# reesAlgebra -- Compute the defining ideal of the Rees Algebra

## Synopsis

• Usage:
A = reesAlgebra M
A = reesAlgebra(M,f)
• Inputs:
• M, , or an ideal of a quotient polynomial ring $R$
• f, , any non-zerodivisor in ideal or the first Fitting ideal of the module. Optional
• Optional inputs:
• BasisElementLimit => ..., default value infinity, Bound the number of Groebner basis elements to compute in the saturation step
• DegreeLimit => ..., default value {}, Bound the degrees considered in the saturation step. Defaults to infinity
• Jacobian (missing documentation) => ..., default value false,
• MinimalGenerators => ..., default value true, Whether the saturation step returns minimal generators
• PairLimit => ..., default value infinity, Bound the number of s-pairs considered in the saturation step
• Strategy => ..., default value null, Choose a strategy for the saturation step
• Variable => ..., default value w, Choose name for variables in the created ring
• Outputs:
• A, a ring, defining the Rees algebra of M

## Description

If $M$ is an ideal or module over a ring $R$, and $F\to M$ is a surjection from a free module, then reesAlgebra(M) returns the ring $Sym(F)/J$, where $J = reesIdeal(M)$.

In the following example, we find the Rees Algebra of a monomial curve singularity. We also demonstrate the use of reesIdeal, symmetricKernel, isLinearType, normalCone, normalCone, specialFiberIdeal.

 i1 : S = QQ[x_0..x_3] o1 = S o1 : PolynomialRing i2 : i = monomialCurveIdeal(S,{3,7,8}) 2 2 3 3 3 2 2 5 4 5 3 o2 = ideal (x x - x x , x x - x x , x x - x x , x - x x , x - x x x ) 0 2 1 3 1 2 0 3 1 2 0 3 2 1 3 1 0 2 3 o2 : Ideal of S i3 : I = reesIdeal i; o3 : Ideal of S[w ..w ] 0 4 i4 : reesIdeal(i, Variable=>v) 3 2 2 2 2 o4 = ideal (x x v - x v + x v , x v + x v - x v , x x v - x v + x v , 1 2 0 0 1 3 2 3 0 2 1 1 3 0 3 0 1 1 2 2 ------------------------------------------------------------------------ 2 2 3 3 2 2 x x v - x v + x v , x v + x x v - x v , x v - x x v + x v , x x v 0 3 0 1 2 2 4 2 0 1 3 1 0 3 1 0 0 2 2 3 4 2 3 0 ------------------------------------------------------------------------ 2 2 2 2 2 2 2 - x v + v v , x x x v - x v + v v , (x x x + x x )v - x x v v + 1 1 2 3 0 1 3 0 2 2 1 4 0 2 3 1 3 0 1 2 1 2 ------------------------------------------------------------------------ v v ) 3 4 o4 : Ideal of S[v ..v ] 0 4 i5 : I=reesIdeal(i,i_0); o5 : Ideal of S[w ..w ] 0 4 i6 : (J=symmetricKernel gens i); o6 : Ideal of S[w ..w ] 0 4 i7 : isLinearType(i,i_0) o7 = false i8 : isLinearType i o8 = false i9 : reesAlgebra (i,i_0) S[w ..w ] 0 4 o9 = --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 2 2 2 3 3 2 2 2 2 2 2 2 2 2 (x x w - x w + x w , x w + x w - x w , x x w - x w + x w , x x w - x w + x w , x w + x x w - x w , x w - x x w + x w , x x w - x w + w w , x x x w - x w + w w , (x x x + x x )w - x x w w + w w ) 1 2 0 0 1 3 2 3 0 2 1 1 3 0 3 0 1 1 2 2 0 3 0 1 2 2 4 2 0 1 3 1 0 3 1 0 0 2 2 3 4 2 3 0 1 1 2 3 0 1 3 0 2 2 1 4 0 2 3 1 3 0 1 2 1 2 3 4 o9 : QuotientRing i10 : trim ideal normalCone (i, i_0) 2 2 3 3 3 2 2 5 4 5 3 o10 = ideal (x x - x x , x x - x x , x x - x x , x - x x , x - x x x ) 0 2 1 3 1 2 0 3 1 2 0 3 2 1 3 1 0 2 3 S[w ..w ] 0 4 o10 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 2 2 2 3 3 2 2 2 2 2 2 2 2 2 (x x w - x w + x w , x w + x w - x w , x x w - x w + x w , x x w - x w + x w , x w + x x w - x w , x w - x x w + x w , x x w - x w + w w , x x x w - x w + w w , (x x x + x x )w - x x w w + w w ) 1 2 0 0 1 3 2 3 0 2 1 1 3 0 3 0 1 1 2 2 0 3 0 1 2 2 4 2 0 1 3 1 0 3 1 0 0 2 2 3 4 2 3 0 1 1 2 3 0 1 3 0 2 2 1 4 0 2 3 1 3 0 1 2 1 2 3 4 i11 : trim ideal associatedGradedRing (i,i_0) 2 2 3 3 3 2 2 5 4 5 3 o11 = ideal (x x - x x , x x - x x , x x - x x , x - x x , x - x x x ) 0 2 1 3 1 2 0 3 1 2 0 3 2 1 3 1 0 2 3 S[w ..w ] 0 4 o11 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 2 2 2 3 3 2 2 2 2 2 2 2 2 2 (x x w - x w + x w , x w + x w - x w , x x w - x w + x w , x x w - x w + x w , x w + x x w - x w , x w - x x w + x w , x x w - x w + w w , x x x w - x w + w w , (x x x + x x )w - x x w w + w w ) 1 2 0 0 1 3 2 3 0 2 1 1 3 0 3 0 1 1 2 2 0 3 0 1 2 2 4 2 0 1 3 1 0 3 1 0 0 2 2 3 4 2 3 0 1 1 2 3 0 1 3 0 2 2 1 4 0 2 3 1 3 0 1 2 1 2 3 4 i12 : trim specialFiberIdeal (i,i_0) o12 = ideal (w w , w w , w w ) 3 4 1 4 2 3 o12 : Ideal of QQ[w ..w ] 0 4

• reesIdeal -- Compute the defining ideal of the Rees Algebra
• symmetricKernel -- Compute the Rees ring of the image of a matrix

## Ways to use reesAlgebra :

• "reesAlgebra(Ideal)"
• "reesAlgebra(Ideal,RingElement)"
• "reesAlgebra(Module)"
• "reesAlgebra(Module,RingElement)"

## For the programmer

The object reesAlgebra is .