Change some tabs to spaces (#2785)

... as per our dev docs

[skip ci]

docs/doc.main

@@ -129,16 +129,16 @@
         "CommutativeAlgebra/ModulesOverMultivariateRings/intro.md",
         "CommutativeAlgebra/ModulesOverMultivariateRings/free_modules.md",
         "CommutativeAlgebra/ModulesOverMultivariateRings/subquotients.md",
-	"CommutativeAlgebra/ModulesOverMultivariateRings/module_operations.md",
-	"CommutativeAlgebra/ModulesOverMultivariateRings/hom_operations.md",
-	"CommutativeAlgebra/ModulesOverMultivariateRings/complexes.md",
+        "CommutativeAlgebra/ModulesOverMultivariateRings/module_operations.md",
+        "CommutativeAlgebra/ModulesOverMultivariateRings/hom_operations.md",
+        "CommutativeAlgebra/ModulesOverMultivariateRings/complexes.md",
      ],
      "CommutativeAlgebra/homological_algebra.md",
 
      "Gröbner/Standard Bases" => [
         "CommutativeAlgebra/GroebnerBases/orderings.md",
         "CommutativeAlgebra/GroebnerBases/groebner_bases.md",
-	"CommutativeAlgebra/GroebnerBases/groebner_bases_integers.md",
+        "CommutativeAlgebra/GroebnerBases/groebner_bases_integers.md",
      ],
      "Miscellaneous" => [
          "CommutativeAlgebra/Miscellaneous/binomial_ideals.md",

docs/src/AlgebraicGeometry/Curves/para_rational_curves.md

@@ -43,11 +43,11 @@ curves only once. Its individual steps are interesting in their own right:
 !!! note
     The defining property of an adjoint curve is that it passes with “sufficiently high” multiplicity through the singularities of $C$.
     There are several concepts of making this precise. For each such concept, there is a corresponding  *adjoint ideal* of $C$,
-	namely the homogeneous ideal formed by the defining polynomials of the adjoint curves. In OSCAR, we follow
-	the concept of Gorenstein which leads to the largest possible adjoint ideal.
-	
+    namely the homogeneous ideal formed by the defining polynomials of the adjoint curves. In OSCAR, we follow
+    the concept of Gorenstein which leads to the largest possible adjoint ideal.
+
 See [Bhm99](@cite) and [BDLP17](@cite) for details and further references.
- 
+
 ## Creating Projective Plane Curves
 
 The data structures for algebraic curves in OSCAR are still under development

docs/src/CommutativeAlgebra/FrameWorks/ring_localizations.md

@@ -1,5 +1,5 @@
 ```@meta
-CurrentModule = Oscar 
+CurrentModule = Oscar
 DocTestSetup = quote
   using Oscar
 end
@@ -36,14 +36,14 @@ Coercing (pairs of) elements of `R` into fractions in `Rloc` must be possible as
    (Rloc::AbsLocalizedRing)(f::RingElem)
    (Rloc::AbsLocalizedRing)(f::RingElem, g::RingElem; check::Bool=true)
 ```
-   
+
 The first constructor maps the element `f` of `R` to the fraction `f//1` in `Rloc`.
 The second constructor takes a pair `f, g` of elements of `R` to the fraction `f//g`
 in `Rloc`. The default `check = true` stands for testing whether `g` is an admissible
 denominator. As this test is often expensive, it may be convenient
 to set `check = false`.
 
-For any concrete instance of type `AbsLocalizedRingElem`, methods for the functions 
+For any concrete instance of type `AbsLocalizedRingElem`, methods for the functions
 `parent`, `numerator`, and `denominator` must be provided. Moreover,
 if a cancellation function for the type of fractions under consideration is
 not yet available, such a function should be implemented and named

docs/src/CommutativeAlgebra/GroebnerBases/groebner_bases.md

@@ -29,7 +29,7 @@ $K[x]_>:= K[x][U^{-1}] = \left\{ \frac{f}{u} \:\bigg|\: f \in K[x], \, u\in U_>\
 
 Then $K[x]\subseteq K[x]_>\subseteq K[x]_{\langle x \rangle},$ where $K[x]_{\langle x \rangle}$
 is the localization of $K[x] $ at the maximal ideal $\langle x \rangle .$ Moreover,
- 
+
 - ``K[x] = K[x]_>`` iff $>$ is global, and
 
 - ``K[x]_> = K[x]_{\langle x \rangle}`` iff $>$ is local.
@@ -43,15 +43,15 @@ The *leading monomial* $\text{LM}_>(f)$, the *leading exponent* $\text{LE}_>(f)$
 !!! note
     Given a monomial ordering $>$ on a free $K[x]$-module $F = K[x]^p$ with basis $e_1, \dots, e_p$,
     the above notation extends naturally to elements of  $K[x]^p$ and $K[x]_>^p$, respectively. There is one particularity:
-	Given an element $f = K[x]^p\setminus \{0\}$ with leading term $\text{LT}(f) = x^\alpha e_i$, we write $\text{LE}_>(f) = (\alpha, i)$.
+    Given an element $f = K[x]^p\setminus \{0\}$ with leading term $\text{LT}(f) = x^\alpha e_i$, we write $\text{LE}_>(f) = (\alpha, i)$.
 
 ## Default Orderings
 
 !!! note
     The OSCAR functions discussed in this section depend on a monomial `ordering` which is entered as a keyword argument.
     Given a polynomial ring $R$, the `default_ordering` for this is `degrevlex` except if $R$ is $\mathbb Z$-graded with
-	positive weights. Then the corresponding `wdegrevlex` ordering is used. Given a free $R$-module $F$, the
-	`default_ordering` is `default_ordering(R)*lex(gens(F))`.
+    positive weights. Then the corresponding `wdegrevlex` ordering is used. Given a free $R$-module $F$, the
+    `default_ordering` is `default_ordering(R)*lex(gens(F))`.
 
 Here are some illustrating OSCAR examples:
 
@@ -222,22 +222,22 @@ still sense to speak of fully reduced remainders. However, even if we start from
 !!! note
     Given a monomial ordering $>$ on a free $K[x]$-module $F = K[x]^p$ with basis $e_1, \dots, e_p$,
     the above notation and the division algorithms extend naturally to $K[x]^p$ and $K[x]_>^p$, respectively.
-	
+
 The OSCAR functions discussed below compute standard representations and polynomial weak standard representations, respectively.
 
 ```@docs
 reduce(g::T, F::Vector{T}; 
-	ordering::MonomialOrdering = default_ordering(parent(F[1]))) where T <: MPolyRingElem
+    ordering::MonomialOrdering = default_ordering(parent(F[1]))) where T <: MPolyRingElem
 ```
 
 ```@docs
 reduce_with_quotients(g::T, F::Vector{T}; 
-	ordering::MonomialOrdering = default_ordering(parent(F[1]))) where T <: MPolyRingElem
+    ordering::MonomialOrdering = default_ordering(parent(F[1]))) where T <: MPolyRingElem
 ```
-		  
+      
 ```@docs
 reduce_with_quotients_and_unit(g::T, F::Vector{T}; 
-	ordering::MonomialOrdering = default_ordering(parent(F[1]))) where T <: MPolyRingElem
+    ordering::MonomialOrdering = default_ordering(parent(F[1]))) where T <: MPolyRingElem
 ```
 
 ## Computing Gröbner/Standard Bases
@@ -279,26 +279,26 @@ computed basis can be retrieved by using the `elements` function or its alias `g
 
 ```@docs
 groebner_basis(I::MPolyIdeal;
-	ord::MonomialOrdering = default_ordering(base_ring(I)),
-	complete_reduction::Bool = false, algorithm::Symbol = :buchberger)
+    ord::MonomialOrdering = default_ordering(base_ring(I)),
+    complete_reduction::Bool = false, algorithm::Symbol = :buchberger)
 ```
 ```@docs
 standard_basis(I::MPolyIdeal;
-	ord::MonomialOrdering = default_ordering(base_ring(I)),
-	complete_reduction::Bool = false, algorithm::Symbol = :buchberger)
+    ord::MonomialOrdering = default_ordering(base_ring(I)),
+    complete_reduction::Bool = false, algorithm::Symbol = :buchberger)
 ```
-	
+
 ### Gröbner Bases with transformation matrix
 
 ```@docs
 groebner_basis_with_transformation_matrix(I::MPolyIdeal;
-	ordering::MonomialOrdering = default_ordering(base_ring(I)),
-	complete_reduction::Bool=false)
+    ordering::MonomialOrdering = default_ordering(base_ring(I)),
+    complete_reduction::Bool=false)
 ```
 ```@docs
 standard_basis_with_transformation_matrix(I::MPolyIdeal;
-	ordering::MonomialOrdering = default_ordering(base_ring(I)),
-	complete_reduction::Bool=false)
+    ordering::MonomialOrdering = default_ordering(base_ring(I)),
+    complete_reduction::Bool=false)
 ```
 
 ### Gröbner Basis Conversion Algorithms
@@ -357,16 +357,16 @@ chosen prime number rather than for $I$ itself.
 !!! note
     If appropriate weights and/or the Hilbert function with respect to appropriate weights
     are already known to the user, this information can be entered when calling the Hilbert
-	driven Gröbner basis algorithm as follows:
-    
+    driven Gröbner basis algorithm as follows:
+
 ```@docs
 groebner_basis_hilbert_driven(I::MPolyIdeal{P};
     destination_ordering::MonomialOrdering,
     complete_reduction::Bool = false,
     weights::Vector{Int} = ones(Int, ngens(base_ring(I))),
     hilbert_numerator::Union{Nothing, ZZPolyRingElem} = nothing) where {P <: MPolyRingElem}
 ```
-	
+
 ### Faugère's F4 Algorithm
 
 !!! warning "Expert function for computing Gröbner bases"

docs/src/CommutativeAlgebra/GroebnerBases/orderings.md

@@ -27,17 +27,17 @@ A monomial ordering $>$ on $\text{Mon}_n(x)$ is called
 !!! note
     - A monomial ordering on $\text{Mon}_n(x)$ is global iff it is a well-ordering.
     - To give a monomial ordering on $\text{Mon}_n(x)$ means to give a total ordering $>$ on $ \N^n$ such that
-	   $\alpha > \beta$ implies $ \gamma + \alpha > \gamma  + \beta$ for all $\alpha , \beta, \gamma \in \N^n.$
+       $\alpha > \beta$ implies $ \gamma + \alpha > \gamma  + \beta$ for all $\alpha , \beta, \gamma \in \N^n.$
        Rather than speaking of a monomial ordering on $\text{Mon}_n(x)$, we may, thus, also speak of a
-	   (global, local, mixed) monomial ordering on $\N^n$.
+       (global, local, mixed) monomial ordering on $\N^n$.
 
 !!! note
     By a result of Robbiano, every monomial ordering can be realized as a matrix ordering.
 
 !!! note
     The lexicograpical monomial ordering specifies the default way of storing and displaying multivariate polynomials in OSCAR (terms are sorted in descending order).
     The other orderings which can be attached to a multivariate polynomial ring are the degree lexicographical ordering  and the degree reverse lexicographical
-	ordering. Independently of the attached orderings, Gröbner bases can be computed with respect to any monomial ordering. See the section on Gröbner bases.
+    ordering. Independently of the attached orderings, Gröbner bases can be computed with respect to any monomial ordering. See the section on Gröbner bases.
 
 In this section, we show how to create monomial orderings in OSCAR. 
 
@@ -108,7 +108,7 @@ cmp(ord::MonomialOrdering, a::MPolyRingElem, b::MPolyRingElem)
 
 Given a matrix $M\in \text{Mat}(k\times n,\mathbb R)$ of rank $n$, with rows $m_1,\dots,m_k$,
 the *matrix ordering* defined by $M$ is obtained by setting
-         
+
 $x^\alpha>_M x^\beta  \Leftrightarrow  \;\exists\; 1\leq i\leq k:  m_1\alpha=m_1\beta,\ldots, 
 m_{i-1}\alpha\ =m_{i-1}\beta,\ m_i\alpha>m_i\beta$
 
@@ -286,7 +286,7 @@ weight_ordering(W::Vector{Int}, ord::MonomialOrdering)
 
 The concept of block orderings (product orderings) allows one to construct new monomial orderings from already given ones: If $>_1$ and $>_2$ are monomial orderings on $\text{Mon}_s(x_1, \ldots, x_s)$ and $\text{Mon}_{n-s}(x_{s+1}, \ldots, x_n)$, respectively, then the *block ordering*
 $> \; = \; (>_1, >_2)$ on $\text{Mon}_n(x)=\text{Mon}_n(x_1, \ldots, x_n)$ is defined by setting
-          
+
 $x^\alpha>x^\beta  \;\Leftrightarrow\;  x_1^{\alpha_1}\cdots x_s^{\alpha_s} >_1 x_1^{\beta_1}\cdots x_s^{\beta_s} \;\text{ or }\;
 \bigl(x_1^{\alpha_1}\cdots x_s^{\alpha_s} = x_1^{\beta_1}\cdots x_s^{\beta_s} \text{ and }  x_{s+1}^{\alpha_{s+1}}\cdots x_n^{\alpha_n} >_2
 x_{s+1}^{\beta_{s+1}}\cdots x_n^{\beta_n}\bigr).$
@@ -295,7 +295,7 @@ x_{s+1}^{\beta_{s+1}}\cdots x_n^{\beta_n}\bigr).$
     The ordering  $(>_1, >_2)$
     - is global (local) iff both $>_1$ and $>_2$ are global (local). Mixed orderings arise by choosing one of $>_1$ and $>_2$ global and the other one local,
     - is an elimination ordering for the first block of variables iff $>_1$ is global.
-	
+
 !!! note
     - The definition of a block ordering above subdivides $x$ into a block of initial variables and its complementary block of variables. 
        Block orderings for a subdivision of $x$ into any block of variables and its complementary block are defined similarly and have similar properties.

docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/intro.md

@@ -18,10 +18,10 @@ modules which naturally  includes both submodules and quotients of free modules.
 !!! note
     Most functions in this section rely on Gröbner (standard) bases techniques. Thus, the functions
     should not be applied to modules over rings other than those from the list above. See the Linear
-	Algebra chapter for module types which are designed to handle modules over Euclidean
-	domains.
+    Algebra chapter for module types which are designed to handle modules over Euclidean
+    domains.
 
 !!! note
     For simplicity of the presentation in what follows, functions are often only illustrated by examples  
     with focus on modules over multivariate polynomial rings, but work similarly for modules over
-	a ring of any of the above types.
+    a ring of any of the above types.

docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/subquotients.md

@@ -36,16 +36,16 @@ and regard $M$ as a submodule of that ambient module, embedded in the natural wa
 !!! note
     Recall from the section on free modules that by a free $R$-module we mean a free
     module of type $R^p$ , where we think of $R^p$ as a free module with a given
-	basis, namely the basis of standard unit vectors. Accordingly, elements of free modules
-	are represented by coordinate vectors, and homomorphisms between free modules by
-	matrices. Here, by convention, vectors are row vectors, and matrices operate by
-	multiplication on the right.
+    basis, namely the basis of standard unit vectors. Accordingly, elements of free modules
+    are represented by coordinate vectors, and homomorphisms between free modules by
+    matrices. Here, by convention, vectors are row vectors, and matrices operate by
+    multiplication on the right.
 
 !!! note
     Over a graded ring $R$, we work with graded free modules $R^s$, $R^p$, $R^t$ and graded
     homomorphisms $a$, $b$. As a consequence, every module involved in the construction of
-	the subquotient defined by $a$ and $b$ carries an induced grading. In particular, the
-	subquotient itself carries an induced grading.
+    the subquotient defined by $a$ and $b$ carries an induced grading. In particular, the
+    subquotient itself carries an induced grading.
 
 ## Types
 
@@ -57,8 +57,8 @@ they are modelled as objects of the concrete type `SubquoModule{T} <: AbstractSu
 !!! note
     Canonical maps such us the canonical projection onto a quotient module arise in many 
     constructions in commutative algebra. The `SubquoModule` type is designed so that it allows
-	for the caching of such maps when executing functions. The `tensor_product`
-	function discussed in this section provides an example.
+    for the caching of such maps when executing functions. The `tensor_product`
+    function discussed in this section provides an example.
 
 ## Constructors
 

docs/src/CommutativeAlgebra/affine_algebras.md

@@ -19,33 +19,33 @@ discuss functionality for handling such algebras in OSCAR.
 !!! note
     Most functions discussed here rely on Gröbner basis techniques. In particular, they typically make use of a Gröbner basis for the
     modulus of the quotient. Nevertheless, the construction of quotients is lazy in the sense that the computation of such a Gröbner
-	basis is delayed until the user performs an operation that indeed requires it (the Gröbner basis is then computed with respect
-	to the `default_ordering` on the underlying polynomial ring; see the section on *Gröbner/Standard Bases* for default orderings in
-	OSCAR). Once computed, the Gröbner basis is cached for later reuse.
-	
+    basis is delayed until the user performs an operation that indeed requires it (the Gröbner basis is then computed with respect
+    to the `default_ordering` on the underlying polynomial ring; see the section on *Gröbner/Standard Bases* for default orderings in
+    OSCAR). Once computed, the Gröbner basis is cached for later reuse.
+
 !!! note
     Recall that Gröbner basis methods are implemented for multivariate polynomial rings over fields (exact fields supported by
     OSCAR) and, where not indicated otherwise, for multivariate polynomial rings over the integers.
 
 !!! note
     In OSCAR, elements of a quotient $A = R/I$ are not necessarily represented by polynomials which are reduced with regard to $I$.
     That is, if $f\in R$ is the internal polynomial representative of an element of $A$, then $f$ may not be the normal form mod $I$
-	with respect to the default ordering on $R$ (see the section on *Gröbner/Standard Bases* for normal forms). Operations involving
-	Gröbner basis computations may lead to (partial) reductions. The function `simplify` discussed in this section computes fully
-	reduced representatives. 
+    with respect to the default ordering on $R$ (see the section on *Gröbner/Standard Bases* for normal forms). Operations involving
+    Gröbner basis computations may lead to (partial) reductions. The function `simplify` discussed in this section computes fully
+    reduced representatives.
 
 !!! note
     Each grading on a multivariate polynomial ring `R`  in OSCAR  descends to a grading on the affine algebra `A = R/I`
     (recall that OSCAR ideals of graded polynomial rings are required to be homogeneous).
     Functionality for dealing with such gradings and our notation for describing this functionality descend accordingly.
-	This applies, in particular, to the functions [`is_graded`](@ref),  [`is_standard_graded`](@ref), [`is_z_graded`](@ref),
-	[`is_zm_graded`](@ref), and [`is_positively_graded`](@ref) which will not be discussed again here. 
+    This applies, in particular, to the functions [`is_graded`](@ref),  [`is_standard_graded`](@ref), [`is_z_graded`](@ref),
+    [`is_zm_graded`](@ref), and [`is_positively_graded`](@ref) which will not be discussed again here.
 
 ## Types
 
 The OSCAR type for quotients of  multivariate polynomial rings is of parametrized form `MPolyQuoRing{T}`,
 with elements of type `MPolyQuoRingElem{T}`. Here, `T` is the element type of the polynomial ring.
-    
+
 ## Constructors
 
 ```@docs
@@ -173,7 +173,7 @@ is_homogeneous(f::MPolyQuoRingElem{<:MPolyDecRingElem})
 
 ### Data associated to Elements of Affine Algebras
 
-Given an element `f` of an affine algebra `A`, 
+Given an element `f` of an affine algebra `A`,
 
 - `parent(f)` refers to `A`.
 
@@ -339,7 +339,7 @@ refer to `R` and `S`, respectively. Given ring homomorphisms `F` : `R` $\to$ `S`
 The OSCAR homomorphism type `AffAlgHom` models ring homomorphisms `R` $\to$ `S` such that
 the type of both `R` and `S`  is a subtype of `Union{MPolyRing{T}, MPolyQuoRing{U}}`, where `T <: FieldElem` and
 `U <: MPolyRingElem{T}`. Functionality for these homomorphism is discussed in what follows.
-       
+
 ### Data Associated to Homomorphisms of Affine Algebras
 
 ```@docs
@@ -583,7 +583,7 @@ well-defined *Hilbert function* of $A$,
 
 $H(A, \underline{\phantom{d}}): G \to \N, \; g\mapsto \dim_K(A_g).$
 
-The *Hilbert series* of $A$ is the generating function 
+The *Hilbert series* of $A$ is the generating function
 
 $H_A(\mathbb t)=\sum_{g\in G} H(A, g) \mathbb t^g$
 
@@ -597,7 +597,7 @@ By a result of Macaulay, if $A = R/I$ is an affine algebra, and $L_{>}(I)$ is th
 ideal of $I$ with respect to a global monomial ordering $>$, then the Hilbert function of $A$
 equals that of $R/L_{>}(I)$ (see Theorem 15.26 in [Eis95](@cite)).
 Thus, using Gröbner bases, the computation of Hilbert series can be reduced to the case where
-the modulus of the affine algebra is a monomial ideal. In the latter case, we face a problem 
+the modulus of the affine algebra is a monomial ideal. In the latter case, we face a problem
 of combinatorial nature, and there are various strategies of how to proceed (see [KR05](@cite)).
 The functions `hilbert_series`, `hilbert_series_reduced`, `hilbert_series_expanded`,
 `hilbert_function`, `hilbert_polynomial`, and `degree` address the case of
@@ -626,7 +626,7 @@ $H_A(t)=\sum_{d\geq 0} H(A, d) t^d\in\mathbb Z[[t]].$
 
 The Hilbert series can be written as a rational function $p(t)/q(t)$, with denominator
 
-$q(t) = (1-t^{w_1})\cdots (1-t^{w_n}).$ 
+$q(t) = (1-t^{w_1})\cdots (1-t^{w_n}).$
 
 In the standard $\mathbb Z$-graded case, where the weights on the variables are all 1, the Hilbert function is of polynomial nature: There exists
  a unique polynomial $P_A(t)\in\mathbb{Q}[t]$, the *Hilbert polynomial*, which satisfies $H(M,d)=P_M(d)$