Tensor-multinomial sums of ideals: Primary decompositions and persistence of associated primes

<p>Given a Noetherian tensor product of two Noetherian algebras over a field and proper ideals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the two algebras, we determine the associated primes of each power of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper I plus upper J"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> <mml:mi>J</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">I+J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of the associated primes of lower powers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We record two applications. First, in case the field is algebraically closed, we construct primary decompositions for powers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper I plus upper J"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> <mml:mi>J</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">I+J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from primary decompositions for powers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Separately, we attack the persistence problem for associated primes of powers of an ideal in case one of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a non-zero normal ideal. </p>