Deformations of finite-dimensional algebras and their idempotents

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite dimensional algebra over an algebraically closed field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If we represent primitive idempotents by points and basis vectors in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Subscript i Baseline upper B e Subscript j"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mi>B</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{e_i}B{e_j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by "arrows" from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Subscript j"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{e_j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Subscript i"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{e_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then any specialization of the algebra acts on this directed graph by coalescing points. This implies that the number of irreducible components in the scheme parametrizing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional algebras is no less than the number of loopless directed graphs with a total of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vertices and arrows. We also show that the condition of having a distributive ideal lattice is open.