is called a determinantal ring. In that case, R is Cohen−Macaulay. Similarly, coordinate rings of determinantal varieties are Cohen-Macaulay.

Some more examples:
 * 1) The ring K[x]/(x²) has dimension 0 and hence is Cohen–Macaulay, but it is not reduced and therefore not regular.
 * 2) The subring K[t2, t3] of the polynomial ring K[t], or its localization or completion at t=0, is a 1-dimensional domain which is Gorenstein, and hence Cohen–Macaulay, but not regular. This ring can also be described as the coordinate ring of the cuspidal cubic curve y2 = x3 over K.
 * 3) The subring K[t3, t4, t5] of the polynomial ring K[t], or its localization or completion at t=0, is a 1-dimensional domain which is Cohen–Macaulay but not Gorenstein.

Rational singularities over a field of characteristic zero are Cohen–Macaulay. Toric varieties over any field are Cohen–Macaulay. The minimal model program makes prominent use of varieties with klt (Kawamata log terminal) singularities; in characteristic zero, these are rational singularities and hence are Cohen–Macaulay, One successful analog of rational singularities in positive characteristic is the notion of F-rational singularities; again, such singularities are Cohen–Macaulay.

Let X be a projective variety of dimension n ≥ 1 over a field, and let L be an ample line bundle on X. Then the section ring of L
 * $$R=\bigoplus_{j\geq 0}H^0(X,L^j)$$

is Cohen–Macaulay if and only if the cohomology group Hi(X, Lj) is zero for all 1 ≤ i ≤ n−1 and all integers j. It follows, for example, that the affine cone Spec R over an abelian variety X is Cohen–Macaulay when X has dimension 1, but not when X has dimension at least 2 (because H1(X, O) is not zero). See also Generalized Cohen–Macaulay ring.

Cohen–Macaulay schemes
We say that a locally Noetherian scheme $$X$$ is Cohen–Macaulay if at each point $$x\in X$$ the local ring $$\mathcal{O}_{X,x}$$ is Cohen–Macaulay.

Cohen–Macaulay curves
Cohen–Macaulay curves are a special case of Cohen–Macaulay schemes, but are useful for compactifying moduli spaces of curves where the boundary of the smooth locus $$\mathcal{M}_g$$ is of Cohen–Macaulay curves. There is a useful criterion for deciding whether or not curves are Cohen–Macaulay. Schemes of dimension $$\leq 1$$ are Cohen–Macaulay if and only if they have no embedded primes. The singularities present in Cohen–Macaulay curves can be classified completely by looking at the plane curve case.

Non-examples
Using the criterion, there are easy examples of non-Cohen–Macaulay curves from constructing curves with embedded points. For example, the scheme
 * $$X = \text{Spec}\left( \frac{\Complex[x,y]}{(x^2,xy)} \right)$$

has the decomposition into prime ideals $$(x)\cdot(x,y)$$. Geometrically it is the $$y$$-axis with an embedded point at the origin, which can be thought of as a fat point. Given a smooth projective plane curve $$C \subset \mathbb{P}^2$$, a curve with an embedded point can be constructed using the same technique: find the ideal $$I_x$$ of a point in $$x \in C$$ and multiply it with the ideal $$I_C$$ of $$C$$. Then
 * $$X = \text{Proj}\left( \frac{\Complex[x,y,z]}{I_C \cdot I_x} \right)$$

is a curve with an embedded point at $$x$$.

Intersection theory
Cohen–Macaulay schemes have a special relation with intersection theory. Precisely, let X be a smooth variety and V, W closed subschemes of pure dimension. Let Z be a proper component of the scheme-theoretic intersection $$V \times_X W$$, that is, an irreducible component of expected dimension. If the local ring A of $$V \times_X W$$ at the generic point of Z is Cohen-Macaulay, then the intersection multiplicity of V and W along Z is given as the length of A:

 * $$i(Z, V \cdot W, X) = \operatorname{length}(A)$$.

In general, that multiplicity is given as a length essentially characterizes Cohen–Macaulay ring; see. Multiplicity one criterion, on the other hand, roughly characterizes a regular local ring as a local ring of multiplicity one.

Example
For a simple example, if we take the intersection of a parabola with a line tangent to it, the local ring at the intersection point is isomorphic to
 * $$ \frac{\Complex[x,y]}{(y - x^2)} \otimes_{\Complex[x,y]}\frac{\Complex[x,y]}{(y)} \cong \frac{\Complex[x]}{(x^2)} $$

which is Cohen–Macaulay of length two, hence the intersection multiplicity is two, as expected.

Miracle flatness or Hironaka's criterion
There is a remarkable characterization of Cohen–Macaulay rings, sometimes called miracle flatness or Hironaka's criterion. Let R be a local ring which is finitely generated as a module over some regular local ring A contained in R. Such a subring exists for any localization R at a prime ideal of a finitely generated algebra over a field, by the Noether normalization lemma; it also exists when R is complete and contains a field, or when R is a complete domain. Then R is Cohen–Macaulay if and only if it is flat as an A-module; it is also equivalent to say that R is free as an A-module.

A geometric reformulation is as follows. Let X be a connected affine scheme of finite type over a field K (for example, an affine variety). Let n be the dimension of X. By Noether normalization, there is a finite morphism f from X to affine space An over K. Then X is Cohen–Macaulay if and only if all fibers of f have the same degree. It is striking that this property is independent of the choice of f.

Finally, there is a version of Miracle Flatness for graded rings. Let R be a finitely generated commutative graded algebra over a field K,
 * $$R=K\oplus R_1 \oplus R_2 \oplus \cdots.$$

There is always a graded polynomial subring A ⊂ R (with generators in various degrees) such that R is finitely generated as an A-module. Then R is Cohen–Macaulay if and only if R is free as a graded A-module. Again, it follows that this freeness is independent of the choice of the polynomial subring A.

Properties

 * A Noetherian local ring is Cohen–Macaulay if and only if its completion is Cohen–Macaulay.
 * If R is a Cohen–Macaulay ring, then the polynomial ring R[x] and the power series ring R x  are Cohen–Macaulay.
 * For a non-zero-divisor u in the maximal ideal of a Noetherian local ring R, R is Cohen–Macaulay if and only if R/(u) is Cohen–Macaulay.
 * The quotient of a Cohen–Macaulay ring by any ideal is universally catenary.
 * If R is a quotient of a Cohen–Macaulay ring, then the locus { p ∈ Spec R | Rp is Cohen–Macaulay } is an open subset of Spec R.
 * Let (R, m, k) be a Noetherian local ring of embedding codimension c, meaning that c = dimk(m/m2) − dim(R). In geometric terms, this holds for a local ring of a subscheme of codimension c in a regular scheme. For c=1, R is Cohen–Macaulay if and only if it is a hypersurface ring. There is also a structure theorem for Cohen–Macaulay rings of codimension 2, the Hilbert–Burch theorem: they are all determinantal rings, defined by the r × r minors of an (r+1) × r matrix for some r.
 * For a Noetherian local ring (R, m), the following are equivalent:
 * R is Cohen–Macaulay.
 * For every parameter ideal Q (an ideal generated by a system of parameters),
 * $$\operatorname{length}(R/Q) = e(Q)$$ := the Hilbert–Samuel multiplicity of Q.
 * For some parameter ideal Q, $$\operatorname{length}(R/Q) = e(Q)$$.
 * (See Generalized Cohen–Macaulay ring as well as Buchsbaum ring for rings that generalize this characterization.)

The unmixedness theorem
An ideal I of a Noetherian ring A is called unmixed in height if the height of I is equal to the height of every associated prime P of A/I. (This is stronger than saying that A/I is equidimensional; see below.)

The unmixedness theorem is said to hold for the ring A if every ideal I generated by a number of elements equal to its height is unmixed. A Noetherian ring is Cohen–Macaulay if and only if the unmixedness theorem holds for it.

The unmixed theorem applies in particular to the zero ideal (an ideal generated by zero elements) and thus it says a Cohen–Macaulay ring is an equidimensional ring; in fact, in the strong sense: there is no embedded component and each component has the same codimension.

See also: quasi-unmixed ring (a ring in which the unmixed theorem holds for integral closure of an ideal).

Counterexamples

 * 1) If K is a field, then the ring R = K[x,y]/(x2,xy) (the coordinate ring of a line with an embedded point) is not Cohen–Macaulay. This follows, for example, by Miracle Flatness: R is finite over the polynomial ring A = K[y], with degree 1 over points of the affine line Spec A with y ≠ 0, but with degree 2 over the point y = 0 (because the K-vector space K[x]/(x2) has dimension 2).
 * 2) If K is a field, then the ring K[x,y,z]/(xy,xz) (the coordinate ring of the union of a line and a plane) is reduced, but not equidimensional, and hence not Cohen–Macaulay. Taking the quotient by the non-zero-divisor x−z gives the previous example.
 * 3) If K is a field, then the ring R = K[w,x,y,z]/(wy,wz,xy,xz) (the coordinate ring of the union of two planes meeting in a point) is reduced and equidimensional, but not Cohen–Macaulay. To prove that, one can use Hartshorne's connectedness theorem: if R is a Cohen–Macaulay local ring of dimension at least 2, then Spec R minus its closed point is connected.

The Segre product of two Cohen-Macaulay rings need not be Cohen-Macaulay.

Grothendieck duality
One meaning of the Cohen–Macaulay condition can be seen in coherent duality theory. A variety or scheme X is Cohen–Macaulay if the "dualizing complex", which a priori lies in the derived category of sheaves on X, is represented by a single sheaf. The stronger property of being Gorenstein means that this sheaf is a line bundle. In particular, every regular scheme is Gorenstein. Thus the statements of duality theorems such as Serre duality or Grothendieck local duality for Gorenstein or Cohen–Macaulay schemes retain some of the simplicity of what happens for regular schemes or smooth varieties.

title: Cohen–Macaulay ring


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List of proposed amendments to the Constitution of the United States

Hundreds of proposed amendments to the United States Constitution are introduced during each session of the United States Congress. From 1789 through January 3, 2019, approximately 11,770 measures have been proposed to amend the United States Constitution. Collectively, members of the House and Senate typically propose around 200 amendments during each two-year term of Congress. Most, however, never get out of the Congressional committees in which they were proposed. Only a fraction of those actually receive enough support to win Congressional approval to go through the constitutional ratification process. Some proposed amendments are introduced over and over again in different sessions of Congress. It is also common for a number of identical resolutions to be offered on issues that have widespread public and congressional support.

Since 1789, Congress has sent 33 constitutional amendments to the states for ratification. Of these, 27 have been ratified. The framers of the Constitution, recognizing the difference between regular legislation and constitutional matters, intended that it be difficult to change the Constitution; but not so difficult as to render it an inflexible instrument of government, as the amendment mechanism in the Articles of Confederation, which required a unanimous vote of thirteen states for ratification, had proven to be. Therefore, a less stringent process for amending the Constitution was established in Article V.

Amending process
Amending the United States Constitution is a two-step process. Proposals to amend it must be properly adopted and ratified before becoming operative. A proposed amendment may be adopted and sent to the states for ratification by either:
 * The United States Congress, whenever a two-thirds majority in both the Senate and the House deem it necessary;
 * or

The latter procedure has never been used. To become part of the Constitution, an adopted amendment must be ratified by either:
 * A national convention, called by Congress for this purpose, on the application of the legislatures of two thirds (presently 34) of the states.
 * The legislatures of three-fourths (presently 38) of the states, within the stipulated time period, if any;
 * or

The decision of which ratification method will be used for any given amendment is Congress' alone to make, as is the decision to set a ratification deadline. Only for the 21st amendment was the latter procedure invoked and followed. Upon being properly ratified, an amendment becomes an operative addition to the Constitution.
 * State ratifying conventions in three-fourths (presently 38) of the states, within the stipulated time period, if any.

19th century
Constitutional amendment proposals considered in but not approved by Congress during the 19th century included:
 * The Dueling Ban Amendment, proposed in 1838 after Representative William Graves killed another Representative, Jonathan Cilley, in a duel, would have prohibited any person involved in a duel from holding federal office.
 * Shortly before his death during the congressional debates leading to the Compromise of 1850, John C. Calhoun proposed constitutional amendments requiring an equal number of slave states and free states and creating two co-Presidents from the North and the South which would have to concur on all legislation. A similar amendment to eliminate the presidency so as to have two elected officials in its place, was proposed by Virginia Representative Albert Jenkins in 1860 shortly before sectional tensions escalated into the American Civil War. Jenkins saw the amendment as a way for both the Northern and the Southern states to be represented equally in the government at a given time.
 * The Christian Amendment, first proposed in February 1863, would have added acknowledgment of the Christian God in the Preamble to the Constitution. Similar amendments were proposed in 1874, 1896, and 1910 with none passing. The last attempt in 1954 did not come to a vote.
 * The Blaine Amendment, proposed in 1875, would have banned public funds from going to religious purposes, in order to prevent Catholics from taking advantage of such funds. Though it failed to pass, many states adopted such provisions.
 * An amendment allowing property-owning unmarried women to vote was proposed by Representative William Mason. The underlying logic behind this amendment was that these single women did not have husbands to represent their interests via the vote. Elizabeth Cady Stanton, a suffragist at the time, stated in her testimony before the U.S. Senate Committee on Woman Suffrage: "Though my coadjutors all believe in universal suffrage, yet I think we should be willing to let you start with spinsters and widows who are householder. Having homes of their own it is fair to suppose that they are industrious, common-sense women, ... women who love their country (having no husbands to love) better than themselves." The Nineteenth Amendment would later establish women's suffrage irrespective of marital status or property in 1920.
 * Representative Lucas M. Miller proposed a Constitutional amendment to change the country's name to "the United States of the Earth."

20th century
Constitutional amendment proposals considered in but not approved by Congress during the 20th century included the following:
 * An amendment abolishing the Senate was proposed by Representative Victor Berger in 1911, due to his belief that it was corrupt as well as useless to the country as a whole. The amendment would have also shielded the unicameral House of Representatives' legislation from presidential veto and judicial review. At the time, Senators were still elected by state legislatures. The year after Berger proposed his amendment, Congress passed an amendment mandating popular election of Senators which was duly ratified by the several states.
 * An Anti-Miscegenation Amendment was proposed by Representative Seaborn Roddenbery, a Southern Democrat from Georgia, in 1912 to forbid interracial marriages nationwide. This was spurred when black boxer Jack Johnson garnered much publicity when he married a white woman, Lucille Cameron. Similar amendments were proposed by Congressman Andrew King, a Missourian Democrat, in 1871 and by Senator Coleman Blease, a South Carolinian Democrat, in 1928. None were passed by Congress, although numerous state legislatures passed laws prohibiting interracial marriage before they were ruled unconstitutional by the Supreme Court's 1967 Loving v. Virginia decision.
 * An Anti-Polygamy Amendment was proposed by Representative Frederick Gillett, a Massachusetts Republican, on January 24, 1914, and supported by former U.S. Senator from Utah, Frank J. Cannon, and by the National Reform Association. Utah was dominated by the LDS Church, which permitted multiple marriages until 1890.
 * The Ludlow Amendment was proposed by Representative Louis Ludlow in 1937. This amendment would have heavily reduced America's ability to be involved in war, requiring a national referendum to confirm any declaration of war. Public support for the amendment was very robust through the 1930s, a period when isolationism was the prevailing mood in the United States.
 * A maximum wage amendment that no person should accumulate more than $1 million was proposed by Representative Wesley Lloyd in 1933. In the wake of the Wall Street Crash and the beginnings of the Great Depression many Americans believed that personal wealth should be limited to avoid such an issue from happening again.
 * An amendment to limit investment income was proposed by Representative J. Buell Snyder in 1933.
 * Multiple attempts to repeal the 21st Amendment ending Prohibition were proposed by Representative Morris Sheppard, introducer of the 18th Amendment originally banning alcoholic beverages, from 1935 to 1938, followed by attempt to outlaw drunkenness after his first proposals failed.
 * The Bricker Amendment, proposed in 1951 by Ohio Senator John W. Bricker, would have limited the federal government's treaty-making power by prohibiting treaties from violating the U.S. Constitution and limiting executive agreements. Opposed by President Dwight Eisenhower, it failed twice to reach the threshold of two-thirds of voting members necessary for passage, the first time by eight votes and the second time by a single vote.
 * A repeal of the Twenty-second Amendment would eliminate term limits for presidents. Presidents Harry S. Truman, Ronald Reagan, Bill Clinton, and Donald Trump all expressed support for some sort of repeal. The first efforts in Congress to repeal the 22nd Amendment were undertaken in 1956, only five years after its ratification. According to the Congressional Research Service, over the ensuing half-century (through 2008) 54 joint resolutions seeking to repeal the two-term presidential election limit were introduced; none were given serious consideration. The most recent attempt was launched by Democratic Representative for New York, José Serrano, in 2013, during the 113th Congress.
 * A School Prayer Amendment to establish that "the people retain the right to pray and to recognize their religious beliefs, heritage, and traditions on public property, including schools" was proposed by Robert Byrd of West Virginia in 1962, 1973, 1979, 1982, 1993, 1995, 1997, and 2006. Representative Ernest Istook, a Republican from Oklahoma's 5th congressional district, proposed the amendment in the House on May 8, 1997. In March 1998, the House Judiciary Committee passed the bill by a 16–11 vote. On June 4, 1998, the full House voted on the amendment, 224–203 in favor. The vote was 61 short of the required two-thirds majority.
 * A Flag Desecration Amendment was first proposed in 1995 to give Congress the power to make acts such as flag burning illegal, seeking to overturn the 1990 Supreme Court Texas v. Johnson decision ruling that such laws were unconstitutional. During each term of Congress from 1995 to 2005, the proposed amendment was passed by the House of Representatives, but never by the Senate, coming closest during voting on June 27, 2006, with 66 in support and 34 opposed (one vote short).
 * The Bayh–Celler amendment was the closest the United States has come to passing an Electoral College abolition amendment. The amendment would have replaced the current Electoral College with a simpler two-round system modeled after French presidential elections. It was proposed during the 91st Congress (1969–1971). The House Judiciary Committee voted 28–6 to approve the proposal and was eventually passed by the full House with bipartisan support on September 18, 1969, by a vote of 339 to 70. The Senate commenced openly debating the proposal and the proposal was quickly filibustered. On September 17, 1970, a motion for cloture, which would have ended the filibuster, received 54 votes to 36 for cloture, failing to receive the then required a two-thirds majority of senators voting. Other proposals were made in 2005, 2009, and 2016, none of which were voted on by committee.
 * The Human Life Amendment, first proposed in 1973, would overturn the Roe v. Wade court ruling and prohibit abortion. A total of 330 proposals using varying texts have been proposed with almost all dying in committee. The only version that reached a formal floor vote, the Hatch–Eagleton Amendment, was rejected by 18 votes in the Senate on June 28, 1983. The Roe v. Wade ruling was later overturned by the Supreme Court in Dobbs v. Jackson Women's Health Organization (2022).
 * A balanced budget amendment, in which Congress and the President are forced to balance the budget every year, has been introduced many times, dating back to the 1930s. No measure passed either body of Congress until 1982, when the Senate took 11 days to consider it and gained the necessary two-thirds majority. The first and only time the House gave two-thirds approval to a balanced budget amendment was in 1995, when Members of the House of Representatives elected in the Republican Revolution voted for the Contract with America. That was also the last time the House held a floor or committee vote.
 * Various proposed amendments for congressional term limits have been made since Supreme Court ruled state term limits on federal officials to be unconstitutional in the U.S. Term Limits, Inc. v. Thornton decision in 1995.
 * The single subject amendment, an amendment first proposed in 1996, would introduce a single-subject rule blocking members of Congress from adding riders to bills.

21st century
Constitutional amendment proposals considered in but not approved by Congress thus far in the 21st century have included:
 * The Equal Opportunity to Govern Amendment, proposed in July 2003 by Senator Orrin Hatch (R-Utah), would repeal the Constitution's natural born citizen clause, thus allowing naturalized citizens – who have been U.S. citizens for at least twenty years – to become President of the United States or Vice President. It was widely seen as an attempt to make California Governor Arnold Schwarzenegger (born in Austria and naturalized in 1983) eligible for the presidency and is sometimes nicknamed the "Arnold Amendment" or "Amend for Arnold".
 * The Federal Marriage Amendment has been introduced in the United States Congress four times: in 2003, 2004, 2005/2006, and 2008 by multiple members of Congress. It would define marriage and prohibit same-sex marriage, even at the state level. The last Congressional vote on the proposed amendment occurred in the House of Representatives on July 18, 2006, when the motion failed 236–187, falling short of the 290 votes required for passage in that body. The Senate has voted only on cloture motions with regard to the proposed amendment, the last of which was on June 7, 2006, when the motion failed 49 to 48, falling short of the 60 votes required to allow the Senate to proceed to consideration of the proposal and the 67 votes required to send the proposed amendment to the states for ratification.
 * Various campaign finance reform amendments have been introduced in Congress since the United States Supreme Court's 2010 Citizens United v. Federal Election Commission ruling declaring that the First Amendment's free speech clause prohibits the federal government from restricting independent expenditures for communications by nonprofit corporations, for-profit corporations, labor unions, and other associations. The most recent version of the amendment was introduced in the 118th Congress by Representative Pramila Jayapal.
 * Senator Marco Rubio and Representative Steven Palazzo proposed constitutional amendments barring the government from imposing taxes on refusal to purchase goods and services. The amendment would have invalidated the individual mandate of the Patient Protection and Affordable Care Act requiring legal persons to purchase health insurance.
 * Representative Randy Neugebauer introduced an amendment requiring a two-thirds majority for Congress to raise the debt ceiling.
 * Various proposals have been made for a constitutional amendment abolishing birthright citizenship for children of foreign nationals as a deterrent against illegal immigration.  President Donald Trump endorsed a similar proposal during the 2016 presidential election.
 * Representative Steve King introduced an amendment repealing the Sixteenth Amendment and abolishing the federal income tax.
 * Representative John Culberson introduced an amendment altering Article V to limit any national convention convoked by the states to amend the Constitution to a single amendment and requiring Congress to authorize such as a convention. This amendment, sometimes called the "Madison Amendment", would prevent a "runaway convention" from drastically altering or replacing the U.S. Constitution.
 * Various proposals were made by Republican members of Congress to base congressional apportionments on the number of citizens in a state rather than residents following the Evenwel v. Abbott decision in 2016.
 * Representative Al Green introduced an amendment prohibiting the President of the United States from issuing a pardon for themselves.
 * Representative Eliot Engel introduced an amendment prohibiting barriers to voting for adult Americans including "undue burden of proof of identity or citizenship", prohibiting foreign interference in elections and "undue or anonymous influence from any person", guaranteeing that electoral districts must be composed of geographically compact and contiguous territory, and designating Election Day as a national holiday unable to be altered by the government. The amendment was proposed after the Shelby County v. Holder case overruled parts of the Voting Rights Act of 1965 and in light of Russian interference in the 2016 United States elections. Many key aspects of the amendment were incorporated into the proposed For the People Act, which passed the U.S. House of Representatives.
 * Representative Cedric Richmond introduced an amendment in the 116th Congress to repeal the penal exception clause from the Thirteenth Amendment, prohibiting unfree labor from being used as a punishment.
 * Representative Tom Marino introduced an amendment to amend Article 1, Section 2 of the Constitution to change the terms of Representatives from two years to four (with elections to be held in non-presidential years), but then resigned afterward. Proposals for extending the terms of House members date back to a similar proposal by Lyndon B. Johnson for 4-year terms, renewable in presidential years, in his 1966 State of the Union Address.
 * In response to revived advocacy among progressive activists for expansion of the Supreme Court beyond the customary nine members (as the Constitution does not specify how many justices should sit on the court), Democratic Representative Collin Peterson and Republican Representative Denver Riggleman filed a proposed amendment in the 116th Congress to limit the Supreme Court membership to nine members maximum. A group of Republican congressmembers including Senators Ted Cruz and Thom Tillis proposed an identical amendment a few weeks later, in an entirely separate effort from that of Peterson and Riggleman.
 * Parental Rights Amendment to the United States Constitution. The amendment's advocates say that it will allow parents' rights to direct the upbringing of their children, protected from federal interference, and the United Nations Convention on the Rights of the Child. The Amendment was first proposed during the 110th Congress as House Joint Resolution 97 in July 2008, but no action was taken during that Congress.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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Riemann–Hurwitz formula

In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves.

Statement
For a compact, connected, orientable surface $$S$$, the Euler characteristic $$\chi(S)$$ is

 * $$\chi(S)=2-2g$$,

where g is the genus (the number of handles). This follows, as the Betti numbers are $$1, 2g, 1, 0, 0, \dots$$.

For the case of an (unramified) covering map of surfaces

 * $$\pi\colon S' \to S$$

that is surjective and of degree $$N$$, we have the formula

 * $$\chi(S') = N\cdot\chi(S).$$

That is because each simplex of $$S$$ should be covered by exactly $$N$$ in $$S'$$, at least if we use a fine enough triangulation of $$S$$, as we are entitled to do since the Euler characteristic is a topological invariant. What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (sheets coming together).

Now assume that $$S$$ and $$S'$$ are Riemann surfaces, and that the map $$\pi$$ is complex analytic. The map $$\pi$$ is said to be ramified at a point P in S′ if there exist analytic coordinates near P and π(P) such that π takes the form π(z) = zn, and n > 1. An equivalent way of thinking about this is that there exists a small neighborhood U of P such that π(P) has exactly one preimage in U, but the image of any other point in U has exactly n preimages in U. The number n is called the ramification index at P and is denoted by eP. In calculating the Euler characteristic of S′ we notice the loss of eP − 1 copies of P above π(P) (that is, in the inverse image of π(P)). Now let us choose triangulations of S and S′ with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics. Then S′ will have the same number of d-dimensional faces for d different from zero, but fewer than expected vertices. Therefore, we find a "corrected" formula

 * $$\chi(S') = N\cdot\chi(S) - \sum_{P\in S'} (e_P -1) $$

or as it is also commonly written, using that $$\chi(X) = 2 - 2g(X)$$ and multiplying through by -1:
 * $$2g(S')-2 = N\cdot(2g(S)-2) +\sum_{P\in S'} (e_P -1) $$

(all but finitely many P have eP = 1, so this is quite safe). This formula is known as the Riemann–Hurwitz formula and also as Hurwitz's theorem.

Another useful form of the formula is:
 * $$\chi(S')- b' = N \cdot (\chi(S) - b) $$

where b is the number of branch points in S (images of ramification points) and b' is the size of the union of the fibers of branch points (this contains all ramification points and perhaps some non-ramified points). Indeed, to obtain this formula, remove disjoint disc neighborhoods of the branch points from S and their preimages in S' so that the restriction of $$ \pi $$ is a covering. Removing a disc from a surface lowers its Euler characterstic by 1 by the formula for connected sum, so we finish by the formula for a non-ramified covering.

We can also see that this formula is equivalent to the usual form, as we have
 * $$N \cdot b - b' = \sum_{P\in S'} (e_P - 1)$$

since for any $$Q \in S$$ we have $$N = \sum_{P \in \pi^{-1}(Q)} e_P$$

Examples
The Weierstrass $\wp$-function, considered as a meromorphic function with values in the Riemann sphere, yields a map from an elliptic curve (genus 1) to the projective line (genus 0). It is a double cover (N = 2), with ramification at four points only, at which e = 2. The Riemann–Hurwitz formula then reads

 * $$0 = 2\cdot2 - 4\cdot(2 - 1)$$

with the summation taken over four ramification points.

The formula may also be used to calculate the genus of hyperelliptic curves.

As another example, the Riemann sphere maps to itself by the function zn, which has ramification index n at 0, for any integer n > 1. There can only be other ramification at the point at infinity. In order to balance the equation

 * $$2 = n\cdot2 - (n - 1) - (e_\infty - 1)$$

we must have ramification index n at infinity, also.

Consequences
Several results in algebraic topology and complex analysis follow.

Firstly, there are no ramified covering maps from a curve of lower genus to a curve of higher genus – and thus, since non-constant meromorphic maps of curves are ramified covering spaces, there are no non-constant meromorphic maps from a curve of lower genus to a curve of higher genus.

As another example, it shows immediately that a curve of genus 0 has no cover with N > 1 that is unramified everywhere: because that would give rise to an Euler characteristic > 2.

Generalizations
For a correspondence of curves, there is a more general formula, Zeuthen's theorem, which gives the ramification correction to the first approximation that the Euler characteristics are in the inverse ratio to the degrees of the correspondence.

An orbifold covering of degree N between orbifold surfaces S' and S is a branched covering, so the Riemann–Hurwitz formula implies the usual formula for coverings
 * $$\chi(S') = N\cdot\chi(S) \,$$

denoting with $$\chi \,$$ the orbifold Euler characteristic.

title: Riemann–Hurwitz formula


text:
At the Mountains of Madness

At the Mountains of Madness is a science fiction-horror novella by American author H. P. Lovecraft, written in February/March 1931. Rejected that year by Weird Tales editor Farnsworth Wright on the grounds of its length, it was originally serialized in the February, March, and April 1936 issues of Astounding Stories. It has been reproduced in numerous collections.

The story details the events of a disastrous expedition to Antarctica in September 1930, and what is found there by a group of explorers led by the narrator, Dr. William Dyer of Miskatonic University. Throughout the story, Dyer details a series of previously untold events in the hope of deterring another group of explorers who wish to return to the continent. These events include the discovery of an ancient civilization older than the human race, and realization of Earth's past told through various sculptures and murals.

The story was inspired by Lovecraft's interest in Antarctic exploration; the continent was still not fully explored in the 1930s. Lovecraft explicitly draws from Edgar Allan Poe's novel The Narrative of Arthur Gordon Pym of Nantucket, and he may have used other stories for inspiration. Many story elements, such as the formless "shoggoth", recur in other Lovecraft works. The story has been adapted and used for graphic novels, video games, and musical works.

Plot
The story is narrated in a first-person perspective by the geologist William Dyer, a professor at Arkham's Miskatonic University, aiming to prevent an important and much publicized scientific expedition to Antarctica. Throughout the course of his explanation, Dyer relates how he led a group of scholars from Miskatonic University on a previous expedition to Antarctica, during which they discovered ancient ruins and a dangerous secret beyond a range of mountains higher than the Himalayas.

A small advance group, led by Professor Lake, discovers the remains of fourteen prehistoric life-forms previously unknown to science, and also unidentifiable as either plants or animals. Six of the specimens have been badly damaged, while another eight have been preserved in pristine condition. The specimens' stratum places them far too early on the geologic time scale for the features of the specimens to have evolved. Some fossils of Cambrian age show signs of the use of tools to carve a specimen for food.

When the main expedition loses contact with Lake's party, Dyer and a graduate student named Danforth investigate. Lake's camp is devastated, with the majority of men and dogs slaughtered, while a man named Gedney and one of the dogs are absent. Near the expedition's campsite, they find six star-shaped snow mounds with one specimen under each. They also discover that the better preserved life-forms have vanished, and that some form of dissection experiment has been done on both an unnamed man and a dog. The missing man Gedney is suspected of having gone utterly insane and having killed and mutilated all the others.

The two men fly an aeroplane across the "mountains", soon revealed to be the outer walls of a vast, abandoned stone city, alien to any human architecture. For their resemblance to creatures of myth mentioned in the Necronomicon, the builders of this lost civilization are dubbed the "Elder Things." By exploring these fantastic structures, the men learn through hieroglyphic murals that the Elder Things first came to Earth shortly after the Moon took form, and built their cities with the help of "shoggoths"—biological entities created to perform any task, assume any form, and reflect any thought. There is a hint that all Earthly life evolved from cellular material left over from the creation of the shoggoths.

As more buildings are explored, the explorers learn about the Elder Things' conflict with both the Star-spawn of Cthulhu and the Mi-Go, who arrived on Earth shortly afterwards. The images also reflect a degradation of their civilization, once the shoggoths gain independence. As more resources are applied in maintaining order, the etchings become haphazard and primitive. The murals also allude to an unnamed evil lurking within an even larger mountain range located beyond the city. This mountain range rose in one night and certain phenomena and incidents deterred the Elder Things from exploring it. When Antarctica became uninhabitable, even for the Elder Things, they soon migrated into a large, subterranean ocean.

Dyer and Danforth eventually realize that the Elder Things missing from the advance party's camp had somehow returned to life and, after slaughtering the explorers, have returned to their city. Dyer and Danforth also discover traces of the Elder Things' earlier exploration, as well as sleds containing the corpses of both Gedney and his missing dog. They are ultimately drawn towards the entrance of a tunnel, into the subterranean region depicted in the murals. Here, they find evidence of various Elder Things killed in a brutal struggle and blind six-foot-tall penguins wandering placidly, apparently used as livestock. They are then confronted by a black, bubbling mass, which they identify as a shoggoth, and escape. Aboard the plane, high above the plateau, Danforth looks back and sees something which causes him to go insane, implied to be the unnamed evil itself.

Dyer concludes the Elder Things are survivors of a bygone era, who slaughtered Lake's group only in self-defense or scientific curiosity. Their civilization was eventually destroyed by the shoggoths, which now prey on the enormous penguins. He warns the planners of the next proposed Antarctic expedition to stay away from the site. Danforth continues to experience manic episodes, whispering of "bizarre conceptions" which Dyer attributes to his being one of the few to have completely read through Miskatonic's copy of the Necronomicon.

Connections to other Lovecraft stories
At the Mountains of Madness has numerous connections to other Lovecraft stories. A few include:
 * The formless Shoggoths later appeared in "The Shadow over Innsmouth" (1931), "The Thing on the Doorstep" (1933), and "The Haunter of the Dark" (1935)
 * The star-headed Elder Things also appear in "The Dreams in the Witch House" (1933), when the main character, Walter Gilman, visits a city of theirs in one of his dreams, and "The Shadow Out of Time", in which an Elder Thing is kept as a fellow prisoner.
 * Dyer is mentioned in "The Shadow Out of Time" as accompanying Nathaniel Wingate Peaslee's expedition to Australia's Great Sandy Desert.
 * The expedition is sponsored by the Nathaniel Derby Pickman Foundation, combining two major names in Lovecraft's fiction: Derby and Pickman. Richard Upton Pickman is the main character in Lovecraft's "Pickman's Model", while Edward Pickman Derby is the protagonist of his "The Thing on the Doorstep", and also one of his literary alter-egos.
 * The Elder Things record the coming of Cthulhu to Earth and the sinking of R'lyeh, events referred to in "The Call of Cthulhu" (1928).
 * The Elder Things' city is identified with the Plateau of Leng first mentioned in Lovecraft's "Celephaïs" (1920).
 * Some members of the expedition have read Miskatonic University's copy of the Necronomicon.
 * Dyer mentions "Kadath in the Cold Waste" while referring to a massive mountain range which even the Elder Things "shunned as vaguely and namelessly evil."
 * At the very end of the story, Danforth links the horror beyond the forbidden mountain range to Yog-Sothoth and "The Colour Out of Space".
 * The Mi-Go are the focus of "The Whisperer in Darkness". Several times throughout, Dyer also makes reference to Albert Wilmarth, the main character of "The Whisperer in Darkness".

Inspiration
Lovecraft had a lifelong interest in Antarctic exploration. "Lovecraft had been fascinated with the Antarctic continent since he was at least 12 years old, when he had written several small treatises on early Antarctic explorers," the biographer S. T. Joshi wrote. At about the age of 9, inspired by W. Clark Russell's 1887 book The Frozen Pirate, Lovecraft had written "several yarns" set in Antarctica.

By the 1920s, Antarctica was "one of the last unexplored regions of the Earth in which large stretches of territory had never seen the tread of human feet. Contemporary maps of the continent show a number of provocative blanks, and Lovecraft could exercise his imagination in filling them in... with little fear of immediate contradiction." However, Lovecraft was basically accurate in presenting the geographic knowledge of Antarctica as it was known at the time, and he referred to continental drift, a theory which was then not widely accepted.

The first expedition of Richard E. Byrd took place between 1928 and 1930, just before the novella was written, and Lovecraft mentioned the explorer repeatedly in his letters and remarked at one point on "geologists of the Byrd expedition having found many fossils indicating a tropical past." In fact, Miskatonic University's expedition was modelled after that of Byrd.

In Lovecraft: A Look Behind the Cthulhu Mythos Lin Carter suggests that one inspiration for At the Mountains of Madness was Lovecraft's own hypersensitivity to cold, as evidenced by an incident in which the writer "collapsed in the street and was carried unconscious into a drug store" because the temperature dropped from 60 degrees to 30 degrees Fahrenheit (15 degrees to -1 degree Celsius). "The loathing and horror that extreme cold evoked in him was carried over into his writing," Carter wrote, "and the pages of Madness convey the blighting, blasting, stifling sensation caused by sub-zero temperatures in a way that even Poe could not suggest." S. T. Joshi called the theory "facile."

Joshi further cites Lovecraft's most obvious literary source for At the Mountains of Madness as Edgar Allan Poe's only novel, The Narrative of Arthur Gordon Pym of Nantucket, whose concluding section is set in Antarctica. Lovecraft twice cites Poe's "disturbing and enigmatic" story in his text and explicitly borrows the mysterious cry Tekeli-li or Takkeli from Poe's work. In a letter to August Derleth, Lovecraft wrote that he was trying to achieve with his ending an effect similar to what Poe accomplished in Pym.

Another proposed inspiration for At the Mountains of Madness is Edgar Rice Burroughs's At the Earth's Core (1914), a novel that posits a highly intelligent reptilian race, the Mahar, living in a hollow Earth. "Consider the similarity of Burroughs' Mahar to Lovecraft's Old Ones, both of whom are presented sympathetically despite their ill-treatment of man," wrote the critic William Fulwiler. "[B]oth are winged, web-footed, dominant races; both are scientific scholarly races with a talent for genetics, engineering, and architecture; and both races use men as cattle." Both stories, Fulwiler points out, involve radical new drilling techniques. In both stories, humans are vivisected by nonhuman scientists. Burroughs' Mahar even employ a species of servants known as Sagoths, possibly the source of Lovecraft's Shoggoth.

Other possible sources include A. Merritt's "The People of the Pit," whose description of an underground city in the Yukon bears some resemblance to that of Lovecraft's Elder Things, and Katharine Metcalf Roof's "A Million Years After," a story about dinosaurs hatching from eggs millions of years old that appeared in the November 1930 edition Weird Tales. In a letter to Frank Belknap Long, Lovecraft declared Metcalf Roof's story to be a "rotten," "cheap," and "puerile" version of an idea he had years earlier, and his dissatisfaction may have provoked him to write his own tale of "the awakening of entities from the dim reaches of Earth's history."

Edward Guimont has argued that At the Mountains of Madness was inspired by contemporary discourse around life on Mars, including Mars-set fictional works and the claims of Martian canals made by Percival Lowell (whom Lovecraft met in 1907). Guimont has also proposed other influences, including contemporary theories about the decline of the Norse Greenlanders and claims of survival of woolly mammoths in Alaska and particularly plot details being inspired by the 1930 discovery of the remains of Andrée's Arctic balloon expedition.

An H.P. Lovecraft Encyclopedia suggest that the long scope of history recounted in the story may have been inspired by Oswald Spengler's The Decline of the West. Some details of the story may also have been taken from M. P. Shiel's 1901 Arctic exploration novel The Purple Cloud, which was republished in 1930.

The title is derived from a line in Edward Plunkett, 18th Baron of Dunsany's short story "The Hashish Man": "And we came at last to those ivory hills that are named the Mountains of Madness...".

Lovecraft's own "The Nameless City" (1921), which also deals with the exploration of an ancient underground city that is apparently abandoned by its nonhuman builders, sets a precedent for At the Mountains of Madness. In both stories, the explorers use the nonhumans' artwork to deduce the history of their species. Lovecraft had also used that device in "The Dream-Quest of Unknown Kadath" (1927)

As for details of the Antarctic setting, the author's description of some of the scenery is in part inspired by the Asian paintings of Nicholas Roerich and the illustrations of Gustave Doré, both of whom are referenced by the story's narrator multiple times.

Publication
Lovecraft submitted the story to Weird Tales, but it was rejected by the editor Farnsworth Wright in July 1931. Lovecraft took the rejection badly and put the story to one side. It was eventually submitted by Lovecraft's literary agent Julius Schwartz in 1935 to F. Orlin Tremaine, the editor of Astounding Stories.

The novella was serialized in the February, March, and April 1936 issues, and Lovecraft received $315 —the most he had ever received for a story. The story, however, was harshly edited, with alterations to spellings, punctuation, and paragraphing, and the end of the story had several lengthy passages omitted. Lovecraft was outraged and called Tremaine "that god-damn'd dung of a hyaena [sic]". Lovecraft's own hand-corrected copies of Astounding Stories formed the basis for the first Arkham House edition, but this still contained over a thousand errors, and a fully restored text was not published until 1985.

Reception
The novella was received negatively during Lovecraft's lifetime; Lovecraft stated that its hostile reception had done "more than anything to end my effective fictional career." Theodore Sturgeon described the novella as "perfect Lovecraft" and "a good deal more lucid than much of the master's work", as well as "first-water, true-blue science fiction." The story popularized ancient astronaut theories, as well as Antarctica's place in the "ancient astronaut mythology". Edward Guimont has argued that At the Mountains of Madness, despite its terrestrial setting, helped influence later hard science fiction depictions of planetary expeditions and the Big Dumb Object trope, particularly those of Arthur C. Clarke, whose 1940 parody "At the Mountains of Murkiness" was one of his first works of fiction.

Adaptations

 * At the Mountains of Madness was adapted into a graphic novel created by I. N. J. Culbard and published in 2010 by SelfMadeHero as part of their Eye Classics line (ISBN 9781906838126). The book was named The Observer Graphic Novel of the Month.
 * Japanese artist Gou Tanabe created a two-volume graphic novel with the same name, translated into English by Dark Horse Manga in 2019.
 * The Mountains of Madness is an audiovisual musical adaptation of the works of H. P. Lovecraft by Tiger Lillies, Danielle de Picciotto and Alexander Hacke.
 * On July 28, 2010, director Guillermo del Toro announced that he would direct a film adaptation of H. P. Lovecraft's At the Mountains of Madness for Universal Pictures, with James Cameron producing. Cameron suggested casting Tom Cruise in the lead role and releasing the film in 3-D. On October 28, 2010, del Toro said that the adaptation probably would not happen at all. He stated, "It doesn't look like I can do it. It's very difficult for the studios to take the step of doing a period-set, R-rated, tentpole movie with a tough ending and no love story. Lovecraft has a readership as big as any best-seller, but it's tough to quantify because his works are in the public domain." Cameron and del Toro put forward the idea to Universal, who greenlit it. In 2012, del Toro posted that, due to the resemblance in premise with the Ridley Scott film Prometheus, the project would probably face a "long pause—if not demise." However, in 2021 del Toro announced that he was again revisiting the concept and revising the script for an animation approach, saying, "... it would be ideal to do Mountains of Madness as stop-motion."

Unofficial sequels and other inspired works

 * David A. McIntee noted similarities between the first half of the 1979 science fiction horror film Alien, particularly in early versions of the script, to At the Mountains of Madness, "not in storyline, but in dread-building mystery", and calls the finished film "the best Lovecraftian movie ever made, without being a Lovecraft adaptation", due to its similarities in tone and atmosphere to Lovecraft's works. In 2009, Alien writer Dan O'Bannon said the film was "strongly influenced, tone-wise, by Lovecraft, and one of the things it proved is that you can't adapt Lovecraft effectively without an extremely strong visual style ... What you need is a cinematic equivalent of Lovecraft's prose." The 2004 Alien sequel, Alien vs. Predator borrows its plot liberally from At the Mountains of Madness.
 * The 1995 Prisoner of Ice video game by Infogrames Multimedia is loosely based on At The Mountains of Madness, particularly the introduction where it involves a group of people discovering two mysterious cargo crates in Antarctica that contain Lovecraftian creatures called "the Prisoners."
 * Chaosium Games released a campaign book titled Beyond the Mountains of Madness for their Call of Cthulhu role-playing game in 1999. This book details the Starkweather-Moore expedition return to the ice to discover the truth about the Miskatonic Expedition. The book incorporates many of the aspects of the original Lovecraft story, including references to the Poe story, The Narrative of Arthur Gordon Pym of Nantucket, Nicholas Roerich, Danforth and Dyer. This book won the Origins Award for "Best Role-Playing Adventure" in 2000.
 * Call of Cthulhu: Beyond the Mountains of Madness was a cancelled video game adaptation by Headfirst Productions. It was announced in 2002 as a planned sequel to then-upcoming Call of Cthulhu: Dark Corners of the Earth (a long-delayed video game based on Lovecraft's The Shadow over Innsmouth, eventually released in 2006).
 * Charles Stross' 2000 novelette "A Colder War" is a direct sequel to Lovecraft's story. Set in the 1980s, it presents an alternate history of the Cold War where the threat of mutual annihilation is not nuclear, but occult, the US and USSR being locked in an arms race based on technologies found by subsequent expeditions to the alien city.
 * Edge of Nowhere, a 2016 video game developed by Insomniac Games, is set in 1952, 20 years after the expedition, and involves Victor Howard (Robin Atkin Downes) as he ventures into the site to locate his fiancée, who was on an expedition down a similar path.

title: At the Mountains of Madness


text:
Cannonball (disambiguation)

A cannonball is round shot ammunition for a cannon.

Cannonball or cannon ball may also refer to:

Biology

 * Couroupita guianensis, an evergreen tree commonly called the cannonball tree
 * Sphaerobolus, a genus commonly known as the cannonball fungi
 * Cannonball jellyfish, a species of jellyfish

Film and television

 * Cannonball (film), a 1976 film inspired by "Cannon Ball" Baker
 * Cannonball (TV series), a TV show about two truck drivers, produced in Canada in 1958–59 and syndicated in the U.S. in 1959–60
 * Cannonball (Australian game show)
 * Cannonball (British game show)
 * Cannonball (American game show)
 * "Cannonball", a 2003 episode of Lilo & Stitch: The Series
 * Hooterville Cannonball, a fictional train in the television series Petticoat Junction

Albums

 * Cannonball (album), by Pat Green, 2006
 * Cannonball!!!, by Bleubird, 2012

Songs

 * "Cannonball" (The Breeders song), 1993
 * "Cannonball" (Damien Rice song), 2002; covered by Little Mix, 2011
 * "Cannonball" (Duane Eddy song), 1958
 * "Cannonball" (Lea Michele song), 2013
 * "Cannonball" (Showtek and Justin Prime song), 2013
 * "Cannonball" (Skylar Grey song), 2015
 * "Cannonball" (Supertramp song), 1985
 * "Cannonball" (Tom Dice song), 2017
 * "Cannonball", by Alestorm from Seventh Rum of a Seventh Rum, 2022
 * "Cannonball", by Brandi Carlile from The Story, 2007
 * "Cannonball", by Dog Eat Dog, 2005
 * "Cannonball", by Five Iron Frenzy from The End Is Near, 2004
 * "Cannonball", by Grouplove from Big Mess, 2016
 * "Cannonball", by Gudda Gudda, featuring Drake, 2009
 * "Cannonball", by ZZ Ward from The Storm, 2017

People and characters

 * Cannonball (nickname), a list of people
 * Cannonball (comics), a Marvel Comics character associated with the X-Men
 * Cannonball (G.I. Joe), a fictional character in the G.I. Joe universe
 * Cannonball (Transformers), the Decepticon space pirate in the Transformers toy line
 * Cannonball, also known as Experiment 520, a fictional alien character in the Lilo & Stitch franchise

Places

 * Cannon Ball, North Dakota, a city
 * Cannonball River, a tributary of the Missouri River
 * Cannonball Cliffs, Alexander Island, Antarctica

Trains

 * The Cannon Ball, a Boston & Maine Railroad train from Boston to Plymouth, New Hampshire
 * The Cannonball (LIRR train), a seasonal train operated by the Long Island Rail Road along its Montauk Branch
 * The Cannon Ball a train operated by the Norfolk & Western Railway with the Pennsylvania Railroad, Richmond, Fredericksburg and Potomac Railroad, and the Atlantic Coast Line Railroad, from New York to Norfolk, Virginia

Other uses

 * Cannonball (diving), a diving maneuver
 * Cannonball (missile), originally intended to be a US Navy anti-ship missile, later developed as an anti-tank missile
 * Cannonball (novel), a novel by Joseph McElroy
 * Cannonball Musical Instruments, a Salt Lake City-based musical instrument manufacturer
 * Cannon Ball roller coaster, a wooden roller coaster at Lake Winnepesaukah, Georgia/Tennessee border
 * Cannonball problem, a mathematical problem
 * Cannon Ball Route, an early auto route between Kansas City and Chicago
 * Operation Cannonball, a Central Intelligence Agency operation
 * Cannonball Motorcycle Club, an outlaw motorcycle club in Finland

title: Cannonball (disambiguation)