Definitions for Local structure of algebraic monoids

QAlgorithms.Papers.Independent.Intakethird100.P79.IsIdempotentPoint

For a scheme $M$ over $\operatorname{Spec}k$ with a monoid-object structure and a $k$-point $e:\operatorname{Spec}k\to M$, this predicate means that the composite of $(e,e):\operatorname{Spec}k\to M\times_k M$ with the multiplication $\mu:M\times_k M\to M$ equals $e$. Equivalently, it is the equality $e\cdot e=e$ of $k$-points.

QAlgorithms.Papers.Independent.Intakethird100.P79.IsMinimalIdempotent

For a monoid scheme $M$ over $k$, write $P=\operatorname{Hom}_k(\operatorname{Spec}k,M)$, with its induced monoid multiplication. For any $e\in P$, this defines the proposition that every $x\in P$ satisfying both $(ex)e=x$ and $x^2=x$ must equal $e$. The definition does not separately require $e^2=e$; its universally quantified implication also includes points $x$ for which either premise fails.

QAlgorithms.Papers.Independent.Intakethird100.P79.IsQuasiProjective

For a scheme $X$ over $\operatorname{Spec}k$, with structure morphism $f:X\to\operatorname{Spec}k$, this predicate means that there exist a natural number $n$, possibly zero, and a scheme morphism $i:X\to\operatorname{Proj}(k[t_0,\ldots,t_n])$ such that $i$ is an immersion and its composite with the specified morphism to $\operatorname{Spec}k$ equals $f$. Here the polynomial ring is graded by total degree, and the specified morphism is the canonical map to the spectrum of its degree-zero component followed by the map induced by the inclusion of constant polynomials from $k$. No nonemptiness condition on $X$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P79.IsSmoothPoint

For a scheme $X$ over $\operatorname{Spec}k$ and a $k$-point $e:\operatorname{Spec}k\to X$, this predicate means that there exists an open subset $U$ of the underlying scheme of $X$ which contains the image under $e$ of the distinguished closed point of $\operatorname{Spec}k$, and for which the composite $U\hookrightarrow X\to\operatorname{Spec}k$ is smooth. The condition concerns smoothness of the structure morphism on such an open neighborhood.

QAlgorithms.Papers.Independent.Intakethird100.P79.IsVariety

For a scheme $X$ over the base ring $k$, with structure morphism $p_X:X\to\operatorname{Spec} k$, this property consists of four requirements: $p_X$ is locally of finite type, quasi-compact, and separated, and the underlying scheme $X$ is reduced. No nonemptiness or irreducibility requirement is included.

QAlgorithms.Papers.Independent.Intakethird100.P79.KScheme

For the base ring $k$, this denotes the category of schemes over $\operatorname{Spec} k$. An object is a scheme $X$ equipped with a structure morphism $p_X:X\to\operatorname{Spec} k$, and a morphism from $(X,p_X)$ to $(Y,p_Y)$ is a scheme morphism $f:X\to Y$ satisfying $p_Y\circ f=p_X$.

QAlgorithms.Papers.Independent.Intakethird100.P79.MonoidSchemePoint

For any scheme $M$ over $\operatorname{Spec}k$, this abbreviates the type of morphisms from the monoidal unit of the category of $k$-schemes to $M$; equivalently, it is the type of $k$-points $\operatorname{Spec}k\to M$, with morphisms required to commute with the structure maps to $\operatorname{Spec}k$. This abbreviation itself requires no monoid-object structure on $M$.

QAlgorithms.Papers.Independent.Intakethird100.P79.affineSpace

For the base commutative ring $k$ and every natural number $n$, including $n=0$, this defines the $k$-scheme $\operatorname{Spec}(k[t_i\mid i\in\{0,\ldots,n-1\}])$, whose structure morphism to $\operatorname{Spec}k$ is induced contravariantly by the ring homomorphism sending each element of $k$ to its constant polynomial. When $n=0$, the polynomial ring has no variables and is canonically $k$, so the resulting $k$-scheme is canonically $\operatorname{Spec}k$.

QAlgorithms.Papers.Independent.Intakethird100.P79.constantMap

For schemes $X$ and $Y$ over the base ring $k$ and a $k$-point $e:\operatorname{Spec} k\to Y$, this defines the morphism $X\to Y$ obtained by composing the unique morphism from $X$ to the terminal object $\operatorname{Spec} k$ with $e$:

$$X\xrightarrow{p_X}\operatorname{Spec} k\xrightarrow{e}Y.$$

QAlgorithms.Papers.Independent.Intakethird100.P79.degreeZeroConstants

For the base commutative ring $k$ and every natural number $n$, including $n=0$, this defines the ring homomorphism from $k$ to the degree-zero homogeneous component of $k[t_0,\ldots,t_n]$ which sends each $a\in k$ to the constant polynomial $a$, regarded as a homogeneous polynomial of degree zero. This map preserves $0$, $1$, addition, and multiplication; the zero polynomial is included in the degree-zero component.

QAlgorithms.Papers.Independent.Intakethird100.P79.groupCentralizerPoints

For a monoid scheme $M$ over $k$, let $P=\operatorname{Hom}_k(\operatorname{Spec}k,M)$, with its induced monoid multiplication. Given any $e\in P$ and $s\in\{0,1,2\}$, this defines the set of points $x\in P$ that have a two-sided multiplicative inverse and satisfy $xe=(ex)e$ if $s=0$, $ex=(ex)e$ if $s=1$, and $xe=ex$ if $s=2$. No idempotence assumption on $e$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P79.groupStabilizer

For a scheme $M$ over $\operatorname{Spec}k$ with a monoid-object structure, a $k$-point $e:\operatorname{Spec}k\to M$, and $s\in\{0,1,2\}$, let $U$ be the equalizer defining pairs $(x,y)$ with $xy=yx=1_M$, and let $p:U\to M$ send $(x,y)$ to $x$. This definition produces a $k$-scheme as follows: for $s=0$, it is the equalizer of $u\mapsto p(u)e$ and the constant map with value $e$; for $s=1$, it is the equalizer of $u\mapsto ep(u)$ and that constant map; for $s=2$, it is the equalizer of $u\mapsto(p(u)e,ep(u))$ and the constant map with value $(e,e)$. These are scheme equalizers, imposing respectively $xe=e$, $ex=e$, or both equations on inverse pairs. No idempotence condition on $e$ is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P79.groupStabilizerInclusion

For a scheme $M$ over $\operatorname{Spec}k$ with a monoid-object structure, a $k$-point $e$, and $s\in\{0,1,2\}$, form the scheme $U$ of inverse pairs $(x,y)$ satisfying $xy=yx=1_M$. Inside $U$, take the scheme equalizer imposing $xe=e$ when $s=0$, $ex=e$ when $s=1$, and both equations when $s=2$. The morphism defined here from this equalizer to $M$ is the composite of its equalizer inclusion into $U$, the equalizer inclusion $U\to M\times_k M$, and the first projection $M\times_k M\to M$. Thus it sends an inverse pair satisfying the selected equations to its first component $x$; $e$ is an arbitrary $k$-point.

QAlgorithms.Papers.Independent.Intakethird100.P79.groupStabilizerPoints

For a monoid scheme $M$ over $k$, let $P=\operatorname{Hom}_k(\operatorname{Spec}k,M)$ be its monoid of $k$-rational points, with multiplication induced by the multiplication of $M$. Given any $e\in P$ and $s\in\{0,1,2\}$, this defines the set of points $x\in P$ that have a two-sided multiplicative inverse and satisfy $xe=e$ if $s=0$, $ex=e$ if $s=1$, and both $xe=e$ and $ex=e$ if $s=2$. No idempotence assumption on $e$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P79.kernelPoints

For a monoid scheme $M$ over $k$, let $P=\operatorname{Hom}_k(\operatorname{Spec}k,M)$, with its induced monoid multiplication. This defines the set of all $x\in P$ such that, for every subset $I\subseteq P$, if $I$ is nonempty and $(ba)c\in I$ for every $a\in I$ and every $b,c\in P$, then $x\in I$. Thus the quantification is over all nonempty subsets stable under multiplication on both sides, with no topological or scheme-theoretic condition on those subsets; the empty subset imposes no requirement.

QAlgorithms.Papers.Independent.Intakethird100.P79.leftIdealPoints

For a monoid scheme $M$ over $k$, let $P=\operatorname{Hom}_k(\operatorname{Spec}k,M)$, with its induced monoid multiplication. For any $e\in P$, this defines the subset $\{ex\mid x\in P\}$ of $P$: a point $y$ belongs to it exactly when there exists $x\in P$ such that $y=ex$. No idempotence assumption on $e$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P79.leftMultiplication

For a monoid scheme $M$ over the base ring $k$, with associative multiplication $\mu:M\times_k M\to M$ and an identity section satisfying the unit laws, and any $k$-point $e:\operatorname{Spec} k\to M$, this defines the endomorphism

$$L_e=\mu\circ\langle e\circ p_M,\operatorname{id}_M\rangle:M\to M,$$

where $p_M$ is the structure morphism. Thus it sends a generalized point $x$ to $ex$, with $e$ pulled back to the same base as $x$.

QAlgorithms.Papers.Independent.Intakethird100.P79.localOpenPoints

For a monoid scheme $M$ over $k$, let $P=\operatorname{Hom}_k(\operatorname{Spec}k,M)$, with its induced monoid multiplication. Given any $e\in P$ and $s\in\{0,1,2\}$, this defines a subset of $P$ as follows: when $s=0$, its members are exactly the $x$ for which there exists a two-sided invertible $u\in P$ satisfying $eu=(eu)e$ and $xe=ue$; when $s=1$, its members are exactly the $x$ for which there exists a two-sided invertible $v\in P$ satisfying $ve=(ev)e$ and $ex=ev$; when $s=2$, both of these existence conditions must hold, with independently chosen witnesses $u$ and $v$. No idempotence assumption on $e$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P79.projectiveSpace

For the base commutative ring $k$ and every natural number $n$, including $n=0$, this defines the scheme $\operatorname{Proj}(k[t_0,\ldots,t_n])$, using the grading by total degree: its degree-$d$ component consists of the homogeneous polynomials of total degree $d$. For $n=0$, this construction uses the polynomial ring in one variable. The value of this definition is a scheme.

QAlgorithms.Papers.Independent.Intakethird100.P79.projectiveSpaceBase

For the base commutative ring $k$ and every natural number $n$, including $n=0$, put $S=k[t_0,\ldots,t_n]$ with its total-degree grading and let $S_0$ be its degree-zero component. This defines the morphism $\operatorname{Proj}S\to\operatorname{Spec}k$ as the composite of the canonical morphism $\operatorname{Proj}S\to\operatorname{Spec}S_0$ with the morphism $\operatorname{Spec}S_0\to\operatorname{Spec}k$ induced by the ring homomorphism $k\to S_0$ sending each scalar to its constant polynomial.

QAlgorithms.Papers.Independent.Intakethird100.P79.reducedOver

For a scheme $X$ over the base ring $k$, with structure morphism $p_X$, this defines the scheme over $k$ whose underlying scheme is the closed subscheme $X_{\mathrm{red}}$ cut out by the nilradical ideal sheaf of $X$, and whose structure morphism is

$$X_{\mathrm{red}}\xrightarrow{\iota_X}X\xrightarrow{p_X}\operatorname{Spec} k,$$

where $\iota_X$ is the canonical closed immersion.

QAlgorithms.Papers.Independent.Intakethird100.P79.reductionInclusion

For a scheme $X$ over the base ring $k$, this defines the morphism over $k$ from its reduction to $X$ whose underlying scheme morphism is the canonical inclusion $\iota_X:X_{\mathrm{red}}\to X$. Here $X_{\mathrm{red}}$ is the closed subscheme cut out by the nilradical ideal sheaf, equipped with the structure morphism $p_X\circ\iota_X$.

QAlgorithms.Papers.Independent.Intakethird100.P79.rightIdealPoints

For a monoid scheme $M$ over $k$, let $P=\operatorname{Hom}_k(\operatorname{Spec}k,M)$, with its induced monoid multiplication. For any $e\in P$, this defines the subset $\{xe\mid x\in P\}$ of $P$: a point $y$ belongs to it exactly when there exists $x\in P$ such that $y=xe$. No idempotence assumption on $e$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P79.rightMultiplication

For a monoid scheme $M$ over the base ring $k$, with associative multiplication $\mu:M\times_k M\to M$ and an identity section satisfying the unit laws, and any $k$-point $e:\operatorname{Spec} k\to M$, this defines the endomorphism

$$R_e=\mu\circ\langle\operatorname{id}_M,e\circ p_M\rangle:M\to M,$$

where $p_M$ is the structure morphism. Thus it sends a generalized point $x$ to $xe$, with $e$ pulled back to the same base as $x$.

QAlgorithms.Papers.Independent.Intakethird100.P79.stabilizer

For a monoid scheme $M$ over the base ring $k$, an arbitrary $k$-point $e:\operatorname{Spec} k\to M$, and an index $s\in\{0,1,2\}$, this defines the reduction of the equalizer imposing $xe=e$ when $s=0$, $ex=e$ when $s=1$, or both equations when $s=2$. Multiplication is the given associative multiplication with identity on $M$, and these are equations of morphisms on generalized points after pulling back $e$ to their base. Explicitly, if $E_s$ is that equalizer, the resulting scheme is the closed subscheme $(E_s)_{\mathrm{red}}$ cut out by its nilradical ideal sheaf, with structure morphism obtained by composing $(E_s)_{\mathrm{red}}\to E_s$ with the structure morphism of $E_s$. No idempotency condition on $e$ is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P79.stabilizerEquations

For a monoid scheme $M$ over the base ring $k$, with associative multiplication and an identity section satisfying the unit laws, an arbitrary $k$-point $e:\operatorname{Spec} k\to M$, and an index $s\in\{0,1,2\}$, this defines a scheme over $k$ by taking the equalizer of the maps $x\mapsto xe$ and $x\mapsto e$ when $s=0$, the equalizer of $x\mapsto ex$ and $x\mapsto e$ when $s=1$, and the equalizer of $x\mapsto(xe,ex)$ and $x\mapsto(e,e)$ when $s=2$. All products are over $k$, and constant maps factor through $\operatorname{Spec} k$. Equivalently, for every scheme $T$ over $k$, its $T$-points are precisely those $x:T\to M$ satisfying $xe_T=e_T$ for $s=0$, $e_Tx=e_T$ for $s=1$, or both equations for $s=2$, where $e_T=e\circ p_T$. No idempotency condition on $e$ is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P79.stabilizerEquationsInclusion

For a monoid scheme $M$ over the base ring $k$, an arbitrary $k$-point $e:\operatorname{Spec} k\to M$, and an index $s\in\{0,1,2\}$, this defines the canonical equalizer morphism into $M$: its source is the equalizer imposing $xe=e$ when $s=0$, $ex=e$ when $s=1$, or both equations when $s=2$. Here multiplication comes from the given associative monoid structure with identity on $M$, and the equations mean equalities of morphisms on generalized points, with $e$ pulled back to their base. In each case the defined map is the equalizer inclusion itself.

QAlgorithms.Papers.Independent.Intakethird100.P79.stabilizerInclusion

For a monoid scheme $M$ over the base ring $k$, an arbitrary $k$-point $e:\operatorname{Spec} k\to M$, and an index $s\in\{0,1,2\}$, let $E_s$ be the equalizer imposing $xe=e$ for $s=0$, $ex=e$ for $s=1$, or both equations for $s=2$, using the given associative multiplication with identity on $M$ and pulling back $e$ to the base of each generalized point. This defines the morphism over $k$

$$(E_s)_{\mathrm{red}}\longrightarrow E_s\longrightarrow M,$$

where the first arrow is the inclusion of the closed subscheme cut out by the nilradical ideal sheaf of $E_s$, and the second is the canonical equalizer inclusion.

QAlgorithms.Papers.Independent.Intakethird100.P79.stabilizerPoints

For a scheme $M$ over $\operatorname{Spec}k$ equipped with a monoid-object structure, a $k$-point $e$, and $s\in\{0,1,2\}$, this defines a set of $k$-points of $M$, using the multiplication on points induced by the scheme multiplication. For $s=0$, the set is $\{x\mid xe=e\}$; for $s=1$, it is $\{x\mid ex=e\}$; and for $s=2$, it is $\{x\mid xe=e\text{ and }ex=e\}$. The point $e$ is arbitrary, with no idempotence assumption, and membership imposes no invertibility condition on $x$.

QAlgorithms.Papers.Independent.Intakethird100.P79.unitPairs

For a monoid scheme $M$ over the base ring $k$, with associative multiplication $\mu:M\times_k M\to M$ and identity section $\eta:\operatorname{Spec} k\to M$ satisfying the unit laws, this defines the equalizer of the two morphisms

$$M\times_k M\rightrightarrows M\times_k M, \qquad (x,y)\longmapsto(xy,yx), \qquad (x,y)\longmapsto(1,1),$$

where $1$ denotes the identity section pulled back along the structure morphism. Thus, for every scheme $T$ over $k$, its $T$-points are pairs $(x,y)$ of $T$-points of $M$ satisfying both $xy=1_T$ and $yx=1_T$, where $1_T=\eta\circ p_T$.

QAlgorithms.Papers.Independent.Intakethird100.P79.unitsInclusion

For a scheme $M$ over $\operatorname{Spec}k$, equipped with an associative multiplication and a unit making it a monoid object, let $U$ be the equalizer of the two morphisms $M\times_k M\to M\times_k M$ given by $(x,y)\mapsto(xy,yx)$ and the constant map with value $(1_M,1_M)$. Thus $U$ is the scheme of pairs satisfying $xy=yx=1_M$. The morphism defined here is $U\hookrightarrow M\times_k M\to M$, where the second arrow is the first projection; it sends an inverse pair $(x,y)$ to $x$.